This **preview** has intentionally **blurred** parts. Sign up to view the full document

**Unformatted Document Excerpt**

Categories i SPSS 17.0 For more information about SPSS Inc. software products, please visit our Web site at http://www.spss.com or contact SPSS Inc. 233 South Wacker Drive, 11th Floor Chicago, IL 60606-6412 Tel: (312) 651-3000 Fax: (312) 651-3668 SPSS is a registered trademark and the other product names are the trademarks of SPSS Inc. for its proprietary computer software. No material describing such software may be produced or distributed without the written permission of the owners of the trademark and license rights in the software and the copyrights in the published materials. The SOFTWARE and documentation are provided with RESTRICTED RIGHTS. Use, duplication, or disclosure by the Government is subject to restrictions as set forth in subdivision (c) (1) (ii) of The Rights in Technical Data and Computer Software clause at 52.227-7013. Contractor/manufacturer is SPSS Inc., 233 South Wacker Drive, 11th Floor, Chicago, IL 60606-6412. Patent No. 7,023,453 General notice: Other product names mentioned herein are used for identication purposes only and may be trademarks of their respective companies. Windows is a registered trademark of Microsoft Corporation. Apple, Mac, and the Mac logo are trademarks of Apple Computer, Inc., registered in the U.S. and other countries. This product uses WinWrap Basic, Copyright 1993-2007, Polar Engineering and Consulting, http://www.winwrap.com. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Preface SPSS Statistics 17.0 is a comprehensive system for analyzing data. The Categories optional add-on module provides the additional analytic techniques described in this manual. The Categories add-on module must be used with the SPSS Statistics 17.0 Base system and is completely integrated into that system. Installation To install the Categories add-on module, run the License Authorization Wizard using the authorization code that you received from SPSS Inc. For more information, see the installation instructions supplied with the Categories add-on module. Compatibility SPSS Statistics is designed to run on many computer systems. See the installation instructions that came with your system for specic information on minimum and recommended requirements. Serial Numbers Your serial number is your identication number with SPSS Inc. You will need this serial number when you contact SPSS Inc. for information regarding support, payment, or an upgraded system. The serial number was provided with your Base system. Customer Service If you have any questions concerning your shipment or account, contact your local ofce, listed on the Web site at http://www.spss.com/worldwide. Please have your serial number ready for identication. iii Training Seminars SPSS Inc. provides both public and onsite training seminars. All seminars feature hands-on workshops. Seminars will be offered in major cities on a regular basis. For more information on these seminars, contact your local ofce, listed on the Web site at http://www.spss.com/worldwide. Technical Support Technical Support services are available to maintenance customers. Customers may contact Technical Support for assistance in using SPSS Statistics or for installation help for one of the supported hardware environments. To reach Technical Support, see the Web site at http://www.spss.com, or contact your local ofce, listed on the Web site at http://www.spss.com/worldwide. Be prepared to identify yourself, your organization, and the serial number of your system. Additional Publications The SPSS Statistical Procedures Companion, by Marija Noruis, has been published by Prentice Hall. A new version of this book, updated for SPSS Statistics 17.0, is planned. The SPSS Advanced Statistical Procedures Companion, also based on SPSS Statistics 17.0, is forthcoming. The SPSS Guide to Data Analysis for SPSS Statistics 17.0 is also in development. Announcements of publications available exclusively through Prentice Hall will be available on the Web site at http://www.spss.com/estore (select your home country, and then click Books). Acknowledgments The optimal scaling procedures and their implementation in SPSS Statistics were developed by the Data Theory Scaling System Group (DTSS), consisting of members of the departments of Education and Psychology of the Faculty of Social and Behavioral Sciences at Leiden University. Willem Heiser, Jacqueline Meulman, Gerda van den Berg, and Patrick Groenen were involved with the original 1990 procedures. Jacqueline Meulman and Peter Neufeglise participated in the development of procedures for categorical regression, correspondence analysis, categorical principal components analysis, and multidimensional scaling. In addition, Anita van der Kooij contributed especially to CATREG, CORRESPONDENCE, and CATPCA. Willem Heiser, Jacques Commandeur, Frank Busing, Gerda van den Berg, and Patrick Groenen participated in the development of the PROXSCAL procedure. Frank Busing, Willem Heiser, Patrick iv Groenen, and Peter Neufeglise participated in the development of the PREFSCAL procedure. v Contents Part I: User's Guide 1 Introduction to Optimal Scaling Procedures for Categorical Data 1 What Is Optimal Scaling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Why Use Optimal Scaling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Optimal Scaling Level and Measurement Level . . . . . . . . . . . . . . . . . . . . . . . 2 Selecting the Optimal Scaling Level . . . . . . . . Transformation Plots . . . . . . . . . . . . . . . . . . . Category Codes . . . . . . . . . . . . . . . . . . . . . . Which Procedure Is Best for Your Application? . . Categorical Regression . . . . . . . . . . . . . . . . . Categorical Principal Components Analysis . Nonlinear Canonical Correlation Analysis . . . Correspondence Analysis . . . . . . . . . . . . . . . Multiple Correspondence Analysis . . . . . . . . Multidimensional Scaling. . . . . . . . . . . . . . . . Multidimensional Unfolding . . . . . . . . . . . . . . Aspect Ratio in Optimal Scaling Charts . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 3 4 5 8 ... 9 . . 10 . . 11 . . 12 . . 13 . . 15 . . 15 . . 16 Recommended Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Categorical Regression (CATREG) 19 Define Scale in Categorical Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Categorical Regression Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 vi Categorical Regression Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Categorical Regression Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Categorical Regression Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Categorical Regression Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Categorical Regression Save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Categorical Regression Transformation Plots . . . . . . . . . . . . . . . . . . . . . . . . 33 CATREG Command Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Categorical Principal Components Analysis (CATPCA) 35 Define Scale and Weight in CATPCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Categorical Principal Components Analysis Discretization . . . . . . . . . . . . . . 40 Categorical Principal Components Analysis Missing Values . . . . . . . . . . . . . 42 Categorical Principal Components Analysis Options . . . . . . . . . . . . . . . . . . . 43 Categorical Principal Components Analysis Output . . . . . . . . . . . . . . . . . . . . 46 Categorical Principal Components Analysis Save . . . . . . . . . . . . . . . . . . . . . 48 Categorical Principal Components Analysis Object Plots. . . . . . . . . . . . . . . . 49 Categorical Principal Components Analysis Category Plots. . . . . . . . . . . . . . 51 Categorical Principal Components Analysis Loading Plots . . . . . . . . . . . . . . 52 CATPCA Command Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Nonlinear Canonical Correlation Analysis (OVERALS) 55 Define Range and Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Define Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 vii Nonlinear Canonical Correlation Analysis Options . . . . . . . . . . . . . . . . . . . . 60 OVERALS Command Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5 Correspondence Analysis 64 Define Row Range in Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . 66 Define Column Range in Correspondence Analysis . . . . . . . . . . . . . . . . . . . . 67 Correspondence Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Correspondence Analysis Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Correspondence Analysis Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 CORRESPONDENCE Command Additional Features. . . . . . . . . . . . . . . . . . . . 74 6 Multiple Correspondence Analysis 76 Define Variable Weight in Multiple Correspondence Analysis . . . . . . . . . . . . 78 Multiple Correspondence Analysis Discretization . . . . . . . . . . . . . . . . . . . . . 79 Multiple Correspondence Analysis Missing Values . . . . . . . . . . . . . . . . . . . . 81 Multiple Correspondence Analysis Options. . . . . . . . . . . . . . . . . . . . . . . . . . 82 Multiple Correspondence Analysis Output . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Multiple Correspondence Analysis Save. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Multiple Correspondence Analysis Object Plots . . . . . . . . . . . . . . . . . . . . . . 88 Multiple Correspondence Analysis Variable Plots . . . . . . . . . . . . . . . . . . . . . 90 MULTIPLE CORRESPONDENCE Command Additional Features . . . . . . . . . . . 91 7 Multidimensional Scaling (PROXSCAL) 92 Proximities in Matrices across Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 viii Proximities in Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Proximities in One Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Create Proximities from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Create Measure from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Define a Multidimensional Scaling Model . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Multidimensional Scaling Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Multidimensional Scaling Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Multidimensional Scaling Plots, Version 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Multidimensional Scaling Plots, Version 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Multidimensional Scaling Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 PROXSCAL Command Additional Features . . . . . . . . . . . . . . . . . . . . . . . . . 109 8 Multidimensional Unfolding (PREFSCAL) 110 Define a Multidimensional Unfolding Model . . . . . . . . . . . . . . . . . . . . . . . . 112 Multidimensional Unfolding Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Multidimensional Unfolding Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Multidimensional Unfolding Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Multidimensional Unfolding Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 PREFSCAL Command Additional Features . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Part II: Examples 9 Categorical Regression 125 Example: Carpet Cleaner Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A Standard Linear Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . 126 A Categorical Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 ix Example: Ozone Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Discretizing Variables . . . . . . . . . Selection of Transformation Type . Optimality of the Quantifications . Effects of Transformations . . . . . . Recommended Readings . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . 149 150 164 167 177 10 Categorical Principal Components Analysis Running the Analysis . . . . . . . . . . . . . . . . . . . Number of Dimensions . . . . . . . . . . . . . . . . . Quantifications . . . . . . . . . . . . . . . . . . . . . . . Object Scores . . . . . . . . . . . . . . . . . . . . . . . . Component Loadings . . . . . . . . . . . . . . . . . . . Additional Dimensions . . . . . . . . . . . . . . . . . . Example: Symptomatology of Eating Disorders . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 179 . . . . . . . . . . . . . . 180 187 188 190 191 193 196 198 212 216 216 218 220 237 Example: Examining Interrelations of Social Systems . . . . . . . . . . . . . . . . . 179 Running the Analysis . . . . . . . . . . . . . . . . . . . . . . Transformation Plots . . . . . . . . . . . . . . . . . . . . . . Model Summary . . . . . . . . . . . . . . . . . . . . . . . . . Component Loadings . . . . . . . . . . . . . . . . . . . . . . Object Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . Examining the Structure of the Course of Illness . Recommended Readings . . . . . . . . . . . . . . . . . . . . . . 11 Nonlinear Canonical Correlation Analysis 239 Example: An Analysis of Survey Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Examining the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Accounting for Similarity between Sets . . . . . . . . . . . . . . . . . . . . . . . . 249 x Component Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Category versus Multiple Category Coordinates . . . . . . . . . . . . Centroids and Projected Centroids. . . . . . . . . . . . . . . . . . . . . . . . . . . . An Alternative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 255 259 260 265 271 272 12 Correspondence Analysis 273 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Example: Smoking Behavior by Job Category . . . . . . . . . . . . . . . . . . . . . . . 275 Running the Analysis . . . . . . . . . . . . . . . . . . . Correspondence Table. . . . . . . . . . . . . . . . . . Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . Biplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Profiles and Distances . . . . . . . . . . . . . . . . . . Row and Column Scores . . . . . . . . . . . . . . . . Permutations of the Correspondence Table . . Confidence Statistics. . . . . . . . . . . . . . . . . . . Supplementary Profiles . . . . . . . . . . . . . . . . . Example: Perceptions of Coffee Brands . . . . . . . . Running the Analysis . . . . . . . . . . . . . Dimensionality . . . . . . . . . . . . . . . . . . Contributions . . . . . . . . . . . . . . . . . . . Plots . . . . . . . . . . . . . . . . . . . . . . . . . Symmetrical Normalization . . . . . . . . Example: Flying Mileage between Cities . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . 276 279 280 281 282 284 286 287 288 294 295 300 301 303 305 307 Correspondence Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Row and Column Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 Recommended Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 xi 13 Multiple Correspondence Analysis Running the Analysis . . . . . . . . . . . . . . . . Model Summary . . . . . . . . . . . . . . . . . . . Object Scores . . . . . . . . . . . . . . . . . . . . . Discrimination Measures. . . . . . . . . . . . . Category Quantifications . . . . . . . . . . . . . A More Detailed Look at Object Scores . . Omission of Outliers. . . . . . . . . . . . . . . . . Recommended Readings . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 320 . . . . . . . . 321 325 325 327 328 330 333 338 Example: Characteristics of Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 14 Multidimensional Scaling 340 341 350 357 362 362 Example: An Examination of Kinship Terms . . . . . . . . . . . . . . . . . . . . . . . . . 340 Choosing the Number of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . A Three-Dimensional Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Three-Dimensional Solution with Nondefault Transformations . . . . . Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Multidimensional Unfolding Producing a Degenerate Solution . . . Measures . . . . . . . . . . . . . . . . . . . . . Common Space . . . . . . . . . . . . . . . . . Running a Nondegenerate Analysis . . Measures . . . . . . . . . . . . . . . . . . . . . Common Space . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 364 . . . . . . 364 368 369 370 372 373 Example: Breakfast Item Preferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 xii Example: Three-Way Unfolding of Breakfast Item Preferences . . . . . . . . . . 374 Running the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . Using a Different Initial Configuration. . . . . . . . . . . . . Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . Example: Examining Behavior-Situation Appropriateness . Running the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proximity Transformations . . . . . . . . . . . . . . . . . . . . . Changing the Proximities Transformation (Ordinal) . . . Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proximity Transformations . . . . . . . . . . . . . . . . . . . . . Recommended Readings . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . 374 380 381 382 385 388 389 390 392 393 398 399 400 400 402 403 404 404 Appendix A Sample Files Bibliography Index 406 420 428 xiii Part I: User's Guide Chapter Introduction to Optimal Scaling Procedures for Categorical Data 1 Categories procedures use optimal scaling to analyze data that are difcult or impossible for standard statistical procedures to analyze. This chapter describes what each procedure does, the situations in which each procedure is most appropriate, the relationships between the procedures, and the relationships of these procedures to their standard statistical counterparts. Note: These procedures and their implementation in SPSS Statistics were developed by the Data Theory Scaling System Group (DTSS), consisting of members of the departments of Education and Psychology, Faculty of Social and Behavioral Sciences, Leiden University. What Is Optimal Scaling? The idea behind optimal scaling is to assign numerical quantications to the categories of each variable, thus allowing standard procedures to be used to obtain a solution on the quantied variables. The optimal scale values are assigned to categories of each variable based on the optimizing criterion of the procedure in use. Unlike the original labels of the nominal or ordinal variables in the analysis, these scale values have metric properties. In most Categories procedures, the optimal quantication for each scaled variable is obtained through an iterative method called alternating least squares in which, after the current quantications are used to nd a solution, the quantications are updated using that solution. The updated quantications are then used to nd a new solution, which is used to update the quantications, and so on, until some criterion is reached that signals the process to stop. 1 2 Chapter 1 Why Use Optimal Scaling? Categorical data are often found in marketing research, survey research, and research in the social and behavioral sciences. In fact, many researchers deal almost exclusively with categorical data. While adaptations of most standard models exist specically to analyze categorical data, they often do not perform well for datasets that feature: Too few observations Too many variables Too many values per variable By quantifying categories, optimal scaling techniques avoid problems in these situations. Moreover, they are useful even when specialized techniques are appropriate. Rather than interpreting parameter estimates, the interpretation of optimal scaling output is often based on graphical displays. Optimal scaling techniques offer excellent exploratory analyses, which complement other SPSS Statistics models well. By narrowing the focus of your investigation, visualizing your data through optimal scaling can form the basis of an analysis that centers on interpretation of model parameters. Optimal Scaling Level and Measurement Level This can be a very confusing concept when you rst use Categories procedures. When specifying the level, you specify not the level at which variables are measured but the level at which they are scaled. The idea is that the variables to be quantied may have nonlinear relations regardless of how they are measured. For Categories purposes, there are three basic levels of measurement: The nominal level implies that a variables values represent unordered categories. Examples of variables that might be nominal are region, zip code area, religious afliation, and multiple choice categories. The ordinal level implies that a variables values represent ordered categories. Examples include attitude scales representing degree of satisfaction or condence and preference rating scores. The numerical level implies that a variables values represent ordered categories with a meaningful metric so that distance comparisons between categories are appropriate. Examples include age in years and income in thousands of dollars. 3 Introduction to Optimal Scaling Procedures for Categorical Data For example, suppose that the variables region, job, and age are coded as shown in the following table. Table 1-1 Coding scheme for region, job, and age 1 2 3 4 Region North South East West 1 2 3 Job intern sales rep manager 20 22 25 27 Age twenty years old twenty-two years old twenty-ve years old twenty-seven years old The values shown represent the categories of each variable. Region would be a nominal variable. There are four categories of region, with no intrinsic ordering. Values 1 through 4 simply represent the four categories; the coding scheme is completely arbitrary. Job, on the other hand, could be assumed to be an ordinal variable. The original categories form a progression from intern to manager. Larger codes represent a job higher on the corporate ladder. However, only the order information is knownnothing can be said about the distance between adjacent categories. In contrast, age could be assumed to be a numerical variable. In the case of age, the distances between the values are intrinsically meaningful. The distance between 20 and 22 is the same as the distance between 25 and 27, while the distance between 22 and 25 is greater than either of these. Selecting the Optimal Scaling Level It is important to understand that there are no intrinsic properties of a variable that automatically predene what optimal scaling level you should specify for it. You can explore your data in any way that makes sense and makes interpretation easier. By analyzing a numerical-level variable at the ordinal level, for example, the use of a nonlinear transformation may allow a solution in fewer dimensions. The following two examples illustrate how the obvious level of measurement might not be the best optimal scaling level. Suppose that a variable sorts objects into age groups. Although age can be scaled as a numerical variable, it may be true that for people younger than 25 safety has a positive relation with age, whereas for people older than 60 safety has a negative relation with age. In this case, it might be better to treat age as a nominal variable. 4 Chapter 1 As another example, a variable that sorts persons by political preference appears to be essentially nominal. However, if you order the parties from political left to political right, you might want the quantication of parties to respect this order by using an ordinal level of analysis. Even though there are no predened properties of a variable that make it exclusively one level or another, there are some general guidelines to help the novice user. With single-nominal quantication, you dont usually know the order of the categories but you want the analysis to impose one. If the order of the categories is known, you should try ordinal quantication. If the categories are unorderable, you might try multiple-nominal quantication. Transformation Plots The different levels at which each variable can be scaled impose different restrictions on the quantications. Transformation plots illustrate the relationship between the quantications and the original categories resulting from the selected optimal scaling level. For example, a linear transformation plot results when a variable is treated as numerical. Variables treated as ordinal result in a nondecreasing transformation plot. Transformation plots for variables treated nominally that are U-shaped (or the inverse) display a quadratic relationship. Nominal variables could also yield transformation plots without apparent trends by changing the order of the categories completely. The following gure displays a sample transformation plot. Transformation plots are particularly suited to determining how well the selected optimal scaling level performs. If several categories receive similar quantications, collapsing these categories into one category may be warranted. Alternatively, if a variable treated as nominal receives quantications that display an increasing trend, an ordinal transformation may result in a similar t. If that trend is linear, numerical treatment may be appropriate. However, if collapsing categories or changing scaling levels is warranted, the analysis will not change signicantly. 5 Introduction to Optimal Scaling Procedures for Categorical Data Figure 1-1 Transformation plot of price (numerical) Category Codes Some care should be taken when coding categorical variables because some coding schemes may yield unwanted output or incomplete analyses. Possible coding schemes for job are displayed in the following table. Table 1-2 Alternative coding schemes for job Scheme Category intern sales rep manager A 1 2 3 B 1 2 7 C 5 6 7 D 1 5 3 Some Categories procedures require that the range of every variable used be dened. Any value outside this range is treated as a missing value. The minimum category value is always 1. The maximum category value is supplied by the user. This value is not the number of categories for a variableit is the largest category value. For 6 Chapter 1 example, in the table, scheme A has a maximum category value of 3 and scheme B has a maximum category value of 7, yet both schemes code the same three categories. The variable range determines which categories will be omitted from the analysis. Any categories with codes outside the dened range are omitted from the analysis. This is a simple method for omitting categories but can result in unwanted analyses. An incorrectly dened maximum category can omit valid categories from the analysis. For example, for scheme B, dening the maximum category value to be 3 indicates that job has categories coded from 1 to 3; the manager category is treated as missing. Because no category has actually been coded 3, the third category in the analysis contains no cases. If you wanted to omit all manager categories, this analysis would be appropriate. However, if managers are to be included, the maximum category must be dened as 7, and missing values must be coded with values above 7 or below 1. For variables treated as nominal or ordinal, the range of the categories does not affect the results. For nominal variables, only the label and not the value associated with that label is important. For ordinal variables, the order of the categories is preserved in the quantications; the category values themselves are not important. All coding schemes resulting in the same category ordering will have identical results. For example, the rst three schemes in the table are functionally equivalent if job is analyzed at an ordinal level. The order of the categories is identical in these schemes. Scheme D, on the other hand, inverts the second and third categories and will yield different results than the other schemes. Although many coding schemes for a variable are functionally equivalent, schemes with small differences between codes are preferred because the codes have an impact on the amount of output produced by a procedure. All categories coded with values between 1 and the user-dened maximum are valid. If any of these categories are empty, the corresponding quantications will be either system-missing or 0, depending on the procedure. Although neither of these assignments affect the analyses, output is produced for these categories. Thus, for scheme B, job has four categories that receive system-missing values. For scheme C, there are also four categories receiving system-missing indicators. In contrast, for scheme A there are no system-missing quantications. Using consecutive integers as codes for variables treated as nominal or ordinal results in much less output without affecting the results. Coding schemes for variables treated as numerical are more restricted than the ordinal case. For these variables, the differences between consecutive categories are important. The following table displays three coding schemes for age. 7 Introduction to Optimal Scaling Procedures for Categorical Data Table 1-3 Alternative coding schemes for age Category 20 22 25 27 A 20 22 25 27 Scheme B 1 3 6 8 C 1 2 3 4 Any recoding of numerical variables must preserve the differences between the categories. Using the original values is one method for ensuring preservation of differences. However, this can result in many categories having system-missing indicators. For example, scheme A employs the original observed values. For all Categories procedures except for Correspondence Analysis, the maximum category value is 27 and the minimum category value is set to 1. The rst 19 categories are empty and receive system-missing indicators. The output can quickly become rather cumbersome if the maximum category is much greater than 1 and there are many empty categories between 1 and the maximum. To reduce the amount of output, recoding can be done. However, in the numerical case, the Automatic Recode facility should not be used. Coding to consecutive integers results in differences of 1 between all consecutive categories, and, as a result, all quantications will be equally spaced. The metric characteristics deemed important when treating a variable as numerical are destroyed by recoding to consecutive integers. For example, scheme C in the table corresponds to automatically recoding age. The difference between categories 22 and 25 has changed from three to one, and the quantications will reect the latter difference. An alternative recoding scheme that preserves the differences between categories is to subtract the smallest category value from every category and add 1 to each difference. Scheme B results from this transformation. The smallest category value, 20, has been subtracted from each category, and 1 was added to each result. The transformed codes have a minimum of 1, and all differences are identical to the original data. The maximum category value is now 8, and the zero quantications before the rst nonzero quantication are all eliminated. Yet, the nonzero quantications corresponding to each category resulting from scheme B are identical to the quantications from scheme A. 8 Chapter 1 Which Procedure Is Best for Your Application? The techniques embodied in four of these procedures (Correspondence Analysis, Multiple Correspondence Analysis, Categorical Principal Components Analysis, and Nonlinear Canonical Correlation Analysis) fall into the general area of multivariate data analysis known as dimension reduction. That is, relationships between variables are represented in a few dimensionssay two or threeas often as possible. This enables you to describe structures or patterns in the relationships that would be too difcult to fathom in their original richness and complexity. In market research applications, these techniques can be a form of perceptual mapping. A major advantage of these procedures is that they accommodate data with different levels of optimal scaling. Categorical Regression describes the relationship between a categorical response variable and a combination of categorical predictor variables. The inuence of each predictor variable on the response variable is described by the corresponding regression weight. As in the other procedures, data can be analyzed with different levels of optimal scaling. Multidimensional Scaling and Multidimensional Unfolding describe relationships between objects in a low-dimensional space, using the proximities between the objects. Following are brief guidelines for each of the procedures: Use Categorical Regression to predict the values of a categorical dependent variable from a combination of categorical independent variables. Use Categorical Principal Components Analysis to account for patterns of variation in a single set of variables of mixed optimal scaling levels. Use Nonlinear Canonical Correlation Analysis to assess the extent to which two or more sets of variables of mixed optimal scaling levels are correlated. Use Correspondence Analysis to analyze two-way contingency tables or data that can be expressed as a two-way table, such as brand preference or sociometric choice data. Use Multiple Correspondence Analysis to analyze a categorical multivariate data matrix when you are willing to make no stronger assumption that all variables are analyzed at the nominal level. Use Multidimensional Scaling to analyze proximity data to nd a least-squares representation of a single set of objects in a low-dimensional space. Use Multidimensional Unfolding to analyze proximity data to nd a least-squares representation of two sets of objects in a low-dimensional space. 9 Introduction to Optimal Scaling Procedures for Categorical Data Categorical Regression The use of Categorical Regression is most appropriate when the goal of your analysis is to predict a dependent (response) variable from a set of independent (predictor) variables. As with all optimal scaling procedures, scale values are assigned to each category of every variable such that these values are optimal with respect to the regression. The solution of a categorical regression maximizes the squared correlation between the transformed response and the weighted combination of transformed predictors. Relation to other Categories procedures. Categorical regression with optimal scaling is comparable to optimal scaling canonical correlation analysis with two sets, one of which contains only the dependent variable. In the latter technique, similarity of sets is derived by comparing each set to an unknown variable that lies somewhere between all of the sets. In categorical regression, similarity of the transformed response and the linear combination of transformed predictors is assessed directly. Relation to standard techniques. In standard linear regression, categorical variables can either be recoded as indicator variables or be treated in the same fashion as interval level variables. In the rst approach, the model contains a separate intercept and slope for each combination of the levels of the categorical variables. This results in a large number of parameters to interpret. In the second approach, only one parameter is estimated for each variable. However, the arbitrary nature of the category codings makes generalizations impossible. If some of the variables are not continuous, alternative analyses are available. If the response is continuous and the predictors are categorical, analysis of variance is often employed. If the response is categorical and the predictors are continuous, logistic regression or discriminant analysis may be appropriate. If the response and the predictors are both categorical, loglinear models are often used. Regression with optimal scaling offers three scaling levels for each variable. Combinations of these levels can account for a wide range of nonlinear relationships for which any single standard method is ill-suited. Consequently, optimal scaling offers greater exibility than the standard approaches with minimal added complexity. In addition, nonlinear transformations of the predictors usually reduce the dependencies among the predictors. If you compare the eigenvalues of the correlation matrix for the predictors with the eigenvalues of the correlation matrix for the optimally scaled predictors, the latter set will usually be less variable than the former. In other words, in categorical regression, optimal scaling makes the larger eigenvalues of the predictor correlation matrix smaller and the smaller eigenvalues larger. 10 Chapter 1 Categorical Principal Components Analysis The use of Categorical Principal Components Analysis is most appropriate when you want to account for patterns of variation in a single set of variables of mixed optimal scaling levels. This technique attempts to reduce the dimensionality of a set of variables while accounting for as much of the variation as possible. Scale values are assigned to each category of every variable so that these values are optimal with respect to the principal components solution. Objects in the analysis receive component scores based on the quantied data. Plots of the component scores reveal patterns among the objects in the analysis and can reveal unusual objects in the data. The solution of a categorical principal components analysis maximizes the correlations of the object scores with each of the quantied variables for the number of components (dimensions) specied. An important application of categorical principal components is to examine preference data, in which respondents rank or rate a number of items with respect to preference. In the usual SPSS Statistics data conguration, rows are individuals, columns are measurements for the items, and the scores across rows are preference scores (on a 0 to 10 scale, for example), making the data row-conditional. For preference data, you may want to treat the individuals as variables. Using the Transpose procedure, you can transpose the data. The raters become the variables, and all variables are declared ordinal. There is no objection to using more variables than objects in CATPCA. Relation to other Categories procedures. If all variables are declared multiple nominal, categorical principal components analysis produces an analysis equivalent to a multiple correspondence analysis run on the same variables. Thus, categorical principal components analysis can be seen as a type of multiple correspondence analysis in which some of the variables are declared ordinal or numerical. Relation to standard techniques. If all variables are scaled on the numerical level, categorical principal components analysis is equivalent to standard principal components analysis. More generally, categorical principal components analysis is an alternative to computing the correlations between non-numerical scales and analyzing them using a standard principal components or factor-analysis approach. Naive use of the usual Pearson correlation coefcient as a measure of association for ordinal data can lead to nontrivial bias in estimation of the correlations. 11 Introduction to Optimal Scaling Procedures for Categorical Data Nonlinear Canonical Correlation Analysis Nonlinear Canonical Correlation Analysis is a very general procedure with many different applications. The goal of nonlinear canonical correlation analysis is to analyze the relationships between two or more sets of variables instead of between the variables themselves, as in principal components analysis. For example, you may have two sets of variables, where one set of variables might be demographic background items on a set of respondents and a second set might be responses to a set of attitude items. The scaling levels in the analysis can be any mix of nominal, ordinal, and numerical. Optimal scaling canonical correlation analysis determines the similarity among the sets by simultaneously comparing the canonical variables from each set to a compromise set of scores assigned to the objects. Relation to other Categories procedures. If there are two or more sets of variables with only one variable per set, optimal scaling canonical correlation analysis is equivalent to optimal scaling principal components analysis. If all variables in a one-variable-per-set analysis are multiple nominal, optimal scaling canonical correlation analysis is equivalent to multiple correspondence analysis. If there are two sets of variables, one of which contains only one variable, optimal scaling canonical correlation analysis is equivalent to categorical regression with optimal scaling. Relation to standard techniques. Standard canonical correlation analysis is a statistical technique that nds a linear combination of one set of variables and a linear combination of a second set of variables that are maximally correlated. Given this set of linear combinations, canonical correlation analysis can nd subsequent independent sets of linear combinations, referred to as canonical variables, up to a maximum number equal to the number of variables in the smaller set. If there are two sets of variables in the analysis and all variables are dened to be numerical, optimal scaling canonical correlation analysis is equivalent to a standard canonical correlation analysis. Although SPSS Statistics does not have a canonical correlation analysis procedure, many of the relevant statistics can be obtained from multivariate analysis of variance. Optimal scaling canonical correlation analysis has various other applications. If you have two sets of variables and one of the sets contains a nominal variable declared as single nominal, optimal scaling canonical correlation analysis results can be interpreted in a similar fashion to regression analysis. If you consider the variable to be multiple nominal, the optimal scaling analysis is an alternative to discriminant analysis. Grouping the variables in more than two sets provides a variety of ways to analyze your data. 12 Chapter 1 Correspondence Analysis The goal of correspondence analysis is to make biplots for correspondence tables. In a correspondence table, the row and column variables are assumed to represent unordered categories; therefore, the nominal optimal scaling level is always used. Both variables are inspected for their nominal information only. That is, the only consideration is the fact that some objects are in the same category while others are not. Nothing is assumed about the distance or order between categories of the same variable. One specic use of correspondence analysis is the analysis of two-way contingency tables. If a table has r active rows and c active columns, the number of dimensions in the correspondence analysis solution is the minimum of r minus 1 or c minus 1, whichever is less. In other words, you could perfectly represent the row categories or the column categories of a contingency table in a space of dimensions. Practically speaking, however, you would like to represent the row and column categories of a two-way table in a low-dimensional space, say two dimensions, for the reason that two-dimensional plots are more easily comprehensible than multidimensional spatial representations. When fewer than the maximum number of possible dimensions is used, the statistics produced in the analysis describe how well the row and column categories are represented in the low-dimensional representation. Provided that the quality of representation of the two-dimensional solution is good, you can examine plots of the row points and the column points to learn which categories of the row variable are similar, which categories of the column variable are similar, and which row and column categories are similar to each other. Relation to other Categories procedures. Simple correspondence analysis is limited to two-way tables. If there are more than two variables of interest, you can combine variables to create interaction variables. For example, for the variables region, job, and age, you can combine region and job to create a new variable rejob with the 12 categories shown in the following table. This new variable forms a two-way table with age (12 rows, 4 columns), which can be analyzed in correspondence analysis. Table 1-4 Combinations of region and job Category code 1 2 3 4 Category denition North, intern North, sales rep North, manager South, intern Category code 7 8 9 10 Category denition East, intern East, sales rep East, manager West, intern 13 Introduction to Optimal Scaling Procedures for Categorical Data Category code 5 6 Category denition South, sales rep South, manager Category code 11 12 Category denition West, sales rep West, manager One shortcoming of this approach is that any pair of variables can be combined. We can combine job and age, yielding another 12-category variable. Or we can combine region and age, which results in a new 16-category variable. Each of these interaction variables forms a two-way table with the remaining variable. Correspondence analyses of these three tables will not yield identical results, yet each is a valid approach. Furthermore, if there are four or more variables, two-way tables comparing an interaction variable with another interaction variable can be constructed. The number of possible tables to analyze can get quite large, even for a few variables. You can select one of these tables to analyze, or you can analyze all of them. Alternatively, the Multiple Correspondence Analysis procedure can be used to examine all of the variables simultaneously without the need to construct interaction variables. Relation to standard techniques. The Crosstabs procedure can also be used to analyze contingency tables, with independence as a common focus in the analyses. However, even in small tables, detecting the cause of departures from independence may be difcult. The utility of correspondence analysis lies in displaying such patterns for two-way tables of any size. If there is an association between the row and column variablesthat is, if the chi-square value is signicantcorrespondence analysis may help reveal the nature of the relationship. Multiple Correspondence Analysis Multiple Correspondence Analysis tries to produce a solution in which objects within the same category are plotted close together and objects in different categories are plotted far apart. Each object is as close as possible to the category points of categories that apply to the object. In this way, the categories divide the objects into homogeneous subgroups. Variables are considered homogeneous when they classify objects in the same categories into the same subgroups. For a one-dimensional solution, multiple correspondence analysis assigns optimal scale values (category quantications) to each category of each variable in such a way that overall, on average, the categories have maximum spread. For a two-dimensional solution, multiple correspondence analysis nds a second set of quantications of the categories of each variable unrelated to the rst set, attempting again to maximize 14 Chapter 1 spread, and so on. Because categories of a variable receive as many scorings as there are dimensions, the variables in the analysis are assumed to be multiple nominal in optimal scaling level. Multiple correspondence analysis also assigns scores to the objects in the analysis in such a way that the category quantications are the averages, or centroids, of the object scores of objects in that category. Relation to other Categories procedures. Multiple correspondence analysis is also known as homogeneity analysis or dual scaling. It gives comparable, but not identical, results to correspondence analysis when there are only two variables. Correspondence analysis produces unique output summarizing the t and quality of representation of the solution, including stability information. Thus, correspondence analysis is usually preferable to multiple correspondence analysis in the two-variable case. Another difference between the two procedures is that the input to multiple correspondence analysis is a data matrix, where the rows are objects and the columns are variables, while the input to correspondence analysis can be the same data matrix, a general proximity matrix, or a joint contingency table, which is an aggregated matrix in which both the rows and columns represent categories of variables. Multiple correspondence analysis can also be thought of as principal components analysis of data scaled at the multiple nominal level. Relation to standard techniques. Multiple correspondence analysis can be thought of as the analysis of a multiway contingency table. Multiway contingency tables can also be analyzed with the Crosstabs procedure, but Crosstabs gives separate summary statistics for each category of each control variable. With multiple correspondence analysis, it is often possible to summarize the relationship between all of the variables with a single two-dimensional plot. An advanced use of multiple correspondence analysis is to replace the original category values with the optimal scale values from the rst dimension and perform a secondary multivariate analysis. Since multiple correspondence analysis replaces category labels with numerical scale values, many different procedures that require numerical data can be applied after the multiple correspondence analysis. For example, the Factor Analysis procedure produces a rst principal component that is equivalent to the rst dimension of multiple correspondence analysis. The component scores in the rst dimension are equal to the object scores, and the squared component loadings are equal to the discrimination measures. The second multiple correspondence analysis dimension, however, is not equal to the second dimension of factor analysis. 15 Introduction to Optimal Scaling Procedures for Categorical Data Multidimensional Scaling The use of Multidimensional Scaling is most appropriate when the goal of your analysis is to nd the structure in a set of distance measures between a single set of objects or cases. This is accomplished by assigning observations to specic locations in a conceptual low-dimensional space so that the distances between points in the space match the given (dis)similarities as closely as possible. The result is a least-squares representation of the objects in that low-dimensional space, which, in many cases, will help you further understand your data. Relation to other Categories procedures. When you have multivariate data from which you create distances and which you then analyze with multidimensional scaling, the results are similar to analyzing the data using categorical principal components analysis with object principal normalization. This kind of PCA is also known as principal coordinates analysis. Relation to standard techniques. The Categories Multidimensional Scaling procedure (PROXSCAL) offers several improvements upon the scaling procedure available in the Base system (ALSCAL). PROXSCAL offers an accelerated algorithm for certain models and allows you to put restrictions on the common space. Moreover, PROXSCAL attempts to minimize normalized raw stress rather than S-stress (also referred to as strain). The normalized raw stress is generally preferred because it is a measure based on the distances, while the S-stress is based on the squared distances. Multidimensional Unfolding The use of Multidimensional Unfolding is most appropriate when the goal of your analysis is to nd the structure in a set of distance measures between two sets of objects (referred to as the row and column objects). This is accomplished by assigning observations to specic locations in a conceptual low-dimensional space so that the distances between points in the space match the given (dis)similarities as closely as possible. The result is a least-squares representation of the row and column objects in that low-dimensional space, which, in many cases, will help you further understand your data. Relation to other Categories procedures. If your data consist of distances between a single set of objects (a square, symmetrical matrix), use Multidimensional Scaling. 16 Chapter 1 Relation to standard techniques. The Categories Multidimensional Unfolding procedure (PREFSCAL) offers several improvements upon the unfolding functionality available in the Base system (through ALSCAL). PREFSCAL allows you to put restrictions on the common space; moreover, PREFSCAL attempts to minimize a penalized stress measure that helps it to avoid degenerate solutions (to which older algorithms are prone). Aspect Ratio in Optimal Scaling Charts Aspect ratio in optimal scaling plots is isotropic. In a two-dimensional plot, the distance representing one unit in dimension 1 is equal to the distance representing one unit in dimension 2. If you change the range of a dimension in a two-dimensional plot, the system changes the size of the other dimension to keep the physical distances equal. Isotropic aspect ratio cannot be overridden for the optimal scaling procedures. Recommended Readings See the following texts for general information on optimal scaling techniques: Barlow, R. E., D. J. Bartholomew, D. J. Bremner, and H. D. Brunk. 1972. Statistical inference under order restrictions. New York: John Wiley and Sons. Benzcri, J. P. 1969. Statistical analysis as a tool to make patterns emerge from data. In: Methodologies of Pattern Recognition, S. Watanabe, ed. New York: Academic Press, 3574. Bishop, Y. M., S. E. Feinberg, and P. W. Holland. 1975. Discrete multivariate analysis: Theory and practice. Cambridge, Mass.: MIT Press. De Leeuw, J. 1984. The Gi system of nonlinear multivariate analysis. In: Data Analysis and Informatics III, E. Diday, et al., ed., 415424. De Leeuw, J. 1990. Multivariate analysis with optimal scaling. In: Progress in Multivariate Analysis, S. Das Gupta, and J. Sethuraman, eds. Calcutta: Indian Statistical Institute. De Leeuw, J., and J. Van Rijckevorsel. 1980. HOMALS and PRINCALSSome generalizations of principal components analysis. In: Data Analysis and Informatics, E. Diday,et al., ed. Amsterdam: North-Holland, 231242. 17 Introduction to Optimal Scaling Procedures for Categorical Data De Leeuw, J., F. W. Young, and Y. Takane. 1976. Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika, 41, 471503. Gi, A. 1990. Nonlinear multivariate analysis. Chichester: John Wiley and Sons. Heiser, W. J., and J. J. Meulman. 1995. Nonlinear methods for the analysis of homogeneity and heterogeneity. In: Recent Advances in Descriptive Multivariate Analysis, W. J. Krzanowski, ed. Oxford: Oxford UniversityPress, 5189. Israls, A. 1987. Eigenvalue techniques for qualitative data. Leiden: DSWO Press. Krzanowski, W. J., and F. H. C. Marriott. 1994. Multivariate analysis: Part I, distributions, ordination and inference. London: Edward Arnold. Lebart, L., A. Morineau, and K. M. Warwick. 1984. Multivariate descriptive statistical analysis. New York: John Wiley and Sons. Max, J. 1960. Quantizing for minimum distortion. Proceedings IEEE (Information Theory), 6, 712. Meulman, J. J. 1986. A distance approach to nonlinear multivariate analysis. Leiden: DSWO Press. Meulman, J. J. 1992. The integration of multidimensional scaling and multivariate analysis with optimal transformations of the variables. Psychometrika, 57, 539565. Nishisato, S. 1980. Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press. Nishisato, S. 1994. Elements of dual scaling: An introduction to practical data analysis. Hillsdale, N.J.: Lawrence Erlbaum Associates, Inc. Rao, C. R. 1973. Linear statistical inference and its applications, 2nd ed. New York: John Wiley and Sons. Rao, C. R. 1980. Matrix approximations and reduction of dimensionality in multivariate statistical analysis. In: Multivariate Analysis, Vol. 5, P. R. Krishnaiah, ed. Amsterdam: North-Holland, 322. Roskam, E. E. 1968. Metric analysis of ordinal data in psychology. Voorschoten: VAM. Shepard, R. N. 1966. Metric structures in ordinal data. Journal of Mathematical Psychology, 3, 287315. 18 Chapter 1 Wolter, K. M. 1985. Introduction to variance estimation. Berlin: Springer-Verlag. Young, F. W. 1981. Quantitative analysis of qualitative data. Psychometrika, 46, 357387. Chapter Categorical Regression (CATREG) 2 Categorical regression quanties categorical data by assigning numerical values to the categories, resulting in an optimal linear regression equation for the transformed variables. Categorical regression is also known by the acronym CATREG, for categorical regression. Standard linear regression analysis involves minimizing the sum of squared differences between a response (dependent) variable and a weighted combination of predictor (independent) variables. Variables are typically quantitative, with (nominal) categorical data recoded to binary or contrast variables. As a result, categorical variables serve to separate groups of cases, and the technique estimates separate sets of parameters for each group. The estimated coefcients reect how changes in the predictors affect the response. Prediction of the response is possible for any combination of predictor values. An alternative approach involves regressing the response on the categorical predictor values themselves. Consequently, one coefcient is estimated for each variable. However, for categorical variables, the category values are arbitrary. Coding the categories in different ways yield different coefcients, making comparisons across analyses of the same variables difcult. CATREG extends the standard approach by simultaneously scaling nominal, ordinal, and numerical variables. The procedure quanties categorical variables so that the quantications reect characteristics of the original categories. The procedure treats quantied categorical variables in the same way as numerical variables. Using nonlinear transformations allow variables to be analyzed at a variety of levels to nd the best-tting model. Example. Categorical regression could be used to describe how job satisfaction depends on job category, geographic region, and amount of travel. You might nd that high levels of satisfaction correspond to managers and low travel. The resulting regression equation could be used to predict job satisfaction for any combination of the three independent variables. 19 20 Chapter 2 Statistics and plots. Frequencies, regression coefcients, ANOVA table, iteration history, category quantications, correlations between untransformed predictors, correlations between transformed predictors, residual plots, and transformation plots. Data. CATREG operates on category indicator variables. The category indicators should be positive integers. You can use the Discretization dialog box to convert fractional-value variables and string variables into positive integers. Assumptions. Only one response variable is allowed, but the maximum number of predictor variables is 200. The data must contain at least three valid cases, and the number of valid cases must exceed the number of predictor variables plus one. Related procedures. CATREG is equivalent to categorical canonical correlation analysis with optimal scaling (OVERALS) with two sets, one of which contains only one variable. Scaling all variables at the numerical level corresponds to standard multiple regression analysis. To Obtain a Categorical Regression E From the menus choose: Analyze Regression Optimal Scaling (CATREG)... 21 Categorical Regression (CATREG) Figure 2-1 Categorical Regression dialog box E Select the dependent variable and independent variable(s). E Click OK. Optionally, change the scaling level for each variable. Define Scale in Categorical Regression You can set the optimal scaling level for the dependent and independent variables. By default, they are scaled as second-degree monotonic splines (ordinal) with two interior knots. Additionally, you can set the weight for analysis variables. 22 Chapter 2 Figure 2-2 Define Scale dialog box Optimal Scaling Level. You can also select the scaling level for quantifying each variable. Spline Ordinal. The order of the categories of the observed variable is preserved in the optimally scaled variable. Category points will be on a straight line (vector) through the origin. The resulting transformation is a smooth monotonic piecewise polynomial of the chosen degree. The pieces are specied by the user-specied number and procedure-determined placement of the interior knots. Spline Nominal. The only information in the observed variable that is preserved in the optimally scaled variable is the grouping of objects in categories. The order of the categories of the observed variable is not preserved. Category points will be on a straight line (vector) through the origin. The resulting transformation is a smooth, possibly nonmonotonic, piecewise polynomial of the chosen degree. The pieces are specied by the user-specied number and procedure-determined placement of the interior knots. Ordinal. The order of the categories of the observed variable is preserved in the optimally scaled variable. Category points will be on a straight line (vector) through the origin. The resulting transformation ts better than the spline ordinal transformation but is less smooth. Nominal. The only information in the observed variable that is preserved in the optimally scaled variable is the grouping of objects in categories. The order of the categories of the observed variable is not preserved. Category points will be on a 23 Categorical Regression (CATREG) straight line (vector) through the origin. The resulting transformation ts better than the spline nominal transformation but is less smooth. Numeric. Categories are treated as ordered and equally spaced (interval level). The order of the categories and the equal distances between category numbers of the observed variable are preserved in the optimally scaled variable. Category points will be on a straight line (vector) through the origin. When all variables are at the numeric level, the analysis is analogous to standard principal components analysis. Categorical Regression Discretization The Discretization dialog box allows you to select a method of recoding your variables. Fractional-value variables are grouped into seven categories (or into the number of distinct values of the variable if this number is less than seven) with an approximately normal distribution unless otherwise specied. String variables are always converted into positive integers by assigning category indicators according to ascending alphanumeric order. Discretization for string variables applies to these integers. Other variables are left alone by default. The discretized variables are then used in the analysis. 24 Chapter 2 Figure 2-3 Discretization dialog box Method. Choose between grouping, ranking, and multiplying. Grouping. Recode into a specied number of categories or recode by interval. Ranking. The variable is discretized by ranking the cases. Multiplying. The current values of the variable are standardized, multiplied by 10, rounded, and have a constant added so that the lowest discretized value is 1. Grouping. The following options are available when discretizing variables by grouping: Number of categories. Specify a number of categories and whether the values of the variable should follow an approximately normal or uniform distribution across those categories. Equal intervals. Variables are recoded into categories dened by these equally sized intervals. You must specify the length of the intervals. 25 Categorical Regression (CATREG) Categorical Regression Missing Values The Missing Values dialog box allows you to choose the strategy for handling missing values in analysis variables and supplementary variables. Figure 2-4 Missing Values dialog box Strategy. Choose to exclude objects with missing values (listwise deletion) or impute missing values (active treatment). Exclude objects with missing values on this variable. Objects with missing values on the selected variable are excluded from the analysis. This strategy is not available for supplementary variables. Impute missing values. Objects with missing values on the selected variable have those values imputed. You can choose the method of imputation. Select Mode to replace missing values with the most frequent category. When there are multiple modes, the one with the smallest category indicator is used. Select Extra category to replace missing values with the same quantication of an extra category. This 26 Chapter 2 implies that objects with a missing value on this variable are considered to belong to the same (extra) category. Categorical Regression Options The Options dialog box allows you to select the initial conguration style, specify iteration and convergence criteria, select supplementary objects, and set the labeling of plots. Figure 2-5 Options dialog box 27 Categorical Regression (CATREG) Supplementary Objects. This allows you to specify the objects that you want to treat as supplementary. Simply type the number of a supplementary object (or specify a range of cases) and click Add. You cannot weight supplementary objects (specied weights are ignored). Initial Configuration. If no variables are treated as nominal, select the Numerical conguration. If at least one variable is treated as nominal, select the Random conguration. Alternatively, if at least one variable has an ordinal or spline ordinal scaling level, the usual model-tting algorithm can result in a suboptimal solution. Choosing Multiple systematic starts with all possible sign patterns to test will always nd the optimal solution, but the necessary processing time rapidly increases as the number of ordinal and spline ordinal variables in the dataset increase. You can reduce the number of test patterns by specifying a percentage of loss of variance threshold, where the higher the threshold, the more sign patterns will be excluded. With this option, obtaining the optimal solution is not garantueed, but the chance of obtaining a suboptimal solution is diminished. Also, if the optimal solution is not found, the chance that the suboptimal solution is very different from the optimal solution is diminished. When multiple systematic starts are requested, the signs of the regression coefcients for each start are written to an external SPSS Statistics data le or dataset in the current session. For more information, see Categorical Regression Save on p. 32. The results of a previous run with multiple systematic starts allows you to Use fixed signs for the regression coefficients. The signs (indicated by 1 and 1) need to be in a row of the specied dataset or le. The integer-valued starting number is the case number of the row in this le that contains the signs to be used. Criteria. You can specify the maximum number of iterations that the regression may go through in its computations. You can also select a convergence criterion value. The regression stops iterating if the difference in total t between the last two iterations is less than the convergence value or if the maximum number of iterations is reached. Label Plots By. Allows you to specify whether variables and value labels or variable names and values will be used in the plots. You can also specify a maximum length for labels. 28 Chapter 2 Categorical Regression Regularization Figure 2-6 Regularization dialog box Method. Regularization methods can improve the predictive error of the model by reducing the variability in the estimates of regression coefcient by shrinking the estimates toward 0. The Lasso and Elastic Net will shrink some coefcient estimates to exactly 0, thus providing a form of variable selection. When a regularization method is requested, the regularized model and coefcients for each penalty coefcient value are written to an external SPSS Statistics data le or dataset in the current session. For more information, see Categorical Regression Save on p. 32. Ridge regression. Ridge regression shrinks coefcients by introducing a penalty term equal to the sum of squared coefcients times a penalty coefcient. This coefcient can range from 0 (no penalty) to 1; the procedure will search for the best value of the penalty if you specify a range and increment. 29 Categorical Regression (CATREG) Lasso. The Lassos penalty term is based on the sum of absolute coefcients, and the specication of a penalty coefcient is similar to that of Ridge regression; however, the Lasso is more computationally intensive. Elastic net. The Elastic Net simply combines the Lasso and Ridge regression penalties, and will search over the grid of values specied to nd the best Lasso and Ridge regression penalty coefcients. For a given pair of Lasso and Ridge regression penalties, the Elastic Net is not much more computationally expensive than the Lasso. Display regularization plots. These are plots of the regression coefcients versus the regularization penalty. When searching a range of values for the best penalty coefcient, it provides a view of how the regression coefcients change over that range. Elastic Net Plots. For the Elastic Net method, separate regularization plots are produced by values of the Ridge regression penalty. All possible plots uses every value in the range determined by the minimum and maximum Ridge regression penalty values specied. For some Ridge penalties allows you to specify a subset of the values in the range determined by the minimum and maximum. Simply type the number of a penalty value (or specify a range of values) and click Add. Categorical Regression Output The Output dialog box allows you to select the statistics to display in the output. 30 Chapter 2 Figure 2-7 Output dialog box Tables. Produces tables for: Multiple R. Includes R2, adjusted R2, and adjusted R2 taking the optimal scaling into account. ANOVA. This option includes regression and residual sums of squares, mean squares, and F. Two ANOVA tables are displayed: one with degrees of freedom for the regression equal to the number of predictor variables and one with degrees of freedom for the regression taking the optimal scaling into account. Coefficients. This option gives three tables: a Coefcients table that includes betas, standard error of the betas, t values, and signicance; a Coefcients-Optimal Scaling table with the standard error of the betas taking the optimal scaling degrees of freedom into account; and a table with the zero-order, part, and partial 31 Categorical Regression (CATREG) correlation, Pratts relative importance measure for the transformed predictors, and the tolerance before and after transformation. Iteration history. For each iteration, including the starting values for the algorithm, the multiple R and regression error are shown. The increase in multiple R is listed starting from the rst iteration. Correlations of original variables. A matrix showing the correlations between the untransformed variables is displayed. Correlations of transformed variables. A matrix showing the correlations between the transformed variables is displayed. Regularized models and coefficients. Displays penalty values, R-square, and the regression coefcients for each regularized model. If a resampling method is specied or if supplementary objects (test cases) are specied, it also displays the prediction error or test MSE. Resampling. Resampling methods give you an estimate of the prediction error of the model. Crossvalidation. Crossvalidation divides the sample into a number of subsamples, or folds. Categorical regression models are then generated, excluding the data from each subsample in turn. The rst model is based on all of the cases except those in the rst sample fold, the second model is based on all of the cases except those in the second sample fold, and so on. For each model, the prediction error is estimated by applying the model to the subsample excluded in generating it. .632 Bootstrap. With the bootstrap, observations are drawn randomly from the data with replacement, repeating this process a number of times to obtain a number bootstrap samples. A model is t for each bootstrap sample, and the prediction error for each model is estimated by this tted model is then applied to the cases not in the bootstrap sample. Category Quantifications. Tables showing the transformed values of the selected variables are displayed. Descriptive Statistics. Tables showing the frequencies, missing values, and modes of the selected variables are displayed. 32 Chapter 2 Categorical Regression Save The Save dialog box allows you to save predicted values, residuals, and transformed values to the active dataset and/or save discretized data, transformed values, regularized models and coefcients, and signs of regression coefcients to an external SPSS Statistics data le or dataset in the current session. Datasets are available during the current session but are not available in subsequent sessions unless you explicitly save them as data les. Dataset names must adhere to variable naming rules. Filenames or dataset names must be different for each type of data saved. Figure 2-8 Save dialog box Regularized models and coefcients are saved whenever a regularization method is selected on the Regularization dialog. By default, the procedure creates a new dataset with a unique name, but you can of course specify a name of your own choosing or write to an external le. 33 Categorical Regression (CATREG) Signs of regression coefcients are saved whenever multiple systematic starts are used as the initial conguration on the Options dialog. By default, the procedure creates a new dataset with a unique name, but you can of course specify a name of your own choosing or write to an external le. Categorical Regression Transformation Plots The Plots dialog box allows you to specify the variables that will produce transformation and residual plots. Figure 2-9 Plots dialog box Transformation Plots. For each of these variables, the category quantications are plotted against the original category values. Empty categories appear on the horizontal axis but do not affect the computations. These categories are identied by breaks in the line connecting the quantications. Residual Plots. For each of these variables, residuals (computed for the dependent variable predicted from all predictor variables except the predictor variable in question) are plotted against category indicators and the optimal category quantications multiplied with beta against category indicators. 34 Chapter 2 CATREG Command Additional Features You can customize your categorical regression if you paste your selections into a syntax window and edit the resulting CATREG command syntax. The command syntax language also allows you to: Specify rootnames for the transformed variables when saving them to the active dataset (with the SAVE subcommand). See the Command Syntax Reference for complete syntax information. Chapter Categorical Principal Components Analysis (CATPCA) 3 This procedure simultaneously quanties categorical variables while reducing the dimensionality of the data. Categorical principal components analysis is also known by the acronym CATPCA, for categorical principal components analysis. The goal of principal components analysis is to reduce an original set of variables into a smaller set of uncorrelated components that represent most of the information found in the original variables. The technique is most useful when a large number of variables prohibits effective interpretation of the relationships between objects (subjects and units). By reducing the dimensionality, you interpret a few components rather than a large number of variables. Standard principal components analysis assumes linear relationships between numeric variables. On the other hand, the optimal-scaling approach allows variables to be scaled at different levels. Categorical variables are optimally quantied in the specied dimensionality. As a result, nonlinear relationships between variables can be modeled. Example. Categorical principal components analysis could be used to graphically display the relationship between job category, job division, region, amount of travel (high, medium, and low), and job satisfaction. You might nd that two dimensions account for a large amount of variance. The rst dimension might separate job category from region, whereas the second dimension might separate job division from amount of travel. You also might nd that high job satisfaction is related to a medium amount of travel. Statistics and plots. Frequencies, missing values, optimal scaling level, mode, variance accounted for by centroid coordinates, vector coordinates, total per variable and per dimension, component loadings for vector-quantied variables, category quantications and coordinates, iteration history, correlations of the transformed variables and eigenvalues of the correlation matrix, correlations of the original 35 36 Chapter 3 variables and eigenvalues of the correlation matrix, object scores, category plots, joint category plots, transformation plots, residual plots, projected centroid plots, object plots, biplots, triplots, and component loadings plots. Data. String variable values are always converted into positive integers by ascending alphanumeric order. User-dened missing values, system-missing values, and values less than 1 are considered missing; you can recode or add a constant to variables with values less than 1 to make them nonmissing. Assumptions. The data must contain at least three valid cases. The analysis is based on positive integer data. The discretization option will automatically categorize a fractional-valued variable by grouping its values into categories with a close to normal distribution and will automatically convert values of string variables into positive integers. You can specify other discretization schemes. Related procedures. Scaling all variables at the numeric level corresponds to standard principal components analysis. Alternate plotting features are available by using the transformed variables in a standard linear principal components analysis. If all variables have multiple nominal scaling levels, categorical principal components analysis is identical to multiple correspondence analysis. If sets of variables are of interest, categorical (nonlinear) canonical correlation analysis should be used. To Obtain a Categorical Principal Components Analysis E From the menus choose: Analyze Dimension Reduction Optimal Scaling... 37 Categorical Principal Components Analysis (CATPCA) Figure 3-1 Optimal Scaling dialog box E Select Some variable(s) not multiple nominal. E Select One set. E Click Define. 38 Chapter 3 Figure 3-2 Categorical Principal Components dialog box E Select at least two analysis variables and specify the number of dimensions in the solution. E Click OK. You may optionally specify supplementary variables, which are tted into the solution found, or labeling variables for the plots. Define Scale and Weight in CATPCA You can set the optimal scaling level for analysis variables and supplementary variables. By default, they are scaled as second-degree monotonic splines (ordinal) with two interior knots. Additionally, you can set the weight for analysis variables. 39 Categorical Principal Components Analysis (CATPCA) Figure 3-3 Define Scale and Weight dialog box Variable weight. You can choose to dene a weight for each variable. The value specied must be a positive integer. The default value is 1. Optimal Scaling Level. You can also select the scaling level to be used to quantify each variable. Spline ordinal. The order of the categories of the observed variable is preserved in the optimally scaled variable. Category points will be on a straight line (vector) through the origin. The resulting transformation is a smooth monotonic piecewise polynomial of the chosen degree. The pieces are specied by the user-specied number and procedure-determined placement of the interior knots. Spline nominal. The only information in the observed variable that is preserved in the optimally scaled variable is the grouping of objects in categories. The order of the categories of the observed variable is not preserved. Category points will be on a straight line (vector) through the origin. The resulting transformation is a smooth, possibly nonmonotonic, piecewise polynomial of the chosen degree. The pieces are specied by the user-specied number and procedure-determined placement of the interior knots. Multiple nominal. The only information in the observed variable that is preserved in the optimally scaled variable is the grouping of objects in categories. The order of the categories of the observed variable is not preserved. Category points will be 40 Chapter 3 in the centroid of the objects in the particular categories. Multiple indicates that different sets of quantications are obtained for each dimension. Ordinal. The order of the categories of the observed variable is preserved in the optimally scaled variable. Category points will be on a straight line (vector) through the origin. The resulting transformation ts better than the spline ordinal transformation but is less smooth. Nominal. The only information in the observed variable that is preserved in the optimally scaled variable is the grouping of objects in categories. The order of the categories of the observed variable is not preserved. Category points will be on a straight line (vector) through the origin. The resulting transformation ts better than the spline nominal transformation but is less smooth. Numeric. Categories are treated as ordered and equally spaced (interval level). The order of the categories and the equal distances between category numbers of the observed variable are preserved in the optimally scaled variable. Category points will be on a straight line (vector) through the origin. When all variables are at the numeric level, the analysis is analogous to standard principal components analysis. Categorical Principal Components Analysis Discretization The Discretization dialog box allows you to select a method of recoding your variables. Fractional-valued variables are grouped into seven categories (or into the number of distinct values of the variable if this number is less than seven) with an approximately normal distribution, unless specied otherwise. String variables are always converted into positive integers by assigning category indicators according to ascending alphanumeric order. Discretization for string variables applies to these integers. Other variables are left alone by default. The discretized variables are then used in the analysis. 41 Categorical Principal Components Analysis (CATPCA) Figure 3-4 Discretization dialog box Method. Choose between grouping, ranking, and multiplying. Grouping. Recode into a specied number of categories or recode by interval. Ranking. The variable is discretized by ranking the cases. Multiplying. The current values of the variable are standardized, multiplied by 10, rounded, and have a constant added such that the lowest discretized value is 1. Grouping. The following options are available when you are discretizing variables by grouping: Number of categories. Specify a number of categories and whether the values of the variable should follow an approximately normal or uniform distribution across those categories. Equal intervals. Variables are recoded into categories dened by these equally sized intervals. You must specify the length of the intervals. 42 Chapter 3 Categorical Principal Components Analysis Missing Values The Missing Values dialog box allows you to choose the strategy for handling missing values in analysis variables and supplementary variables. Figure 3-5 Missing Values dialog box Strategy. Choose to exclude missing values (passive treatment), impute missing values (active treatment), or exclude objects with missing values (listwise deletion). Exclude missing values; for correlations impute after quantification. Objects with missing values on the selected variable do not contribute to the analysis for this variable. If all variables are given passive treatment, then objects with missing values on all variables are treated as supplementary. If correlations are specied in the Output dialog box, then (after analysis) missing values are imputed with the 43 Categorical Principal Components Analysis (CATPCA) most frequent category, or mode, of the variable for the correlations of the original variables. For the correlations of the optimally scaled variables, you can choose the method of imputation. Select Mode to replace missing values with the mode of the optimally scaled variable. Select Extra category to replace missing values with the quantication of an extra category. This implies that objects with a missing value on this variable are considered to belong to the same (extra) category. Impute missing values. Objects with missing values on the selected variable have those values imputed. You can choose the method of imputation. Select Mode to replace missing values with the most frequent category. When there are multiple modes, the one with the smallest category indicator is used. Select Extra category to replace missing values with the same quantication of an extra category. This implies that objects with a missing value on this variable are considered to belong to the same (extra) category. Exclude objects with missing values on this variable. Objects with missing values on the selected variable are excluded from the analysis. This strategy is not available for supplementary variables. Categorical Principal Components Analysis Options The Options dialog box allows you to select the initial conguration, specify iteration and convergence criteria, select a normalization method, choose the method for labeling plots, and specify supplementary objects. 44 Chapter 3 Figure 3-6 Options dialog box Supplementary Objects. Specify the case number of the object, or the rst and last case numbers of a range of objects, that you want to make supplementary and then click Add. Continue until you have specied all of your supplementary objects. If an object is specied as supplementary, then case weights are ignored for that object. Normalization Method. You can specify one of ve options for normalizing the object scores and the variables. Only one normalization method can be used in a given analysis. Variable Principal. This option optimizes the association between variables. The coordinates of the variables in the object space are the component loadings (correlations with principal components, such as dimensions and object scores). 45 Categorical Principal Components Analysis (CATPCA) This is useful when you are primarily interested in the correlation between the variables. Object Principal. This option optimizes distances between objects. This is useful when you are primarily interested in differences or similarities between the objects. Symmetrical. Use this normalization option if you are primarily interested in the relation between objects and variables. Independent. Use this normalization option if you want to examine distances between objects and correlations between variables separately. Custom. You can specify any real value in the closed interval [1, 1]. A value of 1 is equal to the Object Principal method, a value of 0 is equal to the Symmetrical method, and a value of 1 is equal to the Variable Principal method. By specifying a value greater than 1 and less than 1, you can spread the eigenvalue over both objects and variables. This method is useful for making a tailor-made biplot or triplot. Criteria. You can specify the maximum number of iterations the procedure can go through in its computations. You can also select a convergence criterion value. The algorithm stops iterating if the difference in total t between the last two iterations is less than the convergence value or if the maximum number of iterations is reached. Label Plots By. Allows you to specify whether variables and value labels or variable names and values will be used in the plots. You can also specify a maximum length for labels. Plot Dimensions. Allows you to control the dimensions displayed in the output. Display all dimensions in the solution. All dimensions in the solution are displayed in a scatterplot matrix. Restrict the number of dimensions. The displayed dimensions are restricted to plotted pairs. If you restrict the dimensions, you must select the lowest and highest dimensions to be plotted. The lowest dimension can range from 1 to the number of dimensions in the solution minus 1 and is plotted against higher dimensions. The highest dimension value can range from 2 to the number of dimensions in the solution and indicates the highest dimension to be used in plotting the dimension pairs. This specication applies to all requested multidimensional plots. Configuration. You can read data from a le containing the coordinates of a conguration. The rst variable in the le should contain the coordinates for the rst dimension, the second variable should contain the coordinates for the second dimension, and so on. 46 Chapter 3 Initial. The conguration in the le specied will be used as the starting point of the analysis. Fixed. The conguration in the le specied will be used to t in the variables. The variables that are tted in must be selected as analysis variables, but because the conguration is xed, they are treated as supplementary variables (so they do not need to be selected as supplementary variables). Categorical Principal Components Analysis Output The Output dialog box allows you to produce tables for object scores, component loadings, iteration history, correlations of original and transformed variables, the variance accounted for per variable and per dimension, category quantications for selected variables, and descriptive statistics for selected variables. 47 Categorical Principal Components Analysis (CATPCA) Figure 3-7 Output dialog box Object scores. Displays the object scores and has the following options: Include Categories Of. Displays the category indicators of the analysis variables selected. Label Object Scores By. From the list of variables specied as labeling variables, you can select one to label the objects. Component loadings. Displays the component loadings for all variables that were not given multiple nominal scaling levels. Iteration history. For each iteration, the variance accounted for, loss, and increase in variance accounted for are shown. 48 Chapter 3 Correlations of original variables. Shows the correlation matrix of the original variables and the eigenvalues of that matrix. Correlations of transformed variables. Shows the correlation matrix of the transformed (optimally scaled) variables and the eigenvalues of that matrix. Variance accounted for. Displays the amount of variance accounted for by centroid coordinates, vector coordinates, and total (centroid and vector coordinates combined) per variable and per dimension. Category Quantifications. Gives the category quantications and coordinates for each dimension of the variable(s) selected. Descriptive Statistics. Displays frequencies, number of missing values, and mode of the variable(s) selected. Categorical Principal Components Analysis Save The Save dialog box allows you to save discretized data, object scores, transformed values, and approximations to an external SPSS Statistics data le or dataset in the current session. You can also save transformed values, object scores, and approximations to the active dataset. Datasets are available during the current session but are not available in subsequent sessions unless you explicitly save them as data les. Dataset names must adhere to variable naming rules. Filenames or dataset names must be different for each type of data saved. If you save object scores or transformed values to the active dataset, you can specify the number of multiple nominal dimensions. 49 Categorical Principal Components Analysis (CATPCA) Figure 3-8 Save dialog box Categorical Principal Components Analysis Object Plots The Object and Variable Plots dialog box allows you to specify the types of plots desired and the variables for which plots will be produced. 50 Chapter 3 Figure 3-9 Object and Variable Plots dialog box Object points. A plot of the object points is displayed. Objects and variables (biplot). The object points are plotted with your choice of the variable coordinatescomponent loadings or variable centroids. Objects, loadings, and centroids (triplot). The object points are plotted with the centroids of multiple nominal-scaling-level variables and the component loadings of other variables. Biplot and Triplot Variables. You can choose to use all variables for the biplots and triplots or select a subset. 51 Categorical Principal Components Analysis (CATPCA) Label Objects. You can choose to have objects labeled with the categories of selected variables (you may choose category indicator values or value labels in the Options dialog box) or with their case numbers. One plot is produced per variable if Variable is selected. Categorical Principal Components Analysis Category Plots The Category Plots dialog box allows you to specify the types of plots desired and the variables for which plots will be produced. Figure 3-10 Category Plots dialog box 52 Chapter 3 Category Plots. For each variable selected, a plot of the centroid and vector coordinates is plotted. For variables with multiple nominal scaling levels, categories are in the centroids of the objects in the particular categories. For all other scaling levels, categories are on a vector through the origin. Joint Category Plots. This is a single plot of the centroid and vector coordinates of each selected variable. Transformation Plots. Displays a plot of the optimal category quantications versus the category indicators. You can specify the number of dimensions desired for variables with multiple nominal scaling levels; one plot will be generated for each dimension. You can also choose to display residual plots for each variable selected. Project Centroids Of. You may choose a variable and project its centroids onto selected variables. Variables with multiple nominal scaling levels cannot be selected to project on. When this plot is requested, a table with the coordinates of the projected centroids is also displayed. Categorical Principal Components Analysis Loading Plots The Loading Plots dialog box allows you to specify the variables that will be included in the plot, and whether or not to include centroids in the plot. 53 Categorical Principal Components Analysis (CATPCA) Figure 3-11 Loading Plots dialog box Display component loadings. If selected, a plot of the component loadings is displayed. Loading Variables. You can choose to use all variables for the component loadings plot or select a subset. Include centroids. Variables with multiple nominal scaling levels do not have component loadings, but you may choose to include the centroids of those variables in the plot. You can choose to use all multiple nominal variables or select a subset. 54 Chapter 3 CATPCA Command Additional Features You can customize your categorical principal components analysis if you paste your selections into a syntax window and edit the resulting CATPCA command syntax. The command syntax language also allows you to: Specify rootnames for the transformed variables, object scores, and approximations when saving them to the active dataset (with the SAVE subcommand). Specify a maximum length for labels for each plot separately (with the PLOT subcommand). Specify a separate variable list for residual plots (with the PLOT subcommand). See the Command Syntax Reference for complete syntax information. Chapter Nonlinear Canonical Correlation Analysis (OVERALS) 4 Nonlinear canonical correlation analysis corresponds to categorical canonical correlation analysis with optimal scaling. The purpose of this procedure is to determine how similar sets of categorical variables are to one another. Nonlinear canonical correlation analysis is also known by the acronym OVERALS. Standard canonical correlation analysis is an extension of multiple regression, where the second set does not contain a single response variable but instead contain multiple response variables. The goal is to explain as much as possible of the variance in the relationships among two sets of numerical variables in a low dimensional space. Initially, the variables in each set are linearly combined such that the linear combinations have a maximal correlation. Given these combinations, subsequent linear combinations are determined that are uncorrelated with the previous combinations and that have the largest correlation possible. The optimal scaling approach expands the standard analysis in three crucial ways. First, OVERALS allows more than two sets of variables. Second, variables can be scaled as either nominal, ordinal, or numerical. As a result, nonlinear relationships between variables can be analyzed. Finally, instead of maximizing correlations between the variable sets, the sets are compared to an unknown compromise set that is dened by the object scores. Example. Categorical canonical correlation analysis with optimal scaling could be used to graphically display the relationship between one set of variables containing job category and years of education and another set of variables containing region of residence and gender. You might nd that years of education and region of residence discriminate better than the remaining variables. You might also nd that years of education discriminates best on the rst dimension. 55 56 Chapter 4 Statistics and plots. Frequencies, centroids, iteration history, object scores, category quantications, weights, component loadings, single and multiple t, object scores plots, category coordinates plots, component loadings plots, category centroids plots, transformation plots. Data. Use integers to code categorical variables (nominal or ordinal scaling level). To minimize output, use consecutive integers beginning with 1 to code each variable. Variables that are scaled at the numerical level should not be recoded to consecutive integers. To minimize output, for each variable that is scaled at the numerical level, subtract the smallest observed value from every value and add 1. Fractional values are truncated after the decimal. Assumptions. Variables can be classied into two or more sets. Variables in the analysis are scaled as multiple nominal, single nominal, ordinal, or numerical. The maximum number of dimensions that are used in the procedure depends on the optimal scaling level of the variables. If all variables are specied as ordinal, single nominal, or numerical, the maximum number of dimensions is the lesser of the following two values: the number of observations minus 1 or the total number of variables. However, if only two sets of variables are dened, the maximum number of dimensions is the number of variables in the smaller set. If some variables are multiple nominal, the maximum number of dimensions is the total number of multiple nominal categories plus the number of nonmultiple nominal variables minus the number of multiple nominal variables. For example, if the analysis involves ve variables, one of which is multiple nominal with four categories, the maximum number of dimensions is (4 + 4 1), or 7. If you specify a number that is greater than the maximum, the maximum value is used. Related procedures. If each set contains one variable, nonlinear canonical correlation analysis is equivalent to principal components analysis with optimal scaling. If each of these variables is multiple nominal, the analysis corresponds to multiple correspondence analysis. If two sets of variables are involved, and one of the sets contains only one variable, the analysis is identical to categorical regression with optimal scaling. To Obtain a Nonlinear Canonical Correlation Analysis E From the menus choose: Analyze Dimension Reduction Optimal Scaling... 57 Nonlinear Canonical Correlation Analysis (OVERALS) Figure 4-1 Optimal Scaling dialog box E Select either All variables multiple nominal or Some variable(s) not multiple nominal. E Select Multiple sets. E Click Define. 58 Chapter 4 Figure 4-2 Nonlinear Canonical Correlation Analysis (OVERALS) dialog box E Dene at least two sets of variables. Select the variable(s) that you want to include in the rst set. To move to the next set, click Next, and select the variables that you want to include in the second set. You can add additional sets as desired. Click Previous to return to the previously dened variable set. E Dene the value range and measurement scale (optimal scaling level) for each selected variable. E Click OK. E Optionally: Select one or more variables to provide point labels for object scores plots. Each variable produces a separate plot, with the points labeled by the values of that variable. You must dene a range for each of these plot label variables. When you are using the dialog box, a single variable cannot be used both in the analysis and as a labeling variable. If labeling the object scores plot with a variable that is used in the analysis is desired, use the Compute facility (available from the Transform 59 Nonlinear Canonical Correlation Analysis (OVERALS) menu) to create a copy of that variable. Use the new variable to label the plot. Alternatively, command syntax can be used. Specify the number of dimensions that you want in the solution. In general, choose as few dimensions as needed to explain most of the variation. If the analysis involves more than two dimensions, three-dimensional plots of the rst three dimensions are produced. Other dimensions can be displayed by editing the chart. Define Range and Scale Figure 4-3 Define Range and Scale dialog box You must dene a range for each variable. The maximum value that is specied must be an integer. Fractional data values are truncated in the analysis. A category value that is outside of the specied range is ignored in the analysis. To minimize output, use the Automatic Recode facility (available from the Transform menu) to create consecutive categories beginning with 1 for variables that are treated as nominal or ordinal. Recoding to consecutive integers is not recommended for variables that are scaled at the numerical level. To minimize output for variables that are treated as numerical, for each variable, subtract the minimum value from every value and add 1. You must also select the scaling to be used to quantify each variable. Ordinal. The order of the categories of the observed variable is preserved in the quantied variable. Single nominal. In the quantied variable, objects in the same category receive the same score. 60 Chapter 4 Multiple nominal. The quantications can be different for each dimension. Discrete numeric. Categories are treated as ordered and equally spaced. The differences between category numbers and the order of the categories of the observed variable are preserved in the quantied variable. Define Range Figure 4-4 Define Range dialog box You must dene a range for each variable. The maximum value that is specied must be an integer. Fractional data values are truncated in the analysis. A category value that is outside of the specied range is ignored in the analysis. To minimize output, use the Automatic Recode facility (available from the Transform menu) to create consecutive categories beginning with 1. You must also dene a range for each variable that is used to label the object scores plots. However, labels for categories with data values that are outside of the dened range for the variable do appear on the plots. Nonlinear Canonical Correlation Analysis Options The Options dialog box allows you to select optional statistics and plots, save object scores as new variables in the active dataset, specify iteration and convergence criteria, and specify an initial conguration for the analysis. 61 Nonlinear Canonical Correlation Analysis (OVERALS) Figure 4-5 Options dialog box Display. Available statistics include marginal frequencies (counts), centroids, iteration history, weights and component loadings, category quantications, object scores, and single and multiple t statistics. Centroids. Category quantications, and the projected and the actual averages of the object scores for the objects (cases) included in each set for those belonging to the same category of the variable. Weights and component loadings. The regression coefcients in each dimension for every quantied variable in a set, where the object scores are regressed on the quantied variables, and the projection of the quantied variable in the object space. Provides an indication of the contribution each variable makes to the dimension within each set. Single and multiple fit. Measures of goodness of t of the single- and multiple-category coordinates/category quantications with respect to the objects. Category quantifications. Optimal scale values assigned to the categories of a variable. Object scores. Optimal score assigned to an object (case) in a particular dimension. 62 Chapter 4 Plot. You can produce plots of category coordinates, object scores, component loadings, category centroids, and transformations. Save object scores. You can save the object scores as new variables in the active dataset. Object scores are saved for the number of dimensions that are specied in the main dialog box. Use random initial configuration. A random initial conguration should be used if some or all of the variables are single nominal. If this option is not selected, a nested initial conguration is used. Criteria. You can specify the maximum number of iterations that the nonlinear canonical correlation analysis can go through in its computations. You can also select a convergence criterion value. The analysis stops iterating if the difference in total t between the last two iterations is less than the convergence value or if the maximum number of iterations is reached. OVERALS Command Additional Features You can customize your nonlinear canonical correlation analysis if you paste your selections into a syntax window and edit the resulting OVERALS command syntax. The command syntax language also allows you to: Specify the dimension pairs to be plotted, rather than plotting all extracted dimensions (using theNDIM keyword on the PLOT subcommand). Specify the number of value label characters that are used to label points on the plots (with thePLOT subcommand). Designate more than ve variables as labeling variables for object scores plots (with thePLOT subcommand). Select variables that are used in the analysis as labeling variables for the object scores plots (with the PLOT subcommand). Select variables to provide point labels for the quantication score plot (with the PLOT subcommand). Specify the number of cases to be included in the analysis if you do not want to use all cases in the active dataset (with the NOBSERVATIONS subcommand). Specify rootnames for variables created by saving object scores (with the SAVE subcommand). 63 Nonlinear Canonical Correlation Analysis (OVERALS) Specify the number of dimensions to be saved, rather than saving all extracted dimensions (with the SAVE subcommand). Write category quantications to a matrix le (using the MATRIX subcommand). Produce low-resolution plots that may be easier to read than the usual high-resolution plots (using the SET command). Produce centroid and transformation plots for specied variables only (with the PLOT subcommand). See the Command Syntax Reference for complete syntax information. Chapter Correspondence Analysis 5 One of the goals of correspondence analysis is to describe the relationships between two nominal variables in a correspondence table in a low-dimensional space, while simultaneously describing the relationships between the categories for each variable. For each variable, the distances between category points in a plot reect the relationships between the categories with similar categories plotted close to each other. Projecting points for one variable on the vector from the origin to a category point for the other variable describe the relationship between the variables. An analysis of contingency tables often includes examining row and column proles and testing for independence via the chi-square statistic. However, the number of proles can be quite large, and the chi-square test does not reveal the dependence structure. The Crosstabs procedure offers several measures of association and tests of association but cannot graphically represent any relationships between the variables. Factor analysis is a standard technique for describing relationships between variables in a low-dimensional space. However, factor analysis requires interval data, and the number of observations should be ve times the number of variables. Correspondence analysis, on the other hand, assumes nominal variables and can describe the relationships between categories of each variable, as well as the relationship between the variables. In addition, correspondence analysis can be used to analyze any table of positive correspondence measures. Example. Correspondence analysis could be used to graphically display the relationship between staff category and smoking habits. You might nd that with regard to smoking, junior managers differ from secretaries, but secretaries do not differ from senior managers. You might also nd that heavy smoking is associated with junior managers, whereas light smoking is associated with secretaries. Statistics and plots. Correspondence measures, row and column proles, singular values, row and column scores, inertia, mass, row and column score condence statistics, singular value condence statistics, transformation plots, row point plots, column point plots, and biplots. 64 65 Correspondence Analysis Data. Categorical variables to be analyzed are scaled nominally. For aggregated data or for a correspondence measure other than frequencies, use a weighting variable with positive similarity values. Alternatively, for table data, use syntax to read the table. Assumptions. The maximum number of dimensions used in the procedure depends on the number of active rows and column categories and the number of equality constraints. If no equality constraints are used and all categories are active, the maximum dimensionality is one fewer than the number of categories for the variable with the fewest categories. For example, if one variable has ve categories and the other has four, the maximum number of dimensions is three. Supplementary categories are not active. For example, if one variable has ve categories, two of which are supplementary, and the other variable has four categories, the maximum number of dimensions is two. Treat all sets of categories that are constrained to be equal as one category. For example, if a variable has ve categories, three of which are constrained to be equal, that variable should be treated as having three categories when determining the maximum dimensionality. Two of the categories are unconstrained, and the third category corresponds to the three constrained categories. If you specify a number of dimensions greater than the maximum, the maximum value is used. Related procedures. If more than two variables are involved, use multiple correspondence analysis. If the variables should be scaled ordinally, use categorical principal components analysis. To Obtain a Correspondence Analysis E From the menus choose: Analyze Dimension Reduction Correspondence Analysis... 66 Chapter 5 Figure 5-1 Correspondence Analysis dialog box E Select a row variable. E Select a column variable. E Dene the ranges for the variables. E Click OK. Define Row Range in Correspondence Analysis You must dene a range for the row variable. The minimum and maximum values specied must be integers. Fractional data values are truncated in the analysis. A category value that is outside of the specied range is ignored in the analysis. 67 Correspondence Analysis Figure 5-2 Define Row Range dialog box All categories are initially unconstrained and active. You can constrain row categories to equal other row categories, or you can dene a row category as supplementary. Categories must be equal. Categories must have equal scores. Use equality constraints if the obtained order for the categories is undesirable or counterintuitive. The maximum number of row categories that can be constrained to be equal is the total number of active row categories minus 1. To impose different equality constraints on sets of categories, use syntax. For example, use syntax to constrain categories 1 and 2 to be equal and categories 3 and 4 to be equal. Category is supplemental. Supplementary categories do not inuence the analysis but are represented in the space dened by the active categories. Supplementary categories play no role in dening the dimensions. The maximum number of supplementary row categories is the total number of row categories minus 2. Define Column Range in Correspondence Analysis You must dene a range for the column variable. The minimum and maximum values specied must be integers. Fractional data values are truncated in the analysis. A category value that is outside of the specied range is ignored in the analysis. 68 Chapter 5 Figure 5-3 Define Column Range dialog box All categories are initially unconstrained and active. You can constrain column categories to equal other column categories, or you can dene a column category as supplementary. Categories must be equal. Categories must have equal scores. Use equality constraints if the obtained order for the categories is undesirable or counterintuitive. The maximum number of column categories that can be constrained to be equal is the total number of active column categories minus 1. To impose different equality constraints on sets of categories, use syntax. For example, use syntax to constrain categories 1 and 2 to be equal and categories 3 and 4 to be equal. Category is supplemental. Supplementary categories do not inuence the analysis but are represented in the space dened by the active categories. Supplementary categories play no role in dening the dimensions. The maximum number of supplementary column categories is the total number of column categories minus 2. Correspondence Analysis Model The Model dialog box allows you to specify the number of dimensions, the distance measure, the standardization method, and the normalization method. 69 Correspondence Analysis Figure 5-4 Model dialog box Dimensions in solution. Specify the number of dimensions. In general, choose as few dimensions as needed to explain most of the variation. The maximum number of dimensions depends on the number of active categories used in the analysis and on the equality constraints. The maximum number of dimensions is the smaller of: The number of active row categories minus the number of row categories constrained to be equal, plus the number of constrained row category sets. The number of active column categories minus the number of column categories constrained to be equal, plus the number of constrained column category sets. 70 Chapter 5 Distance Measure. You can select the measure of distance among the rows and columns of the correspondence table. Choose one of the following alternatives: Chi-square. Use a weighted prole distance, where the weight is the mass of the rows or columns. This measure is required for standard correspondence analysis. Euclidean. Use the square root of the sum of squared differences between pairs of rows and pairs of columns. Standardization Method. Choose one of the following alternatives: Row and column means are removed. Both the rows and columns are centered. This method is required for standard correspondence analysis. Row means are removed. Only the rows are centered. Column means are removed. Only the columns are centered. Row totals are equalized and means are removed. Before centering the rows, the row margins are equalized. Column totals are equalized and means are removed. Before centering the columns, the column margins are equalized. Normalization Method. Choose one of the following alternatives: Symmetrical. For each dimension, the row scores are the weighted average of the column scores divided by the matching singular value, and the column scores are the weighted average of row scores divided by the matching singular value. Use this method if you want to examine the differences or similarities between the categories of the two variables. Principal. The distances between row points and column points are approximations of the distances in the correspondence table according to the selected distance measure. Use this method if you want to examine differences between categories of either or both variables instead of differences between the two variables. Row principal. The distances between row points are approximations of the distances in the correspondence table according to the selected distance measure. The row scores are the weighted average of the column scores. Use this method if you want to examine differences or similarities between categories of the row variable. Column principal. The distances between column points are approximations of the distances in the correspondence table according to the selected distance measure. The column scores are the weighted average of the row scores. Use this method if 71 Correspondence Analysis you want to examine differences or similarities between categories of the column variable. Custom. You must specify a value between 1 and 1. A value of 1 corresponds to column principal. A value of 1 corresponds to row principal. A value of 0 corresponds to symmetrical. All other values spread the inertia over both the row and column scores to varying degrees. This method is useful for making tailor-made biplots. Correspondence Analysis Statistics The Statistics dialog box allows you to specify the numerical output produced. Figure 5-5 Statistics dialog box Correspondence table. A crosstabulation of the input variables with row and column marginal totals. Overview of row points. For each row category, the scores, mass, inertia, contribution to the inertia of the dimension, and the contribution of the dimension to the inertia of the point. Overview of column points. For each column category, the scores, mass, inertia, contribution to the inertia of the dimension, and the contribution of the dimension to the inertia of the point. Row profiles. For each row category, the distribution across the categories of the column variable. 72 Chapter 5 Column profiles. For each column category, the distribution across the categories of the row variable. Permutations of the correspondence table. The correspondence table reorganized such that the rows and columns are in increasing order according to the scores on the rst dimension. Optionally, you can specify the maximum dimension number for which permuted tables will be produced. A permuted table for each dimension from 1 to the number specied is produced. Confidence Statistics for Row points. Includes standard deviation and correlations for all nonsupplementary row points. Confidence Statistics for Column points. Includes standard deviation and correlations for all nonsupplementary column points. Correspondence Analysis Plots The Plots dialog box allows you to specify which plots are produced. 73 Correspondence Analysis Figure 5-6 Plots dialog box Scatterplots. Produces a matrix of all pairwise plots of the dimensions. Available scatterplots include: Biplot. Produces a matrix of joint plots of the row and column points. If principal normalization is selected, the biplot is not available. Row points. Produces a matrix of plots of the row points. Column points. Produces a matrix of plots of the column points. Optionally, you can specify how many value label characters to use when labeling the points. This value must be a non-negative integer less than or equal to 20. 74 Chapter 5 Line Plots. Produces a plot for every dimension of the selected variable. Available line plots include: Transformed row categories. Produces a plot of the original row category values against their corresponding row scores. Transformed column categories. Produces a plot of the original column category values against their corresponding column scores. Optionally, you can specify how many value label characters to use when labeling the category axis. This value must be a non-negative integer less than or equal to 20. Plot Dimensions. Allows you to control the dimensions displayed in the output. Display all dimensions in the solution. All dimensions in the solution are displayed in a scatterplot matrix. Restrict the number of dimensions. The displayed dimensions are restricted to plotted pairs. If you restrict the dimensions, you must select the lowest and highest dimensions to be plotted. The lowest dimension can range from 1 to the number of dimensions in the solution minus 1, and is plotted against higher dimensions. The highest dimension value can range from 2 to the number of dimensions in the solution, and indicates the highest dimension to be used in plotting the dimension pairs. This specication applies to all requested multidimensional plots. CORRESPONDENCE Command Additional Features You can customize your correspondence analysis if you paste your selections into a syntax window and edit the resulting CORRESPONDENCE command syntax. The command syntax language also allows you to: Specify table data as input instead of using casewise data (using the TABLE = ALL subcommand). Specify the number of value-label characters used to label points for each type of scatterplot matrix or biplot matrix (with the PLOT subcommand). Specify the number of value-label characters used to label points for each type of line plot (with the PLOT subcommand). Write a matrix of row and column scores to a matrix data le (with the OUTFILE subcommand). 75 Correspondence Analysis Write a matrix of condence statistics (variances and covariances) for the singular values and the scores to a matrix data le (with the OUTFILE subcommand). Specify multiple sets of categories to be equal (with the EQUAL subcommand). See the Command Syntax Reference for complete syntax information. Chapter Multiple Correspondence Analysis 6 Multiple Correspondence Analysis quanties nominal (categorical) data by assigning numerical values to the cases (objects) and categories so that objects within the same category are close together and objects in different categories are far apart. Each object is as close as possible to the category points of categories that apply to the object. In this way, the categories divide the objects into homogeneous subgroups. Variables are considered homogeneous when they classify objects in the same categories into the same subgroups. Example. Multiple Correspondence Analysis could be used to graphically display the relationship between job category, minority classication, and gender. You might nd that minority classication and gender discriminate between people but that job category does not. You might also nd that the Latino and African-American categories are similar to each other. Statistics and plots. Object scores, discrimination measures, iteration history, correlations of original and transformed variables, category quantications, descriptive statistics, object points plots, biplots, category plots, joint category plots, transformation plots, and discrimination measures plots. Data. String variable values are always converted into positive integers by ascending alphanumeric order. User-dened missing values, system-missing values, and values less than 1 are considered missing; you can recode or add a constant to variables with values less than 1 to make them nonmissing. Assumptions. All variables have the multiple nominal scaling level. The data must contain at least three valid cases. The analysis is based on positive integer data. The discretization option will automatically categorize a fractional-valued variable by grouping its values into categories with a close-to-normal distribution and will 76 77 Multiple Correspondence Analysis automatically convert values of string variables into positive integers. You can specify other discretization schemes. Related procedures. For two variables, Multiple Correspondence Analysis is analogous to Correspondence Analysis. If you believe that variables possess ordinal or numerical properties, Categorical Principal Components Analysis should be used. If sets of variables are of interest, Nonlinear Canonical Correlation Analysis should be used. To Obtain a Multiple Correspondence Analysis E From the menus choose: Analyze Dimension Reduction Optimal Scaling... Figure 6-1 Optimal Scaling dialog box E Select All variables multiple nominal. E Select One set. E Click Define. 78 Chapter 6 Figure 6-2 Multiple Correspondence Analysis dialog box E Select at least two analysis variables and specify the number of dimensions in the solution. E Click OK. You may optionally specify supplementary variables, which are tted into the solution found, or labeling variables for the plots. Define Variable Weight in Multiple Correspondence Analysis You can set the weight for analysis variables. 79 Multiple Correspondence Analysis Figure 6-3 Define Variable Weight dialog box Variable weight. You can choose to dene a weight for each variable. The value specied must be a positive integer. The default value is 1. Multiple Correspondence Analysis Discretization The Discretization dialog box allows you to select a method of recoding your variables. Fractional-valued variables are grouped into seven categories (or into the number of distinct values of the variable if this number is less than seven) with an approximately normal distribution unless otherwise specied. String variables are always converted into positive integers by assigning category indicators according to ascending alphanumeric order. Discretization for string variables applies to these integers. Other variables are left alone by default. The discretized variables are then used in the analysis. 80 Chapter 6 Figure 6-4 Discretization dialog box Method. Choose between grouping, ranking, and multiplying. Grouping. Recode into a specied number of categories or recode by interval. Ranking. The variable is discretized by ranking the cases. Multiplying. The current values of the variable are standardized, multiplied by 10, rounded, and have a constant added so that the lowest discretized value is 1. Grouping. The following options are available when discretizing variables by grouping: Number of categories. Specify a number of categories and whether the values of the variable should follow an approximately normal or uniform distribution across those categories. Equal intervals. Variables are recoded into categories dened by these equally sized intervals. You must specify the length of the intervals. 81 Multiple Correspondence Analysis Multiple Correspondence Analysis Missing Values The Missing Values dialog box allows you to choose the strategy for handling missing values in analysis variables and supplementary variables. Figure 6-5 Missing Values dialog box Missing Value Strategy. Choose to exclude missing values (passive treatment), impute missing values (active treatment), or exclude objects with missing values (listwise deletion). Exclude missing values; for correlations impute after quantification. Objects with missing values on the selected variable do not contribute to the analysis for this variable. If all variables are given passive treatment, then objects with missing 82 Chapter 6 values on all variables are treated as supplementary. If correlations are specied in the Output dialog box, then (after analysis) missing values are imputed with the most frequent category, or mode, of the variable for the correlations of the original variables. For the correlations of the optimally scaled variables, you can choose the method of imputation. Select Mode to replace missing values with the mode of the optimally scaled variable. Select Extra category to replace missing values with the quantication of an extra category. This implies that objects with a missing value on this variable are considered to belong to the same (extra) category. Impute missing values. Objects with missing values on the selected variable have those values imputed. You can choose the method of imputation. Select Mode to replace missing values with the most frequent category. When there are multiple modes, the one with the smallest category indicator is used. Select Extra category to replace missing values with the same quantication of an extra category. This implies that objects with a missing value on this variable are considered to belong to the same (extra) category. Exclude objects with missing values on this variable. Objects with missing values on the selected variable are excluded from the analysis. This strategy is not available for supplementary variables. Multiple Correspondence Analysis Options The Options dialog box allows you to select the initial conguration, specify iteration and convergence criteria, select a normalization method, choose the method for labeling plots, and specify supplementary objects. 83 Multiple Correspondence Analysis Figure 6-6 Options dialog box Supplementary Objects. Specify the case number of the object (or the rst and last case numbers of a range of objects) that you want to make supplementary, and then click Add. Continue until you have specied all of your supplementary objects. If an object is specied as supplementary, then case weights are ignored for that object. Normalization Method. You can specify one of ve options for normalizing the object scores and the variables. Only one normalization method can be used in a given analysis. Variable Principal. This option optimizes the association between variables. The coordinates of the variables in the object space are the component loadings (correlations with principal components, such as dimensions and object scores). 84 Chapter 6 This is useful when you are interested primarily in the correlation between the variables. Object Principal. This option optimizes distances between objects. This is useful when you are interested primarily in differences or similarities between the objects. Symmetrical. Use this normalization option if you are interested primarily in the relation between objects and variables. Independent. Use this normalization option if you want to examine distances between objects and correlations between variables separately. Custom. You can specify any real value in the closed interval [1, 1]. A value of 1 is equal to the Object Principal method, a value of 0 is equal to the Symmetrical method, and a value of 1 is equal to the Variable Principal method. By specifying a value greater than 1 and less than 1, you can spread the eigenvalue over both objects and variables. This method is useful for making a tailor-made biplot or triplot. Criteria. You can specify the maximum number of iterations the procedure can go through in its computations. You can also select a convergence criterion value. The algorithm stops iterating if the difference in total t between the last two iterations is less than the convergence value or if the maximum number of iterations is reached. Label Plots By. Allows you to specify whether variables and value labels or variable names and values will be used in the plots. You can also specify a maximum length for labels. Plot Dimensions. Allows you to control the dimensions displayed in the output. Display all dimensions in the solution. All dimensions in the solution are displayed in a scatterplot matrix. Restrict the number of dimensions. The displayed dimensions are restricted to plotted pairs. If you restrict the dimensions, you must select the lowest and highest dimensions to be plotted. The lowest dimension can range from 1 to the number of dimensions in the solution minus 1 and is plotted against higher dimensions. The highest dimension value can range from 2 to the number of dimensions in the solution and indicates the highest dimension to be used in plotting the dimension pairs. This specication applies to all requested multidimensional plots. Configuration. You can read data from a le containing the coordinates of a conguration. The rst variable in the le should contain the coordinates for the rst dimension, the second variable should contain the coordinates for the second dimension, and so on. 85 Multiple Correspondence Analysis Initial. The conguration in the le specied will be used as the starting point of the analysis. Fixed. The conguration in the le specied will be used to t in the variables. The variables that are tted in must be selected as analysis variables, but, because the conguration is xed, they are treated as supplementary variables (so they do not need to be selected as supplementary variables). Multiple Correspondence Analysis Output The Output dialog box allows you to produce tables for object scores, discrimination measures, iteration history, correlations of original and transformed variables, category quantications for selected variables, and descriptive statistics for selected variables. 86 Chapter 6 Figure 6-7 Output dialog box Object scores. Displays the object scores, including mass, inertia, and contributions, and has the following options: Include Categories Of. Displays the category indicators of the analysis variables selected. Label Object Scores By. From the list of variables specied as labeling variables, you can select one to label the objects. Discrimination measures. Displays the discrimination measures per variable and per dimension. 87 Multiple Correspondence Analysis Iteration history. For each iteration, the variance accounted for, loss, and increase in variance accounted for are shown. Correlations of original variables. Shows the correlation matrix of the original variables and the eigenvalues of that matrix. Correlations of transformed variables. Shows the correlation matrix of the transformed (optimally scaled) variables and the eigenvalues of that matrix. Category Quantifications and Contributions. Gives the category quantications (coordinates), including mass, inertia, and contributions, for each dimension of the variable(s) selected. Note: the coordinates and contributions (including the mass and inertia) are displayed in separate layers of the pivot table output, with the coordinates shown by default. To display the contributions, activate (double-click) on the table and select Contributions from the Layer dropdown list. Descriptive Statistics. Displays frequencies, number of missing values, and mode of the variable(s) selected. Multiple Correspondence Analysis Save The Save dialog box allows you to save discretized data, object scores, and transformed values to an external SPSS Statistics data le or dataset in the current session. You can also save transformed values and object scores to the active dataset. Datasets are available during the current session but are not available in subsequent sessions unless you explicitly save them as data les. Dataset names must adhere to variable naming rules. Filenames or dataset names must be different for each type of data saved. If you save object scores or transformed values to the active dataset, you can specify the number of multiple nominal dimensions. 88 Chapter 6 Figure 6-8 Save dialog box Multiple Correspondence Analysis Object Plots The Object Plots dialog box allows you to specify the types of plots desired and the variables to be plotted 89 Multiple Correspondence Analysis Figure 6-9 Object Plots dialog box Object points. A plot of the object points is displayed. Objects and centroids (biplot). The object points are plotted with the variable centroids. Biplot Variables. You can choose to use all variables for the biplots or select a subset. Label Objects. You can choose to have objects labeled with the categories of selected variables (you may choose category indicator values or value labels in the Options dialog box) or with their case numbers. One plot is produced per variable if Variable is selected. 90 Chapter 6 Multiple Correspondence Analysis Variable Plots The Variable Plots dialog box allows you to specify the types of plots desired and the variables to be plotted. Figure 6-10 Variable Plots dialog box Category Plots. For each variable selected, a plot of the centroid coordinates is plotted. Categories are in the centroids of the objects in the particular categories. Joint Category Plots. This is a single plot of the centroid coordinates of each selected variable. 91 Multiple Correspondence Analysis Transformation Plots. Displays a plot of the optimal category quantications versus the category indicators. You can specify the number of dimensions desired; one plot will be generated for each dimension. You can also choose to display residual plots for each variable selected. Discrimination Measures. Produces a single plot of the discrimination measures for the selected variables. MULTIPLE CORRESPONDENCE Command Additional Features You can customize your Multiple Correspondence Analysis if you paste your selections into a syntax window and edit the resulting MULTIPLE CORRESPONDENCE command syntax. The command syntax language also allows you to: Specify rootnames for the transformed variables, object scores, and approximations when saving them to the active dataset (with the SAVE subcommand). Specify a maximum length for labels for each plot separately (with the PLOT subcommand). Specify a separate variable list for residual plots (with the PLOT subcommand). See the Command Syntax Reference for complete syntax information. Chapter Multidimensional Scaling (PROXSCAL) 7 Multidimensional scaling attempts to nd the structure in a set of proximity measures between objects. This process is accomplished by assigning observations to specic locations in a conceptual low-dimensional space such that the distances between points in the space match the given (dis)similarities as closely as possible. The result is a least-squares representation of the objects in that low-dimensional space, which, in many cases, will help you to further understand your data. Example. Multidimensional scaling can be very useful in determining perceptual relationships. For example, when considering your product image, you can conduct a survey to obtain a dataset that describes the perceived similarity (or proximity) of your product to those of your competitors. Using these proximities and independent variables (such as price), you can try to determine which variables are important to how people view these products, and you can adjust your image accordingly. Statistics and plots. Iteration history, stress measures, stress decomposition, coordinates of the common space, object distances within the nal conguration, individual space weights, individual spaces, transformed proximities, transformed independent variables, stress plots, common space scatterplots, individual space weight scatterplots, individual spaces scatterplots, transformation plots, Shepard residual plots, and independent variables transformation plots. Data. Data can be supplied in the form of proximity matrices or variables that are converted into proximity matrices. The matrices can be formatted in columns or across columns. The proximities can be treated on the ratio, interval, ordinal, or spline scaling levels. 92 93 Multidimensional Scaling (PROXSCAL) Assumptions. At least three variables must be specied. The number of dimensions cannot exceed the number of objects minus one. Dimensionality reduction is omitted if combined with multiple random starts. If only one source is specied, all models are equivalent to the identity model; therefore, the analysis defaults to the identity model. Related procedures. Scaling all variables at the numerical level corresponds to standard multidimensional scaling analysis. To Obtain a Multidimensional Scaling E From the menus choose: Analyze Scale Multidimensional Scaling (PROXSCAL)... This opens the Data Format dialog box. Figure 7-1 Data Format dialog box 94 Chapter 7 E Specify the format of your data: Data Format. Specify whether your data consist of proximity measures or you want to create proximities from the data. Number of Sources. If your data are proximities, specify whether you have a single source or multiple sources of proximity measures. One Source. If there is one source of proximities, specify whether your dataset is formatted with the proximities in a matrix across the columns or in a single column with two separate variables to identify the row and column of each proximity. The proximities are in a matrix across columns. The proximity matrix is spread across a number of columns equal to the number of objects. This leads to the Proximities in Matrices across Columns dialog box. The proximities are in a single column. The proximity matrix is collapsed into a single column, or variable. Two additional variables, identifying the row and column for each cell, are necessary. This leads to the Proximities in One Column dialog box. Multiple Sources. If there are multiple sources of proximities, specify whether the dataset is formatted with the proximities in stacked matrices across columns, in multiple columns with one source per column, or in a single column. The proximities are in stacked matrices across columns. The proximity matrices are spread across a number of columns equal to the number of objects and are stacked above one another across a number of rows equal to the number of objects times the number of sources. This leads to the Proximities in Matrices across Columns dialog box. The proximities are in columns, one source per column. The proximity matrices are collapsed into multiple columns, or variables. Two additional variables, identifying the row and column for each cell, are necessary. This leads to the Proximities in Columns dialog box. The proximites are stacked in a single column. The proximity matrices are collapsed into a single column, or variable. Three additional variables, identifying the row, column, and source for each cell, are necessary. This leads to the Proximities in One Column dialog box. E Click Define. 95 Multidimensional Scaling (PROXSCAL) Proximities in Matrices across Columns If you select the proximities in matrices data model for either one source or multiple sources in the Data Format dialog box, the main dialog box will appear as follows: Figure 7-2 Proximities in Matrices across Columns dialog box E Select three or more proximities variables. (Be sure that the order of the variables in the list matches the order of the columns of the proximities.) E Optionally, select a number of weights variables equal to the number of proximities variables. (Be sure that the order of the weights matches the order of the proximities that they weight.) E Optionally, if there are multiple sources, select a sources variable. (The number of cases in each proximities variable should equal the number of proximities variables times the number of sources.) 96 Chapter 7 Additionally, you can dene a model for the multidimensional scaling, place restrictions on the common space, set convergence criteria, specify the initial conguration to be used, and choose plots and output. Proximities in Columns If you select the multiple columns model for multiple sources in the Data Format dialog box, the main dialog box will appear as follows: Figure 7-3 Proximities in Columns dialog box E Select two or more proximities variables. (Each variable is assumed to be a matrix of proximities from a separate source.) E Select a rows variable to dene the row locations for the proximities in each proximities variable. E Select a columns variable to dene the column locations for the proximities in each proximities variable. (Cells of the proximity matrix that are not given a row/column designation are treated as missing.) 97 Multidimensional Scaling (PROXSCAL) E Optionally, select a number of weights variables equal to the number of proximities variables. Additionally, you can dene a model for the multidimensional scaling, place restrictions on the common space, set convergence criteria, specify the initial conguration to be used, and choose plots and output. Proximities in One Column If you select the one column model for either one source or multiple sources in the Data Format dialog box, the main dialog box will appear as follows: Figure 7-4 Proximities in One Column dialog box E Select a proximities variable. (t is assumed to be one or more matrices of proximities.) E Select a rows variable to dene the row locations for the proximities in the proximities variable. E Select a columns variable to dene the column locations for the proximities in the proximities variable. E If there are multiple sources, select a sources variable. (For each source, cells of the proximity matrix that are not given a row/column designation are treated as missing.) E Optionally, select a weights variable. 98 Chapter 7 Additionally, you can dene a model for the multidimensional scaling, place restrictions on the common space, set convergence criteria, specify the initial conguration to be used, and choose plots and output. Create Proximities from Data If you choose to create proximities from the data in the Data Format dialog box, the main dialog box will appear as follows: Figure 7-5 Create Proximities from Data dialog box E If you create distances between variables (see the Create Measure from Data dialog box), select at least three variables. These variables will be used to create the proximity matrix (or matrices, if there are multiple sources). If you create distances between cases, only one variable is needed. E If there are multiple sources, select a sources variable. E Optionally, choose a measure for creating proximities. Additionally, you can dene a model for the multidimensional scaling, place restrictions on the common space, set convergence criteria, specify the initial conguration to be used, and choose plots and output. 99 Multidimensional Scaling (PROXSCAL) Create Measure from Data Figure 7-6 Create Measure from Data dialog box Multidimensional scaling uses dissimilarity data to create a scaling solution. If your data are multivariate data (values of measured variables), you must create dissimilarity data in order to compute a multidimensional scaling solution. You can specify the details of creating dissimilarity measures from your data. Measure. Allows you to specify the dissimilarity measure for your analysis. Select one alternative from the Measure group corresponding to your type of data, and then select one of the measures from the drop-down list corresponding to that type of measure. Available alternatives are: Interval. Euclidean distance, Squared Euclidean distance, Chebychev, Block, Minkowski, or Customized. Counts. Chi-square measure or Phi-square measure. Binary. Euclidean distance, Squared Euclidean distance, Size difference, Pattern difference, Variance, or Lance and Williams. 100 Chapter 7 Create Distance Matrix. Allows you to choose the unit of analysis. Alternatives are Between variables or Between cases. Transform Values. In certain cases, such as when variables are measured on very different scales, you want to standardize values before computing proximities (not applicable to binary data). Select a standardization method from the Standardize drop-down list (if no standardization is required, select None). Define a Multidimensional Scaling Model The Model dialog box allows you to specify a scaling model, its minimum and maximum number of dimensions, the structure of the proximity matrix, the transformation to use on the proximities, and whether proximities are transformed within each source separately or unconditionally on the source. Figure 7-7 Model dialog box 101 Multidimensional Scaling (PROXSCAL) Scaling Model. Choose from the following alternatives: Identity. All sources have the same conguration. Weighted Euclidean. This model is an individual differences model. Each source has an individual space in which every dimension of the common space is weighted differentially. Generalized Euclidean. This model is an individual differences model. Each source has an individual space that is equal to a rotation of the common space, followed by a differential weighting of the dimensions. Reduced rank. This model is a generalized Euclidean model for which you can specify the rank of the individual space. You must specify a rank that is greater than or equal to 1 and less than the maximum number of dimensions. Shape. Specify whether the proximities should be taken from the lower-triangular part or the upper-triangular part of the proximity matrix. You can specify that the full matrix be used, in which case the weighted sum of the upper-triangular part and the lower-triangular part will be analyzed. In any case, the complete matrix should be specied, including the diagonal, though only the specied parts will be used. Proximities. Specify whether your proximity matrix contains measures of similarity or dissimilarity. Proximity Transformations. Choose from the following alternatives: Ratio. The transformed proximities are proportional to the original proximities. This is allowed only for positively valued proximities. Interval. The transformed proximities are proportional to the original proximities, plus an intercept term. The intercept assures all transformed proximities to be positive. Ordinal. The transformed proximities have the same order as the original proximities. You specify whether tied proximities should be kept tied or allowed to become untied. Spline. The transformed proximities are a smooth nondecreasing piecewise polynomial transformation of the original proximities. You specify the degree of the polynomial and the number of interior knots. Apply Transformations. Specify whether only proximities within each source are compared with each other or whether the comparisons are unconditional on the source. 102 Chapter 7 Dimensions. By default, a solution is computed in two dimensions (Minimum = 2, Maximum = 2). You choose an integer minimum and maximum from 1 to the number of objects minus 1 (as long as the minimum is less than or equal to the maximum). The procedure computes a solution in the maximum dimensions and then reduces the dimensionality in steps until the lowest is reached. Multidimensional Scaling Restrictions The Restrictions dialog box allows you to place restrictions on the common space. Figure 7-8 Restrictions dialog box Restrictions on Common Space. Specify the type of restriction desired. No restrictions. No restrictions are placed on the common space. 103 Multidimensional Scaling (PROXSCAL) Some coordinates fixed. The rst variable selected contains the coordinates of the objects on the rst dimension, the second variable corresponds to coordinates on the second dimension, and so on. A missing value indicates that a coordinate on a dimension is free. The number of variables selected must equal the maximum number of dimensions requested. Linear combination of independent variables. The common space is restricted to be a linear combination of the variables selected. Restriction Variables. Select the variables that dene the restrictions on the common space. If you specied a linear combination, you specify an interval, nominal, ordinal, or spline transformation for the restriction variables. In either case, the number of cases for each variable must equal the number of objects. Multidimensional Scaling Options The Options dialog box allows you to select the initial conguration style, specify iteration and convergence criteria, and select standard or relaxed updates. 104 Chapter 7 Figure 7-9 Options dialog box Initial Configuration. Choose one of the following alternatives: Simplex. Objects are placed at the same distance from each other in the maximum dimension. One iteration is taken to improve this high-dimensional conguration, followed by a dimension reduction operation to obtain an initial conguration that has the maximum number of dimensions that you specied in the Model dialog box. Torgerson. A classical scaling solution is used as the initial conguration. Single random start. A conguration is chosen at random. 105 Multidimensional Scaling (PROXSCAL) Multiple random starts. Several congurations are chosen at random, and the conguration with the lowest normalized raw stress is used as the initial conguration. Custom. You select variables that contain the coordinates of your own initial conguration. The number of variables selected should equal the maximum number of dimensions specied, with the rst variable corresponding to coordinates on dimension 1, the second variable corresponding to coordinates on dimension 2, and so on. The number of cases in each variable should equal the number of objects. Iteration Criteria. Specify the iteration criteria values. Stress convergence. The algorithm will stop iterating when the difference in consecutive normalized raw stress values is less than the number that is specied here, which must lie between 0.0 and 1.0. Minimum stress. The algorithm will stop when the normalized raw stress falls below the number that is specied here, which must lie between 0.0 and 1.0. Maximum iterations. The algorithm will perform the number of specied iterations, unless one of the above criteria is satised rst. Use relaxed updates. Relaxed updates will speed up the algorithm; these updates cannot be used with models other than the identity model or used with restrictions. Multidimensional Scaling Plots, Version 1 The Plots dialog box allows you to specify which plots will be produced. If you have the Proximities in Columns data format, the following Plots dialog box is displayed. For Individual space weights, Original vs. transformed proximities, and Transformed proximities vs. distances plots, you specify the sources for which the plots should be produced. The list of available sources is the list of proximities variables in the main dialog box. 106 Chapter 7 Figure 7-10 Plots dialog box, version 1 Stress. A plot is produced of normalized raw stress versus dimensions. This plot is produced only if the maximum number of dimensions is larger than the minimum number of dimensions. Common space. A scatterplot matrix of coordinates of the common space is displayed. Individual spaces. For each source, the coordinates of the individual spaces are displayed in scatterplot matrices. This is possible only if one of the individual differences models is specied in the Model dialog box. Individual space weights. A scatterplot is produced of the individual space weights. This is possible only if one of the individual differences models is specied in the Model dialog box. For the weighted Euclidean model, the weights are printed in plots, with one dimension on each axis. For the generalized Euclidean model, one plot is produced per dimension, indicating both rotation and weighting of that dimension. The reduced rank model produces the same plot as the generalized Euclidean model but reduces the number of dimensions for the individual spaces. 107 Multidimensional Scaling (PROXSCAL) Original vs. transformed proximities. Plots are produced of the original proximities versus the transformed proximities. Transformed proximities vs. distances. The transformed proximities versus the distances are plotted. Transformed independent variables. Transformation plots are produced for the independent variables. Variable and dimension correlations. A plot of correlations between the independent variables and the dimensions of the common space is displayed. Multidimensional Scaling Plots, Version 2 The Plots dialog box allows you to specify which plots will be produced. If your data format is anything other than Proximities in Columns, the following Plots dialog box is displayed. For Individual space weights, Original vs. transformed proximities, and Transformed proximities vs. distances plots, you specify the sources for which the plots should be produced. The source numbers entered must be values of the sources variable that is specied in the main dialog box and must range from 1 to the number of sources. Figure 7-11 Plots dialog box, version 2 108 Chapter 7 Multidimensional Scaling Output The Output dialog box allows you to control the amount of displayed output and save some of it to separate les. Figure 7-12 Output dialog box Display. Select one or more of the following items for display: Common space coordinates. Displays the coordinates of the common space. Individual space coordinates. The coordinates of the individual spaces are displayed only if the model is not the identity model. Individual space weights. Displays the individual space weights only if one of the individual differences models is specied. Depending on the model, the space weights are decomposed in rotation weights and dimension weights, which are also displayed. Distances. Displays the distances between the objects in the conguration. Transformed proximities. Displays the transformed proximities between the objects in the conguration. 109 Multidimensional Scaling (PROXSCAL) Input data. Includes the original proximities and, if present, the data weights, the initial conguration, and the xed coordinates of the independent variables. Stress for random starts. Displays the random number seed and normalized raw stress value of each random start. Iteration history. Displays the history of iterations of the main algorithm. Multiple stress measures. Displays different stress values. The table contains values for normalized raw stress, Stress-I, Stress-II, S-Stress, Dispersion Accounted For (DAF), and Tuckers Coefcient of Congruence. Stress decomposition. Displays an objects and sources decomposition of nal normalized raw stress, including the average per object and the average per source. Transformed independent variables. If a linear combination restriction was selected, the transformed independent variables and the corresponding regression weights are displayed. Variable and dimension correlations. If a linear combination restriction was selected, the correlations between the independent variables and the dimensions of the common space are displayed. Save to New File. You can save the common space coordinates, individual space weights, distances, transformed proximities, and transformed independent variables to separate SPSS Statistics data les. PROXSCAL Command Additional Features You can customize your multidimensional scaling of proximities analysis if you paste your selections into a syntax window and edit the resulting PROXSCAL command syntax. The command syntax language also allows you to: Specify separate variable lists for transformations and residuals plots (with the PLOT subcommand). Specify separate source lists for individual space weights, transformations, and residuals plots (with the PLOT subcommand). Specify a subset of the independent variables transformation plots to be displayed (with the PLOT subcommand). See the Command Syntax Reference for complete syntax information. Chapter Multidimensional Unfolding (PREFSCAL) 8 The Multidimensional Unfolding procedure attempts to nd a common quantitative scale that allows you to visually examine the relationships between two sets of objects. Examples. You have asked 21 individuals to rank 15 breakfast items in order of preference, 1 to 15. Using Multidimensional Unfolding, you can determine that the individuals discriminate between breakfast items in two primary ways: between soft and hard breads, and between fattening and non-fattening items. Alternatively, you have asked a group of drivers to rate 26 models of cars on 10 attributes on a 6-point scale ranging from 1=not true at all to 6=very true. Averaged over individuals, the values are taken as similarities. Using Multidimensional Unfolding, you nd clusterings of similar models and the attributes with which they are most closely associated. Statistics and plots. The Multidimensional Unfolding procedure can produce an iteration history, stress measures, stress decomposition, coordinates of the common space, object distances within the nal conguration, individual space weights, individual spaces, transformed proximities, stress plots, common space scatterplots, individual space weight scatterplots, individual spaces scatterplots, transformation plots, and Shepard residual plots. Data. Data are supplied in the form of rectangular proximity matrices. Each column is considered a separate column object. Each row of a proximity matrix is considered a separate row object. When there are multiple sources of proximities, the matrices are stacked. 110 111 Multidimensional Unfolding (PREFSCAL) Assumptions. At least two variables must be specied. The number of dimensions in the solution may not exceed the number of objects minus one. If only one source is specied, all models are equivalent to the identity model; therefore, the analysis defaults to the identity model. To Obtain a Multidimensional Unfolding E From the menus choose: Analyze Scale Multidimensional Unfolding (PREFSCAL)... Figure 8-1 Multidimensional Unfolding main dialog box E Select two or more variables that identify the columns in the rectangular proximity matrix. Each variable represents a separate column object. E Optionally, select a number of weights variables equal to the number of column object variables. The order of the weights variables should match the order of the column objects they weight. E Optionally, select a rows variable. The values (or value labels) of this variable are used to label row objects in the output. 112 Chapter 8 E If there are multiple sources, optionally select a sources variable. The number of cases in the data le should equal the number of row objects times the number of sources. Additionally, you can dene a model for the multidimensional unfolding, place restrictions on the common space, set convergence criteria, specify the initial conguration to be used, and choose plots and output. Define a Multidimensional Unfolding Model The Model dialog box allows you to specify a scaling model, its minimum and maximum number of dimensions, the structure of the proximity matrix, the transformation to use on the proximities, and whether proximities are transformed conditonal upon the row, conditional upon the source, or unconditionally on the source. Figure 8-2 Model dialog box 113 Multidimensional Unfolding (PREFSCAL) Scaling Model. Choose from the following alternatives: Identity. All sources have the same conguration. Weighted Euclidean. This model is an individual differences model. Each source has an individual space in which every dimension of the common space is weighted differentially. Generalized Euclidean. This model is an individual differences model. Each source has an individual space that is equal to a rotation of the common space, followed by a differential weighting of the dimensions. Proximities. Specify whether your proximity matrix contains measures of similarity or dissimilarity. Dimensions. By default, a solution is computed in two dimensions (Minimum = 2, Maximum = 2). You can choose an integer minimum and maximum from 1 to the number of objects minus 1 as long as the minimum is less than or equal to the maximum. The procedure computes a solution in the maximum dimensionality and then reduces the dimensionality in steps until the lowest is reached. Proximity Transformations. Choose from the following alternatives: None. The proximities are not transformed. You can optionally select Include intercept, in which case the proximities can be shifted by a constant term. Linear. The transformed proximities are proportional to the original proximities; that is, the transformation function estimates a slope and the intercept is xed at 0. This is also called a ratio transformation. You can optionally select Include intercept, in which case the proximities can also be shifted by a constant term. This is also called an interval transformation. Spline. The transformed proximities are a smooth nondecreasing piecewise polynomial transformation of the original proximities. You can specify the degree of the polynomial and the number of interior knots. You can optionally select Include intercept, in which case the proximities can also be shifted by a constant term. Smooth. The transformed proximities have the same order as the original proximities, including a restriction that takes the differences between subsequent values into account. The result is a smooth ordinal transformation. You can specify whether tied proximities should be kept tied or allowed to become untied. Ordinal. The transformed proximities have the same order as the original proximities. You can specify whether tied proximities should be kept tied or allowed to become untied. 114 Chapter 8 Apply Transformations. Specify whether only proximities within each row are compared with each other, or only proximities within each source are compared with each other, or the comparisons are unconditional on the row or source; that is, whether the transformations are performed per row, per source, or over all proximities at once. Multidimensional Unfolding Restrictions The Restrictions dialog box allows you to place restrictions on the common space. Figure 8-3 Restrictions dialog box 115 Multidimensional Unfolding (PREFSCAL) Restrictions on Common Space. You can choose to x the coordinates of row and/or column objects in the common space. Row/Column Restriction Variables. Choose the le containing the restrictions and select the variables that dene the restrictions on the common space. The rst variable selected contains the coordinates of the objects on the rst dimension, the second variable corresponds to coordinates on the second dimension, and so on. A missing value indicates that a coordinate on a dimension is free. The number of variables selected must equal the maximum number of dimensions requested. The number of cases for each variable must equal the number of objects. Multidimensional Unfolding Options The Options dialog box allows you to select the initial conguration style, specify iteration and convergence criteria, and set the penalty term for stress. 116 Chapter 8 Figure 8-4 Options dialog box Initial Configuration. Choose one of the following alternatives: Classical. The rectangular proximity matrix is used to supplement the intra-blocks (values between rows and between columns) of the complete symmetrical MDS matrix. Once the complete matrix is formed, a classical scaling solution is used as the initial conguration. The intra-blocks can be lled via imputation using the triangle inequality or Spearman distances. 117 Multidimensional Unfolding (PREFSCAL) Ross-Cliff. The Ross-Cliff start uses the results of a singular value decomposition on the double centered and squared proximity matrix as the initial values for the row and column objects. Correspondence. The correspondence start uses the results of a correspondence analysis on the reversed data (similarities instead of dissimilarities), with symmetric normalization of row and column scores. Centroids. The procedure starts by positioning the row objects in the conguration using an eigenvalue decomposition. Then the column objects are positioned at the centroid of the specied choices. For the number of choices, specify a positive integer between 1 and the number of proximities variables. Multiple random starts. Solutions are computed for several initial congurations chosen at random, and the one with the lowest penalized stress is shown as the best solution. Custom. You can select variables that contain the coordinates of your own initial conguration. The number of variables selected should equal the maximum number of dimensions specied, with the rst variable corresponding to coordinates on dimension 1, the second variable corresponding to coordinates on dimension 2, and so on. The number of cases in each variable should equal the combined number of row and column objects. The row and column coordinates should be stacked, with the column coordinates following the row coordinates. Iteration Criteria. Specify the iteration criteria values. Stress convergence. The algorithm will stop iterating when the relative difference in consecutive penalized stress values is less than the number specied here, which must be non-negative. Minimum stress. The algorithm will stop when the penalized stress falls below the number specied here, which must be non-negative. Maximum iterations. The algorithm will perform the number of iterations specied here unless one of the above criteria is satised rst. Penalty Term. The algorithm attempts to minimize penalized stress, a goodness-of-t measure equal to the product of Kruskals Stress-I and a penalty term based on the coefcient of variation of the transformed proximities. These controls allow you to set the strength and range of the penalty term. 118 Chapter 8 Strength. The smaller the value of the strength parameter, the stronger the penalty. Specify a value between 0.0 and 1.0. Range. This parameter sets the moment at which the penalty becomes active. If set to 0.0, the penalty is inactive. Increasing the value causes the algorithm to search for a solution with greater variation among the transformed proximities. Specify a non-negative value. Multidimensional Unfolding Plots The Plots dialog box allows you to specify which plots will be produced. 119 Multidimensional Unfolding (PREFSCAL) Figure 8-5 Plots dialog box Plots. The following plots are available: Multiple starts. Displays a stacked histogram of penalized stress displaying both stress and penalty. 120 Chapter 8 Initial common space. Displays a scatterplot matrix of the coordinates of the initial common space. Stress per dimension. Produces a lineplot of penalized stress versus dimensionality. This plot is produced only if the maximum number of dimensions is larger than the minimum number of dimensions. Final common space. A scatterplot matrix of coordinates of the common space is displayed. Space weights. A scatterplot is produced of the individual space weights. This is possible only if one of the individual differences models is specied in the Model dialog box. For the weighted Euclidean model, the weights for all sources are displayed in a plot, with one dimension on each axis. For the generalized Euclidean model, one plot is produced per dimension, indicating both rotation and weighting of that dimension for each source. Individual spaces. A scatterplot matrix of coordinates of the individual space of each source is displayed. This is possible only if one of the individual differences models is specied in the Model dialog box. Transformation plots. A scatterplot is produced of the original proximities versus the transformed proximities. Depending on how transformations are applied, a separate color is assigned to each row or source. An unconditional transformation produces a single color. Shepard plots. The original proximities versus both transformed proximities and distances. The distances are indicated by points, and the transformed proximities are indicated by a line. Depending on how transformations are applied, a separate line is produced for each row or source. An unconditional transformation produces one line. Scatterplot of fit. A scatterplot of the transformed proximities versus the distances is displayed. A separate color is assigned to each source if multiple sources are specied. Residuals plots. A scatterplot of the transformed proximities versus the residuals (transformed proximities minus distances) is displayed. A separate color is assigned to each source if multiple sources are specied. Row Object Styles. These give you further control of the display of row objects in plots. The values of the optional colors variable are used to cycle through all colors. The values of the optional markers variable are used to cycle through all possible markers. 121 Multidimensional Unfolding (PREFSCAL) Source Plots. For Individual spaces, Scatterplot of fit, and Residuals plotsand if transformations are applied by source, for Transformation plots and Shepard plotsyou can specify the sources for which the plots should be produced. The source numbers entered must be values of the sources variable specied in the main dialog box and range from 1 to the number of sources. Row Plots. If transformations are applied by row, for Transformation plots and Shepard plots, you can specify the row for which the plots should be produced. The row numbers entered must range from 1 to the number of rows. Multidimensional Unfolding Output The Output dialog box allows you to control the amount of displayed output and save some of it to separate les. Figure 8-6 Output dialog box Display. Select one or more of the following for display: Input data. Includes the original proximities and, if present, the data weights, the initial conguration, and the xed coordinates. 122 Chapter 8 Multiple starts. Displays the random number seed and penalized stress value of each random start. Initial data. Displays the coordinates of the initial common space. Iteration history. Displays the history of iterations of the main algorithm. Fit measures. Displays different measures. The table contains several goodness-of-t, badness-of-t, correlation, variation, and nondegeneracy measures. Stress decomposition. Displays an objects, rows, and sources decomposition of penalized stress, including row, column, and source means and standard deviations. Transformed proximities. Displays the transformed proximities. Final common space. Displays the coordinates of the common space. Space weights. Displays the individual space weights. This option is available only if one of the individual differences models is specied. Depending on the model, the space weights are decomposed in rotation weights and dimension weights, which are also displayed. Individual spaces. The coordinates of the individual spaces are displayed. This option is available only if one of the individual differences models is specied. Fitted distances. Displays the distances between the objects in the conguration. Save to New File. You can save the common space coordinates, individual space weights, distances, and transformed proximities to separate SPSS Statistics data les. PREFSCAL Command Additional Features You can customize your Multidimensional Unfolding of proximities analysis if you paste your selections into a syntax window and edit the resulting PREFSCAL command syntax. The command syntax language also allows you to: Specify multiple source lists for Individual spaces, Scatterplots of t, and Residuals plotsand in the case of matrix conditional transformations, for Transformation plots and Shepard plotswhen multiple sources are available (with the PLOT subcommand). Specify multiple row lists for Transformation plots and Shepard plots in the case of row conditional transformations (with the PLOT subcommand). 123 Multidimensional Unfolding (PREFSCAL) Specify a number of rows instead of a row ID variable (with the INPUT subcommand). Specify a number of sources instead of a source ID variable (with the INPUT subcommand). See the Command Syntax Reference for complete syntax information. Part II: Examples Chapter Categorical Regression 9 The goal of categorical regression with optimal scaling is to describe the relationship between a response variable and a set of predictors. By quantifying this relationship, values of the response can be predicted for any combination of predictors. In this chapter, two examples serve to illustrate the analyses involved in optimal scaling regression. The rst example uses a small data set to illustrate the basic concepts. The second example uses a much larger set of variables and observations in a practical example. Example: Carpet Cleaner Data In a popular example (Green and Wind, 1973), a company interested in marketing a new carpet cleaner wants to examine the inuence of ve factors on consumer preferencepackage design, brand name, price, a Good Housekeeping seal, and a money-back guarantee. There are three factor levels for package design, each one differing in the location of the applicator brush; three brand names (K2R, Glory, and Bissell); three price levels; and two levels (either no or yes) for each of the last two factors. The following table displays the variables used in the carpet-cleaner study, with their variable labels and values. Table 9-1 Explanatory variables in the carpet-cleaner study Variable name package brand price seal money Variable label Package design Brand name Price Good Housekeeping seal Money-back guarantee 125 Value label A*, B*, C* K2R, Glory, Bissell $1.19, $1.39, $1.59 No, yes No, yes 126 Chapter 9 Ten consumers rank 22 proles dened by these factors. The variable Preference contains the rank of the average rankings for each prole. Low rankings correspond to high preference. This variable reects an overall measure of preference for each prole. Using categorical regression, you will explore how the ve factors are related to preference. This data set can be found in carpet.sav. For more information, see Sample Files in Appendix A on p. 406. A Standard Linear Regression Analysis E To produce standard linear regression output, from the menus choose: Analyze Regression Linear... Figure 9-1 Linear Regression dialog box E Select Preference as the dependent variable. E Select Package design through Money-back guarantee as independent variables. 127 Categorical Regression E Click Plots. Figure 9-2 Plots dialog box E Select *ZRESID as the y-axis variable. E Select *ZPRED as the x-axis variable. E Click Continue. E Click Save in the Linear Regression dialog box. 128 Chapter 9 Figure 9-3 Save dialog box E Select Standardized in the Residuals group. E Click Continue. E Click OK in the Linear Regression dialog box. 129 Categorical Regression Model Summary Figure 9-4 Model summary for standard linear regression The standard approach for describing the relationships in this problem is linear regression. The most common measure of how well a regression model ts the data is R2. This statistic represents how much of the variance in the response is explained by the weighted combination of predictors. The closer R2 is to 1, the better the model ts. Regressing Preference on the ve predictors results in an R2 of 0.707, indicating that approximately 71% of the variance in the preference rankings is explained by the predictor variables in the linear regression. Coefficients The standardized coefcients are shown in the table. The sign of the coefcient indicates whether the predicted response increases or decreases when the predictor increases, all other predictors being constant. For categorical data, the category coding determines the meaning of an increase in a predictor. For instance, an increase in Money-back guarantee, Package design, or Good Housekeeping seal will result in a decrease in predicted preference ranking. Money-back guarantee is coded 1 for no money-back guarantee and 2 for money-back guarantee. An increase in Money-back guarantee corresponds to the addition of a money-back guarantee. Thus, adding a money-back guarantee reduces the predicted preference ranking, which corresponds to an increased predicted preference. 130 Chapter 9 Figure 9-5 Regression coefficients The value of the coefcient reects the amount of change in the predicted preference ranking. Using standardized coefcients, interpretations are based on the standard deviations of the variables. Each coefcient indicates the number of standard deviations that the predicted response changes for a one standard deviation change in a predictor, all other predictors remaining constant. For example, a one standard deviation change in Brand name yields an increase in predicted preference of 0.056 standard deviations. The standard deviation of Preference is 6.44, so Preference . Changes in Package design yield the greatest increases by changes in predicted preference. 131 Categorical Regression Residual Scatterplots Figure 9-6 Residuals versus predicted values The standardized residuals are plotted against the standardized predicted values. No patterns should be present if the model ts well. Here you see a U-shape in which both low and high standardized predicted values have positive residuals. Standardized predicted values near 0 tend to have negative residuals. E To produce a scatterplot of the residuals by the predictor Package design, from the menus choose: Graphs Chart Builder... 132 Chapter 9 Figure 9-7 Chart Builder E Select the Scatter/Dot gallery and choose Simple Scatter. E Select Standardized Residual as the y-axis variable and Package design as the x-axis variable. E Click OK. 133 Categorical Regression Figure 9-8 Residuals versus package design The U-shape is more pronounced in the plot of the standardized residuals against package. Every residual for Design B* is negative, whereas all but one of the residuals is positive for the other two designs. Because the linear regression model ts one parameter for each variable, the relationship cannot be captured by the standard approach. A Categorical Regression Analysis The categorical nature of the variables and the nonlinear relationship between Preference and Package design suggest that regression on optimal scores may perform better than standard regression. The U-shape of the residual plots indicates that a 134 Chapter 9 nominal treatment of Package design should be used. All other predictors will be treated at the numerical scaling level. The response variable warrants special consideration. You want to predict the values of Preference. Thus, recovering as many properties of its categories as possible in the quantications is desirable. Using an ordinal or nominal scaling level ignores the differences between the response categories. However, linearly transforming the response categories preserves category differences. Consequently, scaling the response numerically is generally preferred and will be employed here. Running the Analysis E To run a Categorical Regression analysis, from the menus choose: Analyze Regression Optimal Scaling (CATREG)... Figure 9-9 Categorical Regression dialog box E Select Preference as the dependent variable. E Select Package design through Money-back guarantee as independent variables. 135 Categorical Regression E Select Preference and click Define Scale. Figure 9-10 Define Scale dialog box E Select Numeric as the optimal scaling level. E Click Continue. E Select Package design and click Define Scale in the Categorical Regression dialog box. Figure 9-11 Define Scale dialog box E Select Nominal as the optimal scaling level. E Click Continue. 136 Chapter 9 E Select Brand name through Money-back guarantee and click Define Scale in the Categorical Regression dialog box. Figure 9-12 Define Scale dialog box E Select Numeric as the optimal scaling level. E Click Continue. E Click Output in the Categorical Regression dialog box. 137 Categorical Regression Figure 9-13 Output dialog box E Select Correlations of original variables and Correlations of transformed variables. E Deselect ANOVA. E Click Continue. E Click Save in the Categorical Regression dialog box. 138 Chapter 9 Figure 9-14 Save dialog box E Select Save residuals to the active dataset. E Select Save transformed variables to the active dataset in the Transformed Variables group. E Click Continue. E Click Plots in the Categorical Regression dialog box. 139 Categorical Regression Figure 9-15 Plots dialog box E Choose to create transformation plots for Package design and Price. E Click Continue. E Click OK in the Categorical Regression dialog box. Intercorrelations The intercorrelations among the predictors are useful for identifying multicollinearity in the regression. Variables that are highly correlated will lead to unstable regression estimates. However, due to their high correlation, omitting one of them from the model only minimally affects prediction. The variance in the response that can be explained by the omitted variable is still explained by the remaining correlated variable. However, zero-order correlations are sensitive to outliers and also cannot identify multicollinearity due to a high correlation between a predictor and a combination of other predictors. 140 Chapter 9 Figure 9-16 Original predictor correlations Figure 9-17 Transformed predictor correlations The intercorrelations of the predictors for both the untransformed and transformed predictors are displayed. All values are near 0, indicating that multicollinearity between individual variables is not a concern. Notice that the only correlations that change involve Package design. Because all other predictors are treated numerically, the differences between the categories and the order of the categories are preserved for these variables. Consequently, the correlations cannot change. Model Fit and Coefficients The Categorical Regression procedure yields an R2 of 0.948, indicating that almost 95% of the variance in the transformed preference rankings is explained by the regression on the optimally transformed predictors. Transforming the predictors improves the t over the standard approach. 141 Categorical Regression Figure 9-18 Model summary for categorical regression The following table shows the standardized regression coefcients. Categorical regression standardizes the variables, so only standardized coefcients are reported. These values are divided by their corresponding standard errors, yielding an F test for each variable. However, the test for each variable is contingent upon the other predictors being in the model. In other words, the test determines if omission of a predictor variable from the model with all other predictors present signicantly worsens the predictive capabilities of the model. These values should not be used to omit several variables at one time for a subsequent model. Moreover, alternating least squares optimizes the quantications, implying that these tests must be interpreted conservatively. Figure 9-19 Standardized coefficients for transformed predictors The largest coefcient occurs for Package design. A one standard deviation increase in Package design yields a 0.748 standard deviation decrease in predicted preference ranking. However, Package design is treated nominally, so an increase in the quantications need not correspond to an increase in the original category codes. Standardized coefcients are often interpreted as reecting the importance of each predictor. However, regression coefcients cannot fully describe the impact of a predictor or the relationships between the predictors. Alternative statistics must be used in conjunction with the standardized coefcients to fully explore predictor effects. 142 Chapter 9 Correlations and Importance To interpret the contributions of the predictors to the regression, it is not sufcient to only inspect the regression coefcients. In addition, the correlations, partial correlations, and part correlations should be inspected. The following table contains these correlational measures for each variable. The zero-order correlation is the correlation between the transformed predictor and the transformed response. For this data, the largest correlation occurs for Package design. However, if you can explain some of the variation in either the predictor or the response, you will get a better representation of how well the predictor is doing. Figure 9-20 Zero-order, part, and partial correlations (transformed variables) Other variables in the model can confound the performance of a given predictor in predicting the response. The partial correlation coefcient removes the linear effects of other predictors from both the predictor and the response. This measure equals the correlation between the residuals from regressing the predictor on the other predictors and the residuals from regressing the response on the other predictors. The squared partial correlation corresponds to the proportion of the variance explained relative to the residual variance of the response remaining after removing the effects of the other variables. For example, Package design has a partial correlation of 0.955. Removing the effects of the other variables, Package design explains (0.955)2 = 0.91 = 91% of the variation in the preference rankings. Both Price and Good Housekeeping seal also explain a large portion of variance if the effects of the other variables are removed. As an alternative to removing the effects of variables from both the response and a predictor, you can remove the effects from just the predictor. The correlation between the response and the residuals from regressing a predictor on the other predictors is the part correlation. Squaring this value yields a measure of the proportion of variance explained relative to the total variance of response. If you remove the effects of Brand 143 Categorical Regression name, Good Housekeeping seal, Money back guarantee, and Price from Package design, the remaining part of Package design explains (0.733)2 = 0.54 = 54% of the variation in preference rankings. Importance In addition to the regression coefcients and the correlations, Pratts measure of relative importance (Pratt, 1987) aids in interpreting predictor contributions to the regression. Large individual importances relative to the other importances correspond to predictors that are crucial to the regression. Also, the presence of suppressor variables is signaled by a low importance for a variable that has a coefcient of similar size to the important predictors. In contrast to the regression coefcients, this measure denes the importance of the predictors additivelythat is, the importance of a set of predictors is the sum of the individual importances of the predictors. Pratts measure equals the product of the regression coefcient and the zero-order correlation for a predictor. These products add to R2, so they are divided by R2, yielding a sum of 1. The set of predictors Package design and Brand name, for example, have an importance of 0.654. The largest importance corresponds to Package design, with Package design, Price, and Good Housekeeping seal accounting for 95% of the importance for this combination of predictors. Multicollinearity Large correlations between predictors will dramatically reduce a regression models stability. Correlated predictors result in unstable parameter estimates. Tolerance reects how much the independent variables are linearly related to one another. This measure is the proportion of a variables variance not accounted for by other independent variables in the equation. If the other predictors can explain a large amount of a predictors variance, that predictor is not needed in the model. A tolerance value near 1 indicates that the variable cannot be predicted very well from the other predictors. In contrast, a variable with a very low tolerance contributes little information to a model, and can cause computational problems. Moreover, large negative values of Pratts importance measure indicate multicollinearity. All of the tolerance measures are very high. None of the predictors are predicted very well by the other predictors and multicollinearity is not present. 144 Chapter 9 Transformation Plots Plotting the original category values against their corresponding quantications can reveal trends that might not be noticed in a list of the quantications. Such plots are commonly referred to as transformation plots. Attention should be given to categories that receive similar quantications. These categories affect the predicted response in the same manner. However, the transformation type dictates the basic appearance of the plot. Variables treated as numerical result in a linear relationship between the quantications and the original categories, corresponding to a straight line in the transformation plot. The order and the difference between the original categories is preserved in the quantications. The order of the quantications for variables treated as ordinal correspond to the order of the original categories. However, the differences between the categories are not preserved. As a result, the transformation plot is nondecreasing but need not be a straight line. If consecutive categories correspond to similar quantications, the category distinction may be unnecessary and the categories could be combined. Such categories result in a plateau on the transformation plot. However, this pattern can also result from imposing an ordinal structure on a variable that should be treated as nominal. If a subsequent nominal treatment of the variable reveals the same pattern, combining categories is warranted. Moreover, if the quantications for a variable treated as ordinal fall along a straight line, a numerical transformation may be more appropriate. For variables treated as nominal, the order of the categories along the horizontal axis corresponds to the order of the codes used to represent the categories. Interpretations of category order or of the distance between the categories is unfounded. The plot can assume any nonlinear or linear form. If an increasing trend is present, an ordinal treatment should be attempted. If the nominal transformation plot displays a linear trend, a numerical transformation may be more appropriate. The following gure displays the transformation plot for Price, which was treated as numerical. Notice that the order of the categories along the straight line correspond to the order of the original categories. Also, the difference between the quantications for $1.19 and $1.39 (1.173 and 0) is the same as the difference between the quantications for $1.39 and $1.59 (0 and 1.173). The fact that categories 1 and 3 are the same distance from category 2 is preserved in the quantications. 145 Categorical Regression Figure 9-21 Transformation plot of Price (numerical) The nominal transformation of Package design yields the following transformation plot. Notice the distinct nonlinear shape in which the second category has the largest quantication. In terms of the regression, the second category decreases predicted preference ranking, whereas the rst and third categories have the opposite effect. 146 Chapter 9 Figure 9-22 Transformation plot of Package design (nominal) Residual Analysis Using the transformed data and residuals that you saved to the active dataset allows you to create a scatterplot of the predicted values by the transformed values of Package design. To obtain such a scatterplot, recall the Chart Builder and click Reset to clear your previous selections and restore the default options. 147 Categorical Regression Figure 9-23 Chart Builder E Select the Scatter/Dot gallery and choose Simple Scatter. E Select Residual as the y-axis variable. E Select Package design Quantication as the x-axis variable. E Click OK. 148 Chapter 9 The scatterplot shows the standardized residuals plotted against the optimal scores for Package design. All of the residuals are within two standard deviations of 0. A random scatter of points replaces the U-shape present in the scatterplot from the standard linear regression. Predictive abilities are improved by optimally quantifying the categories. Figure 9-24 Residuals for Categorical Regression Example: Ozone Data In this example, you will use a larger set of data to illustrate the selection and effects of optimal scaling transformations. The data include 330 observations on six meteorological variables previously analyzed by Breiman and Friedman (Breiman and Friedman, 1985), and Hastie and Tibshirani (Hastie and Tibshirani, 1990), among others. The following table describes the original variables. Your categorical 149 Categorical Regression regression attempts to predict the ozone concentration from the remaining variables. Previous researchers found nonlinearities among these variables, which hinder standard regression approaches. Table 9-2 Original variables Variable ozon ibh dpg vis temp doy Description daily ozone level; categorized into one of 38 categories inversion base height pressure gradient (mm Hg) visibility (miles) temperature (degrees F) day of the year This dataset can be found in ozone.sav.For more information, see Sample Files in Appendix A on p. 406. Discretizing Variables If a variable has more categories than is practically interpretable, you should modify the categories using the Discretization dialog box to reduce the category range to a more manageable number. The variable Day of the year has a minimum value of 3 and a maximum value of 365. Using this variable in a categorical regression corresponds to using a variable with 365 categories. Similarly, Visibility (miles) ranges from 0 to 350. To simplify interpretation of analyses, discretize these variables into equal intervals of length 10. The variable Inversion base height ranges from 111 to 5000. A variable with this many categories results in very complex relationships. However, discretizing this variable into equal intervals of length 100 yields roughly 50 categories. Using a 50-category variable rather than a 5000-category variable simplies interpretations signicantly. Pressure gradient (mm Hg) ranges from 69 to 107. The procedure omits any categories coded with negative numbers from the analysis, but discretizing this variable into equal intervals of length 10 yields roughly 19 categories. Temperature (degrees F) ranges from 25 to 93 on the Fahrenheit scale. In order to analyze the data as if it were on the Celsius scale, discretize this variable into equal intervals of length 1.8. 150 Chapter 9 Different discretizations for variables may be desired. The choices used here are purely subjective. If you desire fewer categories, choose larger intervals. For example, Day of the year could have been divided into months of the year or seasons. Selection of Transformation Type Each variable can be analyzed at one of several different levels. However, because prediction of the response is the goal, you should scale the response as is by employing the numerical optimal scaling level. Consequently, the order and the differences between categories will be preserved in the transformed variable. E To run a Categorical Regression analysis, from the menus choose: Analyze Regression Optimal Scaling (CATREG)... Figure 9-25 Categorical Regression dialog box E Select Daily ozone level as the dependent variable. 151 Categorical Regression E Select Inversion base height through Day of the year as independent variables. E Select Daily ozone level and click Define Scale. Figure 9-26 Define Scale dialog box E Select Numeric as the optimal scaling level. E Click Continue. E Select Inversion base height through Day of the year, and click Define Scale in the Categorical Regression dialog box. 152 Chapter 9 Figure 9-27 Define Scale dialog box E Select Nominal as the optimal scaling level. E Click Continue. E Click Discretize in the Categorical Regression dialog box. 153 Categorical Regression Figure 9-28 Discretization dialog box E Select ibh. E Select Equal intervals and type 100 as the interval length. E Click Change. E Select dpg, vis, and doy. E Type 10 as the interval length. E Click Change. E Select temp. E Type 1.8 as the interval length. E Click Change. 154 Chapter 9 E Click Continue. E Click Plots in the Categorical Regression dialog box. Figure 9-29 Plots dialog box E Select transformation plots for Inversion base height through Day of the year. E Click Continue. E Click OK in the Categorical Regression dialog box. Figure 9-30 Model summary Treating all predictors as nominal yields an R2 of 0.886. This large amount of variance accounted for is not surprising because nominal treatment imposes no restrictions on the quantications. However, interpreting the results can be quite difcult. 155 Categorical Regression Figure 9-31 Regression coefficients (all predictors nominal) This table shows the standardized regression coefcients of the predictors. A common mistake made when interpreting these values involves focusing on the coefcients while neglecting the quantications. You cannot assert that the large positive value of the Temperature coefcient implies that as temperature increases, predicted Ozone increases. Similarly, the negative coefcient for Inversion base height does not suggest that as Inversion base height increases, predicted Ozone decreases. All interpretations must be relative to the transformed variables. As the quantications for Temperature increase, or as the quantications for Inversion base height decrease, predicted Ozone increases. To examine the effects of the original variables, you must relate the categories to the quantications. 156 Chapter 9 Figure 9-32 Transformation plot of Inversion base height (nominal) The transformation plot of Inversion base height shows no apparent pattern. As evidenced by the jagged nature of the plot, moving from low categories to high categories yields uctuations in the quantications in both directions. Thus, describing the effects of this variable requires focusing on the individual categories. Imposing ordinal or linear restrictions on the quantications for this variable might signicantly reduce the t. 157 Categorical Regression Figure 9-33 Transformation plot of Pressure gradient (nominal) This gure displays the transformation plot of Pressure gradient. The initial discretized categories (1 through 6) receive small quantications and thus have minimal contributions to the predicted response. The next three categories receive somewhat higher, positive values, resulting in a moderate increase in predicted ozone. The quantications decrease up to category 16, where Pressure gradient has its greatest decreasing effect on predicted ozone. Although the line increases after this category, using an ordinal scaling level for Pressure gradient may not signicantly reduce the t, while simplifying the interpretations of the effects. However, the importance measure of 0.04 and the regression coefcient for Pressure gradient indicates that this variable is not very useful in the regression. 158 Chapter 9 Figure 9-34 Transformation plot of Visibility (nominal) The transformation plot of Visibility, like that for Inversion base height, shows no apparent pattern. Imposing ordinal or linear restrictions on the quantications for this variable might signicantly reduce the t. 159 Categorical Regression Figure 9-35 Transformation plot of Temperature (nominal) The transformation plot of Temperature displays an alternative pattern. As the categories increase, the quantications tend to increase. As a result, as Temperature increases, predicted ozone tends to increase. This pattern suggests scaling Temperature at the ordinal level. 160 Chapter 9 Figure 9-36 Transformation plot of Day of the year (nominal) This gure shows the transformation plot of Day of the year. The quantications tend to decrease up to category 19, at which point they tend to increase, yielding a U-shape. Considering the sign of the regression coefcient for Day of the year, the initial categories (1 through 5) receive quantications that have a decreasing effect on predicted ozone. From category 6 onward, the effect of the quantications on predicted ozone gets more increasing, reaching a maximum around category 19. Beyond category 19, the quantications tend to decrease the predicted ozone. Although the line is quite jagged, the general shape is still identiable. Thus, the transformation plots suggest scaling Temperature at the ordinal level while keeping all other predictors nominally scaled. 161 Categorical Regression To recompute the regression, scaling Temperature at the ordinal level, recall the Categorical Regression dialog box. Figure 9-37 Define Scale dialog box E Select Temperature and click Define Scale. E Select Ordinal as the optimal scaling level. E Click Continue. E Click Save in the Categorical Regression dialog box. 162 Chapter 9 Figure 9-38 Save dialog box E Select Save transformed variables to the active dataset in the Transformed Variables group. E Click Continue. E Click OK in the Categorical Regression dialog box. Figure 9-39 Model summary for regression with Temperature (ordinal) This model results in an R2 of 0.875, so the variance accounted for decreases negligibly when the quantications for Temperature are restricted to be ordered. 163 Categorical Regression Figure 9-40 Regression coefficients with Temperature (ordinal) This table displays the coefcients for the model in which Temperature is scaled as ordinal. Comparing the coefcients to those for the model in which Temperature is scaled as nominal, no large changes occur. Figure 9-41 Correlations, importance, and tolerance Moreover, the importance measures suggest that Temperature is still much more important to the regression than the other variables. Now, however, as a result of the ordinal scaling level of Temperature and the positive regression coefcient, you can assert that as Temperature increases, predicted ozone increases. 164 Chapter 9 Figure 9-42 Transformation plot of Temperature (ordinal) The transformation plot illustrates the ordinal restriction on the quantications for Temperature. The jagged line from the nominal transformation is replaced here by a smooth ascending line. Moreover, no long plateaus are present, indicating that collapsing categories is not needed. Optimality of the Quantifications The transformed variables from a categorical regression can be used in a standard linear regression, yielding identical results. However, the quantications are optimal only for the model that produced them. Using a subset of the predictors in linear regression does not correspond to an optimal scaling regression on the same subset. For example, the categorical regression that you have computed has an R2 of 0.875. You have saved the transformed variables, so in order to t a linear regression using only Temperature, Pressure gradient, and Inversion base height as predictors, from the menus choose: Analyze Regression Linear... 165 Categorical Regression Figure 9-43 Linear Regression dialog box E Select Daily ozone level Quantication as the dependent variable. E Select Inversion base height Quantication, Pressure gradient (mm Hg) Quantication, and Temperature (degrees F) Quantication as independent variables. E Click OK. 166 Chapter 9 Figure 9-44 Model summary for regression with subset of optimally scaled predictors Using the quantications for the response, Temperature, Pressure gradient, and Inversion base height in a standard linear regression results in a t of 0.733. To compare this to the t of a categorical regression using just those three predictors, recall the Categorical Regression dialog box. Figure 9-45 Categorical Regression dialog box E Deselect Visibility (miles) and Day of the year as independent variables. E Click OK. 167 Categorical Regression Figure 9-46 Model summary for categorical regression on three predictors The categorical regression analysis has a t of 0.798, which is better than the t of 0.733. This demonstrates the property of the scalings that the quantications obtained in the original regression are only optimal when all ve variables are included in the model. Effects of Transformations Transforming the variables makes a nonlinear relationship between the original response and the original set of predictors linear for the transformed variables. However, when there are multiple predictors, pairwise relationships are confounded by the other variables in the model. To focus your analysis on the relationship between Daily ozone level and Day of the year, begin by looking at a scatterplot. From the menus choose: Graphs Chart Builder... 168 Chapter 9 Figure 9-47 Chart Builder dialog box E Select the Scatter/Dot gallery and choose Simple Scatter. E Select Daily ozone level as the y-axis variable and Day of the year as the x-axis variable. E Click OK. 169 Categorical Regression Figure 9-48 Scatterplot of Daily ozone level and Day of the year This gure illustrates the relationship between Daily ozone level and Day of the year. As Day of the year increases to approximately 200, Daily ozone level increases. However, for Day of the year values greater than 200, Daily ozone level decreases. This inverted U pattern suggests a quadratic relationship between the two variables. A linear regression cannot capture this relationship. E To see a best-t line overlaid on the points in the scatterplot, activate the graph by double-clicking on it. E Select a point in the Chart Editor. E Click the Add Fit Line at Total tool, and close the Chart Editor. 170 Chapter 9 Figure 9-49 Scatterplot showing best-fit line A linear regression of Daily ozone level on Day of the year yields an R2 of 0.004. This t suggests that Day of the year has no predictive value for Daily ozone level. This is not surprising, given the pattern in the gure. By using optimal scaling, however, you can linearize the quadratic relationship and use the transformed Day of the year to predict the response. 171 Categorical Regression Figure 9-50 Categorical Regression dialog box To obtain a categorical regression of Daily ozone level on Day of the year, recall the Categorical Regression dialog box. E Deselect Inversion base height through Temperature (degrees F) as independent variables. E Select Day of the year as an independent variable. E Click Define Scale. 172 Chapter 9 Figure 9-51 Define Scale dialog box E Select Nominal as the optimal scaling level. E Click Continue. E Click Discretize in the Categorical Regression dialog box. 173 Categorical Regression Figure 9-52 Discretization dialog box E Select doy. E Select Equal intervals. E Type 10 as the interval length. E Click Change. E Click Continue. E Click Plots in the Categorical Regression dialog box. 174 Chapter 9 Figure 9-53 Plots dialog box E Select doy for transformation plots. E Click Continue. E Click OK in the Categorical Regression dialog box. Figure 9-54 Model summary for categorical regression of Daily ozone level on Day of the year The optimal scaling regression treats Daily ozone level as numerical and Day of the year as nominal. This results in an R2 of 0.549. Although only 55% of the variation in Daily ozone level is accounted for by the categorical regression, this is a substantial improvement over the original regression. Transforming Day of the year allows for the prediction of Daily ozone level. 175 Categorical Regression Figure 9-55 Transformation plot of Day of the year (nominal) This gure displays the transformation plot of Day of the year. The extremes of Day of the year both receive negative quantications, whereas the central values have positive quantications. By applying this transformation, the low and high Day of the year values have similar effects on predicted Daily ozone level. 176 Chapter 9 Figure 9-56 Chart Builder To see a scatterplot of the transformed variables, recall the Chart Builder, and click Reset to clear your previous selections. E Select the Scatter/Dot gallery and choose Simple Scatter. E Select Daily ozone level Quantication [TRA1_3] as the y-axis variable and Day of the year Quantication [TRA2_3] as the x-axis variable. E Click OK. 177 Categorical Regression Figure 9-57 Scatterplot of the transformed variables This gure depicts the relationship between the transformed variables. An increasing trend replaces the inverted U. The regression line has a positive slope, indicating that as transformed Day of the year increases, predicted Daily ozone level increases. Using optimal scaling linearizes the relationship and allows interpretations that would otherwise go unnoticed. Recommended Readings See the following texts for more information on categorical regression: Buja, A. 1990. Remarks on functional canonical variates, alternating least squares methods and ACE. Annals of Statistics, 18, 10321069. 178 Chapter 9 Hastie, T., R. Tibshirani, and A. Buja. 1994. Flexible discriminant analysis. Journal of the American Statistical Association, 89, 12551270. Hayashi, C. 1952. On the prediction of phenomena from qualitative data and the quantication of qualitative data from the mathematico-statistical point of view. Annals of the Institute of Statitical Mathematics, 2, 9396. Kruskal, J. B. 1965. Analysis of factorial experiments by estimating monotone transformations of the data. Journal of the Royal Statistical Society Series B, 27, 251263. Meulman, J. J. 2003. Prediction and classication in nonlinear data analysis: Something old, something new, something borrowed, something blue. Psychometrika, 4, 493517. Ramsay, J. O. 1989. Monotone regression splines in action. Statistical Science, 4, 425441. Van der Kooij, A. J., and J. J. Meulman. 1997. MURALS: Multiple regression and optimal scaling using alternating least squares. In: Softstat 97, F. Faulbaum, and W. Bandilla, eds. Stuttgart: Gustav Fisher, 99106. Winsberg, S., and J. O. Ramsay. 1980. Monotonic transformations to additivity using splines. Biometrika, 67, 669674. Winsberg, S., and J. O. Ramsay. 1983. Monotone spline transformations for dimension reduction. Psychometrika, 48, 575595. Young, F. W., J. De Leeuw, and Y. Takane. 1976. Regression with qualitative and quantitative variables: An alternating least squares method with optimal scaling features. Psychometrika, 41, 505528. Categorical Principal Components Analysis 10 Chapter Categorical principal components analysis can be thought of as a method of dimension reduction. A set of variables is analyzed to reveal major dimensions of variation. The original data set can then be replaced by a new, smaller data set with minimal loss of information. The method reveals relationships among variables, among cases, and among variables and cases. The criterion used by categorical principal components analysis for quantifying the observed data is that the object scores (component scores) should have large correlations with each of the quantied variables. A solution is good to the extent that this criterion is satised. Two examples of categorical principal components analysis will be presented. The rst employs a rather small data set useful for illustrating the basic concepts and interpretations associated with the procedure. The second example examines a practical application. Example: Examining Interrelations of Social Systems This example examines Guttmans (Guttman, 1968) adaptation of a table by Bell (Bell, 1961). The data are also discussed by Lingoes (Lingoes, 1968). Bell presented a table to illustrate possible social groups. Guttman used a portion of this table, in which ve variables describing things such as social interaction, feelings of belonging to a group, physical proximity of members, and formality of the relationship were crossed with seven theoretical social groups, including crowds (for example, people at a football game), audiences (for example, people at a theater or classroom lecture), public (for example, newspaper or television audiences), mobs (like a crowd but with much more intense interaction), primary groups (intimate), secondary 179 180 Chapter 10 groups (voluntary), and the modern community (loose confederation resulting from close physical proximity and a need for specialized services). The following table shows the variables in the dataset resulting from the classication into seven social groups used in the Guttman-Bell data, with their variable labels and the value labels (categories) associated with the levels of each variable. This dataset can be found in guttman.sav. For more information, see Sample Files in Appendix A on p. 406. In addition to selecting variables to be included in the computation of the categorical principal components analysis, you can select variables that are used to label objects in plots. In this example, the rst ve variables in the data are included in the analysis, while cluster is used exclusively as a labeling variable. When you specify a categorical principal components analysis, you must specify the optimal scaling level for each analysis variable. In this example, an ordinal level is specied for all analysis variables. Table 10-1 Variables in the Guttman-Bell dataset Variable name intnsity frquency blonging proxmity formlity cluster Variable label Intensity of interaction Frequency of interaction Feeling of belonging Physical proximity Formality of relationship Value label Slight, low, moderate, high Slight, nonrecurring, infrequent, frequent None, slight, variable, high Distant, close No relationship, formal, informal Crowds, audiences, public, mobs, primary groups, secondary groups, modern community Running the Analysis E To produce categorical principal components output for this dataset, from the menus choose: Analyze Dimension Reduction Optimal Scaling... 181 Categorical Principal Components Analysis Figure 10-1 Optimal Scaling dialog box E Select Some variable(s) not multiple nominal in the Optimal Scaling Level group. E Click Define. 182 Chapter 10 Figure 10-2 Categorical Principal Components dialog box E Select Intensity of interaction through Formality of relationship as analysis variables. E Click Define Scale and Weight. 183 Categorical Principal Components Analysis Figure 10-3 Define Scale and Weight dialog box E Select Ordinal in the Optimal Scaling Level group. E Click Continue. E Select cluster as a labeling variable in the Categorical Principal Components dialog box. E Click Output. 184 Chapter 10 Figure 10-4 Output dialog box E Select Object scores and deselect Correlations of transformed variables in the Tables group. E Choose to produce category quantications for intnsity (Intensity of interaction) through formlity (Formality of relationship). E Choose to label object scores by cluster. E Click Continue. E Click Object in the Plots group of the Categorical Principal Components dialog box. 185 Categorical Principal Components Analysis Figure 10-5 Object and Variable Plots dialog box E Select Objects and variables (biplot) in the Plots group. E Choose to label objects by Variable in the Label Objects group, and then select cluster as the variable to label objects by. E Click Continue. E Click Category in the Plots group of the Categorical Principal Components dialog box. 186 Chapter 10 Figure 10-6 Category Plots dialog box E Choose to produce joint category plots for intnsity (Intensity of interaction) through formlity (Formality of relationship). E Click Continue. E Click OK in the Categorical Principal Components dialog box. 187 Categorical Principal Components Analysis Number of Dimensions These gures show some of the initial output for the categorical principal components analysis. After the iteration history of the algorithm, the model summary, including the eigenvalues of each dimension, is displayed. These eigenvalues are equivalent to those of classical principal components analysis. They are measures of how much variance is accounted for by each dimension. Figure 10-7 Iteration history Figure 10-8 Model summary The eigenvalues can be used as an indication of how many dimensions are needed. In this example, the default number of dimensions, 2, was used. Is this the right number? As a general rule, when all variables are either single nominal, ordinal, or numerical, the eigenvalue for a dimension should be larger than 1. Since the two-dimensional solution accounts for 94.52% of the variance, a third dimension probably would not add much more information. For multiple nominal variables, there is no easy rule of thumb to determine the appropriate number of dimensions. If the number of variables is replaced by the total number of categories minus the number of variables, the above rule still holds. But this rule alone would probably allow more dimensions than are needed. When choosing the 188 Chapter 10 number of dimensions, the most useful guideline is to keep the number small enough so that meaningful interpretations are possible. The model summary table also shows Cronbachs alpha (a measure of reliability), which is maximized by the procedure. Quantifications For each variable, the quantications, the vector coordinates, and the centroid coordinates for each dimension are presented. The quantications are the values assigned to each category. The centroid coordinates are the average of the object scores of objects in the same category. The vector coordinates are the coordinates of the categories when they are required to be on a line, representing the variable in the object space. This is required for variables with the ordinal and numerical scaling level. Figure 10-9 Quantifications for Intensity of interaction Glancing at the quantications in the joint plot of the category points, you can see that some of the categories of some variables were not clearly separated by the categorical principal components analysis as cleanly as would have been expected if the level had been truly ordinal. Variables Intensity of interaction and Frequency of interaction, for example, have equal or almost equal quantications for their two middle categories. This kind of result might suggest trying alternative categorical principal components analyses, perhaps with some categories collapsed, or perhaps with a different level of analysis, such as (multiple) nominal. 189 Categorical Principal Components Analysis Figure 10-10 Joint plot category points The joint plot of category points resembles the plot for the component loadings, but it also shows where the endpoints are located that correspond to the lowest quantications (for example, slight for Intensity of interaction and none for Feeling of belonging). The two variables measuring interaction, Intensity of interaction and Frequency of interaction, appear very close together and account for much of the variance in dimension 1. Formality of Relationship also appears close to Physical Proximity. By focusing on the category points, you can see the relationships even more clearly. Not only are Intensity of interaction and Frequency of interaction close, but the directions of their scales are similar; that is, slight intensity is close to slight frequency, and frequent interaction is near high intensity of interaction. You also see that close physical proximity seems to go hand-in-hand with an informal type of relationship, and physical distance is related to no relationship. 190 Chapter 10 Object Scores You can also request a listing and plot of object scores. The plot of the object scores can be useful for detecting outliers, detecting typical groups of objects, or revealing some special patterns. The object scores table shows the listing of object scores labeled by social group for the Guttman-Bell data. By examining the values for the object points, you can identify specic objects in the plot. Figure 10-11 Object scores The rst dimension appears to separate CROWDS and PUBLIC, which have relatively large negative scores, from MOBS and PRIMARY GROUPS, which have relatively large positive scores. The second dimension has three clumps: PUBLIC and SECONDARY GROUPS with large negative values, CROWDS with large positive values, and the other social groups in between. This is easier to see by inspecting the plot of the object scores. 191 Categorical Principal Components Analysis Figure 10-12 Object scores plot In the plot, you see PUBLIC and SECONDARY GROUPS at the bottom, CROWDS at the top, and the other social groups in the middle. Examining patterns among individual objects depends on the additional information available for the units of analysis. In this case, you know the classication of the objects. In other cases, you can use supplementary variables to label the objects. You can also see that the categorical principal components analysis does not separate MOBS from PRIMARY GROUPS. Although most people usually dont think of their families as mobs, on the variables used, these two groups received the same score on four of the ve variables! Obviously, you might want to explore possible shortcomings of the variables and categories used. For example, high intensity of interaction and informal relationships probably mean different things to these two groups. Alternatively, you might consider a higher dimensional solution. Component Loadings This gure shows the plot of component loadings. The vectors (lines) are relatively long, indicating again that the rst two dimensions account for most of the variance of all of the quantied variables. On the rst dimension, all variables have high (positive) component loadings. The second dimension is correlated mainly with the quantied 192 Chapter 10 variables Feeling of belonging and Physical Proximity, in opposite directions. This means that objects with a large negative score in dimension 2 will have a high score in feeling of belonging and a low score in physical proximity. The second dimension, therefore, reveals a contrast between these two variables while having little relation with the quantied variables Intensity of interaction and Frequency of interaction. Figure 10-13 Component loadings To examine the relation between the objects and the variables, look at the biplot of objects and component loadings. The vector of a variable points into the direction of the highest category of the variable. For example, for Physical Proximity and Feeling of belonging the highest categories are close and high, respectively. Therefore, CROWDS 193 Categorical Principal Components Analysis are characterized by close physical proximity and no feeling of belonging, and SECONDARY GROUPS, by distant physical proximity and a high feeling of belonging. Figure 10-14 Biplot Additional Dimensions Increasing the number of dimensions will increase the amount of variation accounted for and may reveal differences concealed in lower dimensional solutions. As noted previously, in two dimensions MOBS and PRIMARY GROUPS cannot be separated. However, increasing the dimensionality may allow the two groups to be differentiated. Running the Analysis E To obtain a three-dimensional solution, recall the Categorical Principal Components dialog box. E Type 3 as the number of dimensions in the solution. E Click OK in the Categorical Principal Components dialog box. 194 Chapter 10 Model Summary Figure 10-15 Model summary A three-dimensional solution has eigenvalues of 3.424, 0.844, and 0.732, accounting for nearly all of the variance. Object Scores The object scores for the three-dimensional solution are plotted in a scatterplot matrix. In a scatterplot matrix, every dimension is plotted against every other dimension in a series of two-dimensional scatterplots. Note that the rst two eigenvalues in three dimensions are not equal to the eigenvalues in the two-dimensional solution; in other words, the solutions are not nested. Because the eigenvalues in dimensions 2 and 3 are now smaller than 1 (giving a Cronbachs alpha that is negative), you should prefer the two-dimensional solution. The three-dimensional solution is included for purposes of illustration. 195 Categorical Principal Components Analysis Figure 10-16 Three-dimensional object scores scatterplot matrix The top row of plots reveals that the rst dimension separates PRIMARY GROUPS and MOBS from the other groups. Notice that the order of the objects along the vertical axis does not change in any of the plots in the top row; each of these plots employs dimension 1 as the y axis. The middle row of plots allows for interpretation of dimension 2. The second dimension has changed slightly from the two-dimensional solution. Previously, the second dimension had three distinct clumps, but now the objects are more spread out along the axis. The third dimension helps to separate MOBS from PRIMARY GROUPS, which did not occur in the two-dimensional solution. Look more closely at the dimension 2 versus dimension 3 and dimension 1 versus dimension 2 plots. On the plane dened by dimensions 2 and 3, the objects form a rough rectangle, with CROWDS, MODERN COMMUNITY, SECONDARY GROUPS, and PUBLIC at the vertices. On this plane, MOBS and PRIMARY GROUPS appear to be convex combinations of PUBLIC-CROWDS and SECONDARY GROUPS-MODERN COMMUNITY, respectively. However, as previously mentioned, they are separated from the other groups along dimension 1. AUDIENCES is not separated from the other groups along dimension 1 and appears to be a combination of CROWDS and MODERN COMMUNITY. 196 Chapter 10 Component Loadings Figure 10-17 Three-dimensional component loadings Knowing how the objects are separated does not reveal which variables correspond to which dimensions. This is accomplished using the component loadings. The rst dimension corresponds primarily to Feeling of belonging, Intensity of interaction, and Formality of Relationship; the second dimension separates Frequency of interaction and Physical Proximity; and the third dimension separates these from the others. Example: Symptomatology of Eating Disorders Eating disorders are debilitating illnesses associated with disturbances in eating behavior, severe body image distortion, and an obsession with weight that affects the mind and body simultaneously. Millions of people are affected each year, with adolescents particularly at risk. Treatments are available and most are helpful when the condition is identied early. A health professional can attempt to diagnose an eating disorder through a psychological and medical evaluation. However, it can be difcult to assign a patient to one of several different classes of eating disorders because there is no standardized symptomatology of anorectic/bulimic behavior. Are there symptoms that clearly differentiate patients into the four groups? Which symptoms do they have in common? In order to try to answer these questions, researchers (Van der Ham, Meulman, Van Strien, and Van Engeland, 1997) made a study of 55 adolescents with known eating disorders, as shown in the following table. 197 Categorical Principal Components Analysis Table 10-2 Patient diagnoses Diagnosis Anorexia nervosa Anorexia with bulimia nervosa Bulimia nervosa after anorexia Atypical eating disorder Total Number of Patients 25 9 14 7 55 Each patient was seen four times over four years, for a total of 220 observations. At each observation, the patients were scored for each of the 16 symptoms outlined in the following table. Symptom scores are missing for patient 71 at time 2, patient 76 at time 2, and patient 47 at time 3, leaving 217 valid observations. The data can be found in anorectic.sav.For more information, see Sample Files in Appendix A on p. 406. Table 10-3 Modified Morgan-Russell subscales measuring well-being Variable name weight mens fast binge vomit purge hyper fami eman frie school satt sbeh mood Variable label Body weight Menstruation Restriction of food intake (fasting) Binge eating Vomiting Purging Hyperactivity Family relations Emancipation from family Friends School/employment record Sexual attitude Sexual behavior Mental state (mood) Lower end (score1) Outside normal range Amenorrhea Less than 1200 calories Greater than once a week Greater than once a week Greater than once a week Not able to be at rest Poor Very dependent No good friends Stopped school/work Inadequate Inadequate Very depressed Upper end (score 3 or 4) Normal Regular periods Normal/regular meals No bingeing No vomiting No purging No hyperactivity Good Adequate Two or more good friends Moderate to good record Adequate Can enjoy sex Normal 198 Chapter 10 Variable name preo body Variable label Preoccupation with food and weight Body perception Lower end (score1) Complete Disturbed Upper end (score 3 or 4) No preoccupation Normal Principal components analysis is ideal for this situation, since the purpose of the study is to ascertain the relationships between symptoms and the different classes of eating disorders. Moreover, categorical principal components analysis is likely to be more useful than classical principal components analysis because the symptoms are scored on an ordinal scale. Running the Analysis In order to properly examine the structure of the course of illness for each diagnosis, you will want to make the results of the projected centroids table available as data for scatterplots. You can accomplish this using the Output Management System. E To begin an OMS request, from the menus choose: Utilities OMS Control Panel... 199 Categorical Principal Components Analysis Figure 10-18 Output Management System Control Panel E Select Tables as the output type. E Select CATPCA as the command. E Select Projected Centroids as the table type. E Select File in the Output Destinations group and type projected_centroids.sav as the lename. E Click Options. 200 Chapter 10 Figure 10-19 Options dialog box E Select SPSS Statistics Data File as the output format. E Type TableNumber_1 as the table number variable. E Click Continue. 201 Categorical Principal Components Analysis Figure 10-20 Output Management System Control Panel E Click Add. E Click OK, and then click OK to conrm the OMS session. The Output Management System is now set to write the results of the projected centroids table to the le projected_centroids.sav. E To produce categorical principal components output for this dataset, from the menus choose: Analyze Dimension Reduction Optimal Scaling... 202 Chapter 10 Figure 10-21 Optimal Scaling dialog box E Select Some variable(s) not multiple nominal in the Optimal Scaling Level group. E Click Define. 203 Categorical Principal Components Analysis Figure 10-22 Categorical Principal Components dialog box E Select Body weight through Body perception as analysis variables. E Click Define Scale and Weight. 204 Chapter 10 Figure 10-23 Define Scale and Weight dialog box E Select Ordinal as the optimal scaling level. E Click Continue. E Select Time/diagnosis interaction as a supplementary variable and click Define Scale in the Categorical Principal Components dialog box. 205 Categorical Principal Components Analysis Figure 10-24 Define Scale dialog box E Select Multiple nominal as the optimal scaling level. E Click Continue. 206 Chapter 10 Figure 10-25 Categorical Principal Components dialog box E Select Time of interview through Patient number as labeling variables. E Click Options. 207 Categorical Principal Components Analysis Figure 10-26 Options dialog box E Choose to label plots by Variable names or values. E Click Continue. E Click Output in the Categorical Principal Components dialog box. 208 Chapter 10 Figure 10-27 Output dialog box E Select Object scores in the Tables group. E Request category quantications for tidi. E Choose to include categories of time, diag, and number. E Click Continue. E Click Save in the Categorical Principal Components dialog box. 209 Categorical Principal Components Analysis Figure 10-28 Save dialog box E In the Transformed Variables group, select Save to the active dataset. E Click Continue. E Click Object in the Categorical Principal Components dialog box. 210 Chapter 10 Figure 10-29 Object and Variable Plots dialog box E Choose to label objects by Variable. E Select time and diag as the variables to label objects by. E Click Continue. E Click Category in the Categorical Principal Components dialog box. 211 Categorical Principal Components Analysis Figure 10-30 Category Plots dialog box E Request category plots for tidi. E Request transformation plots for weight through body. E Choose to project centroids of tidi onto binge, satt, and preo. E Click Continue. E Click OK in the Categorical Principal Components dialog box. 212 Chapter 10 The procedure results in scores for the subjects (with mean 0 and unit variance) and quantications of the categories that maximize the mean squared correlation of the subject scores and the transformed variables. In the present analysis, the category quantications were constrained to reect the ordinal information. Finally, to write the projected centroids table information to projected_centroids.sav, you need to end the OMS request. Recall the OMS Control Panel. Figure 10-31 Output Management System Control Panel E Click End. E Click OK, and then click OK to conrm. Transformation Plots The transformation plots display the original category number on the horizontal axes; the vertical axes give the optimal quantications. 213 Categorical Principal Components Analysis Figure 10-32 Transformation plot for menstruation Some variables, like Menstruation, obtained nearly linear transformations, so in this analysis you may interpret them as numerical. 214 Chapter 10 Figure 10-33 Transformation plot for School/employment record The quantications for other variables like School/employment record did not obtain linear transformations and should be interpreted at the ordinal scaling level. The difference between the second and third categories is much more important than that between the rst and second categories. 215 Categorical Principal Components Analysis Figure 10-34 Transformation plot for Binge eating An interesting case arises in the quantications for Binge eating. The transformation obtained is linear for categories 1 through 3, but the quantied values for categories 3 and 4 are equal. This result shows that scores of 3 and 4 do not differentiate between patients and suggests that you could use the numerical scaling level in a two-component solution by recoding 4s as 3s. 216 Chapter 10 Model Summary Figure 10-35 Model summary To see how well your model ts the data, look at the model summary. About 47% of the total variance is explained by the two-component model, 35% by the rst dimension and 12% by the second. So, almost half of the variability on the individual objects level is explained by the two-component model. Component Loadings To begin to interpret the two dimensions of your solution, look at the component loadings. All variables have a positive component loading in the rst dimension, which means that there is a common factor that correlates positively with all of the variables. 217 Categorical Principal Components Analysis Figure 10-36 Component loadings plot The second dimension separates the variables. The variables Binge eating, Vomiting, and Purging form a bundle having large positive loadings in the second dimension. These symptoms are typically considered to be representative of bulimic behavior. The variables Emancipation from family, School/employment record, Sexual attitude, Body weight, and Menstruation form another bundle, and you can include Restriction of food intake (fasting) and Family relations in this bundle, because their vectors are close to the main cluster, and these variables are considered to be anorectic symptoms (fasting, weight, menstruation) or are psychosocial in nature (emancipation, school/work record, sexual attitude, family relations). The vectors of this bundle are orthogonal (perpendicular) to the vectors of binge, vomit, and purge, which means that this set of variables is uncorrelated with the set of bulimic variables. The variables Friends, Mental state (mood), and Hyperactivity do not appear to t very well into the solution. You can see this in the plot by observing the lengths of each vector. The length of a given variables vector corresponds to its t, and these variables have the shortest vectors. Based on a two-component solution, you would probably drop these variables from a proposed symptomatology for eating disorders. They may, however, t better in a higher dimensional solution. 218 Chapter 10 The variables Sexual behavior, Preoccupation with food and weight, and Body perception form another theoretic group of symptoms, pertaining to how the patient experiences his or her body. While correlated with the two orthogonal bundles of variables, these variables have fairly long vectors and are strongly associated with the rst dimension and therefore may provide some useful information about the common factor. Object Scores The following gure shows a plot of the object scores, in which the subjects are labeled with their diagnosis category. Figure 10-37 Object scores plot labeled by diagnosis This plot does not help to interpret the rst dimension, because patients are not separated by diagnosis along it. However, there is some information about the second dimension. Anorexia subjects (1) and patients with atypical eating disorder (4) form a group, located above subjects with some form of bulimia (2 and 3). Thus, the second dimension separates bulimic patients from others, as you have also seen in the previous section (the variables in the bulimic bundle have large positive component loadings 219 Categorical Principal Components Analysis in the second dimension). This makes sense, given that the component loadings of the symptoms that are traditionally associated with bulimia have large values in the second dimension. This gure shows a plot of the object scores, in which the subjects are labeled with their time of diagnosis. Figure 10-38 Object scores labeled by time of interview Labeling the object scores by time reveals that the rst dimension has a relation to time because there seems to be a progression of times of diagnosis from the 1s mostly to the left and others to the right. Note that you can connect the time points in this plot by saving the object scores and creating a scatterplot using the dimension 1 scores on the x axis, the dimension 2 scores on the y axis, and setting the markers using the patient numbers. Comparing the object scores plot labeled by time with the one labeled by diagnosis can give you some insight into unusual objects. For example, in the plot labeled by time, there is a patient whose diagnosis at time 4 lies to the left of all other points in the plot. This is unusual because the general trend of the points is for the later times to lie further to the right. Interestingly, this point that seems out of place in time also has an unusual diagnosis, in that the patient is an anorectic whose scores place the patient in the bulimic cluster. By looking in the table of object scores, you nd that 220 Chapter 10 this is patient 43, diagnosed with anorexia nervosa, whose object scores are shown in the following table. Table 10-4 Object scores for patient 43 Time 1 2 3 4 Dimension 1 2.031 2.067 1.575 2.405 Dimension 2 1.250 0.131 1.467 1.807 The patients scores at time 1 are prototypical for anorectics, with the large negative score in dimension 1 corresponding to poor body image and the positive score in dimension 2 corresponding to anorectic symptoms or poor psychosocial behavior. However, unlike the majority of patients, there is little or no progress in dimension 1. In dimension 2, there is apparently some progress toward normal (around 0, between anorectic and bulimic behavior), but then the patient shifts to exhibit bulimic symptoms. Examining the Structure of the Course of Illness To nd out more about how the two dimensions were related to the four diagnosis categories and the four time points, a supplementary variable Time/diagnosis interaction was created by a cross-classication of the four categories of Patient diagnosis and the four categories of Time of interview. Thus, Time/diagnosis interaction has 16 categories, where the rst category indicates the anorexia nervosa patients at their rst visit. The fth category indicates the anorexia nervosa patients at time point 2, and so on, with the sixteenth category indicating the atypical eating disorder patients at time point 4. The use of the supplementary variable Time/diagnosis interaction allows for the study of the courses of illness for the different groups over time. The variable was given a multiple nominal scaling level, and the category points are displayed in the following gure. 221 Categorical Principal Components Analysis Figure 10-39 Category points for time/diagnosis interaction Some of the structure is apparent from this plot: the diagnosis categories at time point 1 clearly separate anorexia nervosa and atypical eating disorder from anorexia nervosa with bulimia nervosa and bulimia nervosa after anorexia nervosa in the second dimension. After that, its a little more difcult to see the patterns. However, you can make the patterns more easily visible by creating a scatterplot based on the quantications. To do this, from the menus choose: Graphs Chart Builder... 222 Chapter 10 Figure 10-40 Scatter/Dot gallery E Select the Scatter/Dot gallery and choose Grouped Scatter. 223 Categorical Principal Components Analysis Figure 10-41 Chart Builder E Select Time/diagnosis interaction Quantication dimension 2 as the y-axis variable and Time/diagnosis interaction Quantication dimension 1 as the x-axis variable. E Choose to set color by Patient Diagnosis. E Click OK. 224 Chapter 10 Figure 10-42 Structures of the courses of illness E Then, to connect the points, double-click on the graph, and click the Add interpolation line tool in the Chart Editor. E Close the Chart Editor. 225 Categorical Principal Components Analysis Figure 10-43 Structures of the courses of illness By connecting the category points for each diagnostic category across time, the patterns immediately suggest that the rst dimension is related to time and the second, to diagnosis, as you previously determined from the object scores plots. However, this plot further shows that, over time, the illnesses tend to become more alike. Moreover, for all groups, the progress is greatest between time points 1 and 2; the anorectic patients show some more progress from 2 to 3, but the other groups show little progress. 226 Chapter 10 Differential Development for Selected Variables One variable from each bundle of symptoms identied by the component loadings was selected as representative of the bundle. Binge eating was selected from the bulimic bundle; sexual attitude, from the anorectic/psychosocial bundle; and body preoccupation, from the third bundle. In order to examine the possible differential courses of illness, the projections of Time/diagnosis interaction on Binge eating, Sexual attitude, and Preoccupation with food and weight were computed and plotted in the following gure. Figure 10-44 Projected centroids of Time/diagnosis interaction on Binge eating, Sexual attitude, and Preoccupation with food and weight This plot shows that at the rst time point, the symptom binge eating separates bulimic patients (2 and 3) from others (1 and 4); sexual attitude separates anorectic and atypical patients (1 and 4) from others (2 and 3); and body preoccupation does not really separate the patients. In many applications, this plot would be sufcient to describe the relationship between the symptoms and diagnosis, but because of the complication of multiple time points, the picture becomes muddled. 227 Categorical Principal Components Analysis In order to view these projections over time, you need to be able to plot the contents of the projected centroids table. This is made possible by the OMS request that saved this information to projected_centroids.sav. Figure 10-45 Projected_centroids.sav The variables Bingeeating, Sexualattitude, and Preoccupationwithfoodandweight contain the values of the centroids projected on each of the symptoms of interest. The case number (1 through 16) corresponds to the time/diagnosis interaction. You will need to compute new variables that separate out the Time and Diagnosis values. E From the menus choose: Transform Compute Variable... 228 Chapter 10 Figure 10-46 Compute Variable dialog box E Type time as the target variable. E Type trunc(($casenum-1)/4) + 1 as the numeric expression. E Click OK. 229 Categorical Principal Components Analysis Figure 10-47 Compute Variable dialog box E Recall the Compute Variable dialog box. E Type diagnosis as the target variable. E Type mod($casenum-1, 4) + 1 as the numeric expression. E Click OK. 230 Chapter 10 Figure 10-48 Projected_centroids.sav In the Variable View, change the measure for diagnosis from Scale to Nominal. 231 Categorical Principal Components Analysis Figure 10-49 Chart Builder E Finally, to view the projected centroids of time of diagnosis on binging over time, recall the Chart Builder and click Reset to clear your previous selections. E Select the Scatter/Dot gallery and choose Grouped Scatter. E Select Centroids Projected on Binge eating as the y-axis variable and time as the x-axis variable. E Choose to set colors by diagnosis. E Click OK. 232 Chapter 10 Figure 10-50 Projected centroids of Time of diagnosis on Binge eating over time E Then, to connect the points, double-click on the graph, and click the Add interpolation line tool in the Chart Editor. E Close the Chart Editor. 233 Categorical Principal Components Analysis With respect to binge eating, it is clear that the anorectic groups have different starting values from the bulimic groups. This difference shrinks over time, as the anorectic groups hardly change, while the bulimic groups show progress. Figure 10-51 Chart Builder E Recall the Chart Builder. E Deselect Centroids Projected on Binge eating as the y-axis variable and select Centroids Projected on Sexual attitude as the y-axis variable. E Click OK. 234 Chapter 10 Figure 10-52 Projected centroids of Time of diagnosis on Sexual attitude over time E Then, to connect the points, double-click on the graph, and click the Add interpolation line tool in the Chart Editor. E Close the Chart Editor. 235 Categorical Principal Components Analysis With respect to sexual attitude, the four trajectories are more or less parallel over time, and all groups show progress. The bulimic groups, however, have higher (better) scores than the anorectic group. Figure 10-53 Chart Builder E Recall the Chart Builder. E Deselect Centroids Projected on Sexual attitude as the y-axis variable and select Centroids Projected on Preoccupation with food and weight as the y-axis variable. E Click OK. 236 Chapter 10 Figure 10-54 Projected centroids of Time of diagnosis on Body preoccupation over time E Then, to connect the points, double-click on the graph, and click the Add interpolation line tool in the Chart Editor. E Close the Chart Editor. 237 Categorical Principal Components Analysis Body preoccupation is a variable that represents the core symptoms, which are shared by the four different groups. Apart from the atypical eating disorder patients, the anorectic group and the two bulimic groups have very similar levels both at the beginning and at the end. Recommended Readings See the following texts for more information on categorical principal components analysis: De Haas, M., J. A. Algera, H. F. J. M. Van Tuijl, and J. J. Meulman. 2000. Macro and micro goal setting: In search of coherence. Applied Psychology, 49, 579595. De Leeuw, J. 1982. Nonlinear principal components analysis. In: COMPSTAT Proceedings in Computational Statistics, Vienna: Physica Verlag, 7789. Eckart, C., and G. Young. 1936. The approximation of one matrix by another one of lower rank. Psychometrika, 1, 211218. Gabriel, K. R. 1971. The biplot graphic display of matrices with application to principal components analysis. Biometrika, 58, 453467. Gi, A. 1985. PRINCALS. Research Report UG-85-02. Leiden: Department of Data Theory, University of Leiden. Gower, J. C., and J. J. Meulman. 1993. The treatment of categorical information in physical anthropology. International Journal of Anthropology, 8, 4351. Heiser, W. J., and J. J. Meulman. 1994. Homogeneity analysis: Exploring the distribution of variables and their nonlinear relationships. In: Correspondence Analysis in the Social Sciences: Recent Developments and Applications, M. Greenacre, and J. Blasius, eds. New York: Academic Press, 179209. Kruskal, J. B. 1978. Factor analysis and principal components analysis: Bilinear methods. In: International Encyclopedia of Statistics, W. H. Kruskal, and J. M. Tanur, eds. New York: The Free Press, 307330. Kruskal, J. B., and R. N. Shepard. 1974. A nonmetric variety of linear factor analysis. Psychometrika, 39, 123157. 238 Chapter 10 Meulman, J. J. 1993. Principal coordinates analysis with optimal transformations of the variables: Minimizing the sum of squares of the smallest eigenvalues. British Journal of Mathematical and Statistical Psychology, 46, 287300. Meulman, J. J., and P. Verboon. 1993. Points of view analysis revisited: Fitting multidimensional structures to optimal distance components with cluster restrictions on the variables. Psychometrika, 58, 735. Meulman, J. J., A. J. Van der Kooij, and A. Babinec. 2000. New features of categorical principal components analysis for complicated data sets, including data mining. In: Classication, Automation and New Media, W. Gaul, and G. Ritter, eds. Berlin: Springer-Verlag, 207217. Meulman, J. J., A. J. Van der Kooij, and W. J. Heiser. 2004. Principal components analysis with nonlinear optimal scaling transformations for ordinal and nominal data. In: Handbook of Quantitative Methodology for the Social Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: Sage Publications, Inc., 4970. Theunissen, N. C. M., J. J. Meulman, A. L. Den Ouden, H. M. Koopman, G. H. Verrips, S. P. Verloove-Vanhorick, and J. M. Wit. 2003. Changes can be studied when the measurement instrument is different at different time points. Health Services and Outcomes Research Methodology, 4, 109126. Tucker, L. R. 1960. Intra-individual and inter-individual multidimensionality. In: Psychological Scaling: Theory & Applications, H. Gulliksen, and S. Messick, eds. NewYork: John Wiley and Sons, 155167. Vlek, C., and P. J. Stallen. 1981. Judging risks and benets in the small and in the large. Organizational Behavior and Human Performance, 28, 235271. Wagenaar, W. A. 1988. Paradoxes of gambling behaviour. London: Lawrence Erlbaum Associates, Inc. Young, F. W., Y. Takane, and J. De Leeuw. 1978. The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features. Psychometrika, 43, 279281. Zeijl, E., Y. te Poel, M. du Bois-Reymond, J. Ravesloot, and J. J. Meulman. 2000. The role of parents and peers in the leisure activities of young adolescents. Journal of Leisure Research, 32, 281302. Nonlinear Canonical Correlation Analysis 11 Chapter The purpose of nonlinear canonical correlation analysis is to determine how similar two or more sets of variables are to one another. As in linear canonical correlation analysis, the aim is to account for as much of the variance in the relationships among the sets as possible in a low-dimensional space. Unlike linear canonical correlation analysis, however, nonlinear canonical correlation analysis does not assume an interval level of measurement or assume that the relationships are linear. Another important difference is that nonlinear canonical correlation analysis establishes the similarity between the sets by simultaneously comparing linear combinations of the variables in each set to an unknown setthe object scores. Example: An Analysis of Survey Results The example in this chapter is from a survey (Verdegaal, 1985). The responses of 15 subjects to 8 variables were recorded. The variables, variable labels, and value labels (categories) in the dataset are shown in the following table. Table 11-1 Survey data Variable name age Variable label Age in years marital pet news Marital status Pets owned Newspaper read most often Value label 2025, 2630, 3135, 3640, 4145, 4650, 5155, 5660, 6165, 6670 Single, Married, Other No, Cat(s), Dog(s), Other than cat or dog, Various domestic animals None, Telegraaf, Volkskrant, NRC, Other 239 240 Chapter 11 Variable name music live math language Variable label Music preferred Neighborhood preference Math test score Language test score Value label Classical, New wave, Popular, Variety, Dont like music Town, Village, Countryside 05, 610, 1115 05, 610, 1115, 1620 This dataset can be found in verd1985.sav. For more information, see Sample Files in Appendix A on p. 406. The variables of interest are the rst six variables, and they are divided into three sets. Set 1 includes age and marital, set 2 includes pet and news, and set 3 includes music and live. Pet is scaled as multiple nominal, and age is scaled as ordinal; all other variables are scaled as single nominal. This analysis requests a random initial conguration. By default, the initial conguration is numerical. However, when some of the variables are treated as single nominal with no possibility of ordering, it is best to choose a random initial conguration. This is the case with most of the variables in this study. Examining the Data E To obtain a nonlinear canonical correlation analysis for this dataset, from the menus choose: Analyze Dimension Reduction Optimal Scaling... 241 Nonlinear Canonical Correlation Analysis Figure 11-1 Optimal Scaling dialog box E Select Some variable(s) not multiple nominal in the Optimal Scaling Level group. E Select Multiple sets in the Number of Sets of Variables group. E Click Define. 242 Chapter 11 Figure 11-2 Nonlinear Canonical Correlation Analysis dialog box E Select Age in years and Marital status as variables for the rst set. E Select age and click Define Range and Scale. Figure 11-3 Define Range and Scale dialog box E Type 10 as the maximum value for this variable. 243 Nonlinear Canonical Correlation Analysis E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, select marital and click Define Range and Scale. Figure 11-4 Define Range and Scale dialog box E Type 3 as the maximum value for this variable. E Select Single nominal as the measurement scale. E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, click Next to dene the next variable set. 244 Chapter 11 Figure 11-5 Nonlinear Canonical Correlation Analysis dialog box E Select Pets owned and Newspaper read most often as variables for the second set. E Select pet and click Define Range and Scale. Figure 11-6 Define Range and Scale dialog box E Type 5 as the maximum value for this variable. 245 Nonlinear Canonical Correlation Analysis E Select Multiple nominal as the measurement scale. E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, select news and click Define Range and Scale. Figure 11-7 Define Range and Scale dialog box E Type 5 as the maximum value for this variable. E Select Single nominal as the measurement scale. E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, click Next to dene the last variable set. 246 Chapter 11 Figure 11-8 Nonlinear Canonical Correlation Analysis dialog box E Select Music preferred and Neighborhood preference as variables for the third set. E Select music and click Define Range and Scale. Figure 11-9 Define Range and Scale dialog box E Type 5 as the maximum value for this variable. 247 Nonlinear Canonical Correlation Analysis E Select Single nominal as the measurement scale. E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, select live and click Define Range and Scale. Figure 11-10 Define Range and Scale dialog box E Type 3 as the maximum value for this variable. E Select Single nominal as the measurement scale. E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, click Options. 248 Chapter 11 Figure 11-11 Options dialog box E Deselect Centroids and select Weights and component loadings in the Display group. E Select Category centroids and Transformations in the Plot group. E Select Use random initial configuration. E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, click OK. After a list of the variables with their levels of optimal scaling, categorical canonical correlation analysis with optimal scaling produces a table showing the frequencies of objects in categories. This table is especially important if there are missing data, since almost-empty categories are more likely to dominate the solution. In this example, there are no missing data. A second preliminary check is to examine the plot of object scores for outliers. Outliers have such different quantications from the other objects that they will be at the boundaries of the plot, thus dominating one or more dimensions. 249 Nonlinear Canonical Correlation Analysis If you nd outliers, you can handle them in one of two ways. You can simply eliminate them from the data and run the nonlinear canonical correlation analysis again. Alternatively, you can try recoding the extreme responses of the outlying objects by collapsing (merging) some categories. As shown in the plot of object scores, there were no outliers for the survey data. Figure 11-12 Object scores Accounting for Similarity between Sets There are several ways to measure the association between sets in a nonlinear canonical correlation analysis (each way being detailed in a separate table or set of tables). 250 Chapter 11 Summary of Analysis The t and loss values tell you how well the nonlinear canonical correlation analysis solution ts the optimally quantied data with respect to the association between the sets. The summary of analysis table shows the t value, loss values, and eigenvalues for the survey example. Figure 11-13 Summary of analysis Loss is partitioned across dimensions and sets. For each dimension and set, loss represents the proportion of variation in the object scores that cannot be accounted for by the weighted combination of variables in the set. The average loss is labeled Mean. In this example, the average loss over sets is 0.464. Notice that more loss occurs for the second dimension than for the rst dimension. The eigenvalue for each dimension equals 1 minus the average loss for the dimension and indicates how much of the relationship is shown by each dimension. The eigenvalues add up to the total t. For Verdegaals data, 0.801 / 1.536 = 52% of the actual t is accounted for by the rst dimension. The maximum t value equals the number of dimensions and, if obtained, indicates that the relationship is perfect. The average loss value over sets and dimensions tells you the difference between the maximum t and the actual t. Fit plus the average loss equals the number of dimensions. Perfect similarity rarely happens and usually capitalizes on trivial aspects in the data. Another popular statistic with two sets of variables is the canonical correlation. Since the canonical correlation is related to the eigenvalue and thus provides no additional information, it is not included in the nonlinear canonical correlation analysis output. For two sets of variables, the canonical correlation per dimension is obtained by the following formula: 251 Nonlinear Canonical Correlation Analysis where d is the dimension number and E is the eigenvalue. You can generalize the canonical correlation for more than two sets with the following formula: where d is the dimension number, K is the number of sets, and E is the eigenvalue. For our example, and Weights and Component Loadings Another measure of association is the multiple correlation between linear combinations from each set and the object scores. If no variables in a set are multiple nominal, you can compute this measure by multiplying the weight and component loading of each variable within the set, adding these products, and taking the square root of the sum. Figure 11-14 Weights 252 Chapter 11 Figure 11-15 Component loadings These gures give the weights and component loadings for the variables in this example. The multiple correlation (R) is as follows for the rst weighted sum of optimally scaled variables (Age in years and Marital status) with the rst dimension of object scores: For each dimension, 1 loss = R2. For example, from the Summary of analysis table, 1 0.238 = 0.762, which is 0.873 squared (plus some rounding error). Consequently, small loss values indicate large multiple correlations between weighted sums of optimally scaled variables and dimensions. Weights are not unique for multiple nominal variables. For multiple nominal variables, use 1 loss per set. 253 Nonlinear Canonical Correlation Analysis Partitioning Fit and Loss The loss of each set is partitioned by the nonlinear canonical correlation analysis in several ways. The t table presents the multiple t, single t, and single loss tables produced by the nonlinear canonical correlation analysis for the survey example. Note that multiple t minus single t equals single loss. Figure 11-16 Partitioning fit and loss Single loss indicates the loss resulting from restricting variables to one set of quantications (that is, single nominal, ordinal, or nominal). If single loss is large, it is better to treat the variables as multiple nominal. In this example, however, single t and multiple t are almost equal, which means that the multiple coordinates are almost on a straight line in the direction given by the weights. Multiple t equals the variance of the multiple category coordinates for each variable. These measures are analogous to the discrimination measures that are found in homogeneity analysis. You can examine the multiple t table to see which variables discriminate best. For example, look at the multiple t table for Marital status and Newspaper read most often. The t values, summed across the two dimensions, are 1.122 for Marital status and 0.911 for Newspaper read most often. This information tells us that a persons marital status provides greater discriminatory power than the newspaper they subscribe to. Single t corresponds to the squared weight for each variable and equals the variance of the single category coordinates. As a result, the weights equal the standard deviations of the single category coordinates. By examining how the single t is 254 Chapter 11 broken down across dimensions, we see that the variable Newspaper read most often discriminates mainly on the rst dimension, and we see that the variable Marital status discriminates almost totally on the second dimension. In other words, the categories of Newspaper read most often are further apart in the rst dimension than in the second, whereas the pattern is reversed for Marital status. In contrast, Age in years discriminates in both the rst and second dimensions; thus, the spread of the categories is equal along both dimensions. Component Loadings The following gure shows the plot of component loadings for the survey data. When there are no missing data, the component loadings are equivalent to the Pearson correlations between the quantied variables and the object scores. The distance from the origin to each variable point approximates the importance of that variable. The canonical variables are not plotted but can be represented by horizontal and vertical lines drawn through the origin. Figure 11-17 Component loadings 255 Nonlinear Canonical Correlation Analysis The relationships between variables are apparent. There are two directions that do not coincide with the horizontal and vertical axes. One direction is determined by Age in years, Newspaper read most often, and Neighborhood preference. The other direction is dened by the variables Marital status, Music preferred, and Pets owned. The Pets owned variable is a multiple nominal variable, so there are two points plotted for it. Each quantication is interpreted as a single variable. Transformation Plots The different levels at which each variable can be scaled impose restrictions on the quantications. Transformation plots illustrate the relationship between the quantications and the original categories resulting from the selected optimal scaling level. The transformation plot for Neighborhood preference, which was treated as nominal, displays a U-shaped pattern, in which the middle category receives the lowest quantication, and the extreme categories receive values that are similar to each other. This pattern indicates a quadratic relationship between the original variable and the transformed variable. Using an alternative optimal scaling level is not suggested for Neighborhood preference. 256 Chapter 11 Figure 11-18 Transformation plot for Neighborhood preference (nominal) The quantications for Newspaper read most often, in contrast, correspond to an increasing trend across the three categories that have observed cases. The rst category receives the lowest quantication, the second category receives a higher value, and the 257 Nonlinear Canonical Correlation Analysis third category receives the highest value. Although the variable is scaled as nominal, the category order is retrieved in the quantications. Figure 11-19 Transformation plot for Newspaper read most often (nominal) 258 Chapter 11 Figure 11-20 Transformation plot for Age in years (ordinal) The transformation plot for Age in years displays an S-shaped curve. The four youngest observed categories all receive the same negative quantication, whereas the two oldest categories receive similar positive values. Consequently, collapsing all younger ages into one common category (that is, below 50) and collapsing the two oldest categories into one category may be attempted. However, the exact equality of the quantications for the younger groups indicates that restricting the order of the quantications to the order of the original categories may not be desirable. Because the quantications for the 2630, 3640, and 4145 groups cannot be lower than the quantication for the 2025 group, these values are set equal to the boundary value. Allowing these values to be smaller than the quantication for the youngest group (that is, treating age as nominal) may improve the t. So although age may be considered an ordinal variable, treating it as such does not appear appropriate in this case. Moreover, treating age as numerical, and thus maintaining the distances between the categories, would substantially reduce the t. 259 Nonlinear Canonical Correlation Analysis Single Category versus Multiple Category Coordinates For every variable treated as single nominal, ordinal, or numerical, quantications, single category coordinates, and multiple category coordinates are determined. These statistics for Age in years are presented. Figure 11-21 Coordinates for Age in years Every category for which no cases were recorded receives a quantication of 0. For Age in years, this includes the 3135, 4650, and 5155 categories. These categories are not restricted to be ordered with the other categories and do not affect any computations. For multiple nominal variables, each category receives a different quantication on each dimension. For all other transformation types, a category has only one quantication, regardless of the dimensionality of the solution. Each set of single category coordinates represents the location of the category on a line in the object space. The coordinates for a given category equal the quantication multiplied by the variable dimension weights. For example, in the table for Age in years, the single category coordinates for category 56-60 (-0.142, -0.165) are the quantication (-0.209) multiplied by the dimension weights (0.680, 0.789). The multiple category coordinates for variables that are treated as single nominal, ordinal, or numerical represent the coordinates of the categories in the object space before ordinal or linear constraints are applied. These values are unconstrained 260 Chapter 11 minimizers of the loss. For multiple nominal variables, these coordinates represent the quantications of the categories. The effects of imposing constraints on the relationship between the categories and their quantications are revealed by comparing the single category coordinates with the multiple category coordinates. On the rst dimension, the multiple category coordinates for Age in years decrease to category 2 and remain relatively at the same level until category 9, at which point a dramatic increase occurs. A similar pattern is evidenced for the second dimension. These relationships are removed in the single category coordinates, in which the ordinal constraint is applied. On both dimensions, the coordinates are now nondecreasing. The differing structure of the two sets of coordinates suggests that a nominal treatment may be more appropriate. Centroids and Projected Centroids The plot of centroids labeled by variables should be interpreted in the same way as the category quantications plot in homogeneity analysis or the multiple category coordinates in nonlinear principal components analysis. By itself, such a plot shows how well variables separate groups of objects (the centroids are in the center of gravity of the objects). Notice that the categories for Age in years are not separated very clearly. The younger age categories are grouped together at the left of the plot. As suggested previously, ordinal may be too strict a scaling level to impose on Age in years. 261 Nonlinear Canonical Correlation Analysis Figure 11-22 Centroids labeled by variables 262 Chapter 11 When you request centroid plots, individual centroid and projected centroid plots for each variable that is labeled by value labels are also produced. The projected centroids are on a line in the object space. Figure 11-23 Centroids and projected centroids for Newspaper read most often The actual centroids are projected onto the vectors that are dened by the component loadings. These vectors have been added to the centroid plots to aid in distinguishing the projected centroids from the actual centroids. The projected centroids fall into one of four quadrants formed by extending two perpendicular reference lines through the origin. The interpretation of the direction of single nominal, ordinal, or numerical variables is obtained from the position of the projected centroids. For example, the variable Newspaper read most often is specied as single nominal. The projected centroids show that Volkskrant and NRC are contrasted with Telegraaf. 263 Nonlinear Canonical Correlation Analysis Figure 11-24 Centroids and projected centroids for Age in years The problem with Age in years is evident from the projected centroids. Treating Age in years as ordinal implies that the order of the age groups must be preserved. To satisfy this restriction, all age groups below age 45 are projected into the same point. Along the direction dened by Age in years, Newspaper read most often, and Neighborhood preference, there is no separation of the younger age groups. Such a nding suggests treating the variable as nominal. 264 Chapter 11 Figure 11-25 Centroids and projected centroids for Neighborhood preference To understand the relationships among variables, nd out what the specic categories (values) are for clusters of categories in the centroid plots. The relationships among Age in years, Newspaper read most often, and Neighborhood preference can be described by looking at the upper right and lower left of the plots. In the upper right, the age groups are the older respondents; they read the newspaper Telegraaf and prefer living in a village. Looking at the lower left corner of each plot, you see that the younger to middle-aged respondents read the Volkskrant or NRC and want to live in the country or in a town. However, separating the younger groups is very difcult. The same types of interpretations can be made about the other direction (Music preferred, Marital status, and Pets owned) by focusing on the upper left and the lower right of the centroid plots. In the upper left corner, we nd that single people tend to have dogs and like new wave music. The married people and other categories for marital have cats; the former group prefers classical music, and the latter group does not like music. 265 Nonlinear Canonical Correlation Analysis An Alternative Analysis The results of the analysis suggest that treating Age in years as ordinal does not appear appropriate. Although Age in years is measured at an ordinal level, its relationships with other variables are not monotonic. To investigate the effects of changing the optimal scaling level to single nominal, you may rerun the analysis. To Run the Analysis E Recall the Nonlinear Canonical Correlation Analysis dialog box and navigate to the rst set. E Select age and click Define Range and Scale. E In the Dene Range and Scale dialog box, select Single nominal as the scaling range. E Click Continue. E In the Nonlinear Canonical Correlation Analysis dialog box, click OK. The eigenvalues for a two-dimensional solution are 0.806 and 0.757, respectively, with a total t of 1.564. Figure 11-26 Eigenvalues for the two-dimensional solution 266 Chapter 11 The multiple t and single t tables show that Age in years is still a highly discriminating variable, as evidenced by the sum of the multiple t values. In contrast to the earlier results, however, examination of the single t values reveals the discrimination to be almost entirely along the second dimension. Figure 11-27 Partitioning fit and loss Turn to the transformation plot for Age in years. The quantications for a nominal variable are unrestricted, so the nondecreasing trend that was displayed when Age in years was treated ordinally is no longer present. There is a decreasing trend until the age of 40 and an increasing trend thereafter, corresponding to a U-shaped (quadratic) 267 Nonlinear Canonical Correlation Analysis relationship. The two older categories still receive similar scores, and subsequent analyses may involve combining these categories. Figure 11-28 Transformation plot for Age in years (nominal) 268 Chapter 11 The transformation plot for Neighborhood preference is shown here. Treating Age in years as nominal does not affect the quantications for Neighborhood preference to any signicant degree. The middle category receives the smallest quantication, with the extreme categories receiving large positive values. Figure 11-29 Transformation plot for Neighborhood preference (age nominal) 269 Nonlinear Canonical Correlation Analysis A change is found in the transformation plot for Newspaper read most often. Previously, an increasing trend was present in the quantications, possibly suggesting an ordinal treatment for this variable. However, treating Age in years as nominal removes this trend from the news quantications. Figure 11-30 Transformation plot for Newspaper read most often (age nominal) 270 Chapter 11 This plot is the centroid plot for Age in years. Notice that the categories do not fall in chronological order along the line joining the projected centroids. The 2025 group is situated in the middle rather than at the end. The spread of the categories is much improved over the ordinal counterpart that was presented previously. Figure 11-31 Centroids and projected centroids for Age in years (nominal) Interpretation of the younger age groups is now possible from the centroid plot. The Volkskrant and NRC categories are also further apart than in the previous analysis, allowing for separate interpretations of each. The groups between the ages of 26 and 45 read the Volkskrant and prefer country living. The 2025 and 5660 age groups read the NRC; the former group prefers to live in a town, and the latter group prefers country living. The oldest groups read the Telegraaf and prefer village living. 271 Nonlinear Canonical Correlation Analysis Interpretation of the other direction (Music preferred, Marital status, and Pets owned) is basically unchanged from the previous analysis. The only obvious difference is that people with a marital status of Other have either cats or no pets. Figure 11-32 Centroids labeled by variables (age nominal) General Suggestions After you have examined the initial results, you will probably want to rene your analysis by changing some of the specications for the nonlinear canonical correlation analysis. Here are some tips for structuring your analysis: Create as many sets as possible. Put an important variable that you want to predict in a separate set by itself. Put variables that you consider predictors together in a single set. If there are many predictors, try to partition them into several sets. 272 Chapter 11 Put each multiple nominal variable in a separate set by itself. If variables are highly correlated to each other, and you dont want this relationship to dominate the solution, put those variables together in the same set. Recommended Readings See the following texts for more information about nonlinear canonical correlation analysis: Carroll, J. D. 1968. Generalization of canonical correlation analysis to three or more sets of variables. In: Proceedings of the 76th Annual Convention of the American Psychological Association, 3, Washington, D.C.: American Psychological Association, 227228. De Leeuw, J. 1984. Canonical analysis of categorical data, 2nd ed. Leiden: DSWO Press. Horst, P. 1961. Generalized canonical correlations and their applications to experimental data. Journal of Clinical Psychology, 17, 331347. Horst, P. 1961. Relations among m sets of measures. Psychometrika, 26, 129149. Kettenring, J. R. 1971. Canonical analysis of several sets of variables. Biometrika, 58, 433460. Van der Burg, E. 1988. Nonlinear canonical correlation and some related techniques. Leiden: DSWO Press. Van der Burg, E., and J. De Leeuw. 1983. Nonlinear canonical correlation. British Journal of Mathematical and Statistical Psychology, 36, 5480. Van der Burg, E., J. De Leeuw, and R. Verdegaal. 1988. Homogeneity analysis with k sets of variables: An alternating least squares method with optimal scaling features. Psychometrika, 53, 177197. Verboon, P., and I. A. Van der Lans. 1994. Robust canonical discriminant analysis. Psychometrika, 59, 485507. Correspondence Analysis 12 Chapter A correspondence table is any two-way table whose cells contain some measurement of correspondence between the rows and the columns. The measure of correspondence can be any indication of the similarity, afnity, confusion, association, or interaction between the row and column variables. A very common type of correspondence table is a crosstabulation, where the cells contain frequency counts. Such tables can be obtained easily with the Crosstabs procedure. However, a crosstabulation does not always provide a clear picture of the nature of the relationship between the two variables. This is particularly true if the variables of interest are nominal (with no inherent order or rank) and contain numerous categories. Crosstabulation may tell you that the observed cell frequencies differ signicantly crosstabulation of occupation and breakfast cereal, from the expected values in a but it may be difcult to discern which occupational groups have similar tastes or what those tastes are. Correspondence Analysis allows you to examine the relationship between two nominal variables graphically in a multidimensional space. It computes row and column scores and produces plots based on the scores. Categories that are similar to each other appear close to each other in the plots. In this way, it is easy to see which categories of a variable are similar to each other or which categories of the two variables are related. The Correspondence Analysis procedure also allows you to t supplementary points into the space dened by the active points. If the ordering of the categories according to their scores is undesirable or counterintuitive, order restrictions can be imposed by constraining the scores for some categories to be equal. For example, suppose that you expect the variable smoking behavior, with categories none, light, medium, and heavy, to have scores that correspond to this ordering. However, if the analysis orders the categories none, light, heavy, and medium, constraining the scores for heavy and medium to be equal preserves the ordering of the categories in their scores. 273 274 Chapter 12 The interpretation of correspondence analysis in terms of distances depends on the normalization method used. The Correspondence Analysis procedure can be used to analyze either the differences between categories of a variable or the differences between variables. With the default normalization, it analyzes the differences between the row and column variables. The correspondence analysis algorithm is capable of many kinds of analyses. Centering the rows and columns and using chi-square distances corresponds to standard correspondence analysis. However, using alternative centering options combined with Euclidean distances allows for an alternative representation of a matrix in a low-dimensional space. Three examples will be presented. The rst employs a relatively small correspondence table and illustrates the concepts inherent in correspondence analysis. The second example demonstrates a practical marketing application. The nal example uses a table of distances in a multidimensional scaling approach. Normalization Normalization is used to distribute the inertia over the row scores and column scores. Some aspects of the correspondence analysis solution, such as the singular values, the inertia per dimension, and the contributions, do not change under the various normalizations. The row and column scores and their variances are affected. Correspondence analysis has several ways to spread the inertia. The three most common include spreading the inertia over the row scores only, spreading the inertia over the column scores only, or spreading the inertia symmetrically over both the row scores and the column scores. Row principal. In row principal normalization, the Euclidean distances between the row points approximate chi-square distances between the rows of the correspondence table. The row scores are the weighted average of the column scores. The column scores are standardized to have a weighted sum of squared distances to the centroid of 1. Since this method maximizes the distances between row categories, you should use row principal normalization if you are primarily interested in seeing how categories of the row variable differ from each other. Column principal. On the other hand, you might want to approximate the chi-square distances between the columns of the correspondence table. In that case, the column scores should be the weighted average of the row scores. The row scores are 275 Correspondence Analysis standardized to have a weighted sum of squared distances to the centroid of 1. This method maximizes the distances between column categories and should be used if you are primarily concerned with how categories of the column variable differ from each other. Symmetrical. You can also treat the rows and columns symmetrically. This normalization spreads inertia equally over the row and column scores. Note that neither the distances between the row points nor the distances between the column points are approximations of chi-square distances in this case. Use this method if you are primarily interested in the differences or similarities between the two variables. Usually, this is the preferred method to make biplots. Principal. A fourth option is called principal normalization, in which the inertia is spread twice in the solutiononce over the row scores and once over the column scores. You should use this method if you are interested in the distances between the row points and the distances between the column points separately but not in how the row and column points are related to each other. Biplots are not appropriate for this normalization option and are therefore not available if you have specied the principal normalization method. Example: Smoking Behavior by Job Category The aim of correspondence analysis is to show the relationships between the rows and columns of a correspondence table. You will use a hypothetical table introduced by Greenacre (Greenacre, 1984) to illustrate the basic concepts. This information is collected in smoking.sav. For more information, see Sample Files in Appendix A on p. 406. The table of interest is formed by the crosstabulation of smoking behavior by job category. The variable Staff Group contains the job categories Sr Managers, Jr Managers, Sr Employees, Jr Employees, and Secretaries, which will be used to create the solution, plus the category National Average, which can be used as supplementary to the analysis. The variable Smoking contains the behaviors None, Light, Medium, and Heavy, which will be used to create the solution, plus the categories No Alcohol and Alcohol, which can be used as supplementary to the analysis. 276 Chapter 12 Running the Analysis E Before you can run the Correspondence Analysis procedure, the setup of the data requires that the cases be weighted by the variable count. To do this, from the menus choose: Data Weight Cases... Figure 12-1 Weight Cases dialog box E Weight cases by count. E Click OK. E Then, to obtain a correspondence analysis in two dimensions using row principal normalization, from the menus choose: Analyze Dimension Reduction Correspondence Analysis... 277 Correspondence Analysis Figure 12-2 Correspondence Analysis dialog box E Select Staff Group as the row variable. E Click Define Range. Figure 12-3 Define Row Range dialog box E Type 1 as the minimum value. E Type 5 as the maximum value. 278 Chapter 12 E Click Update. E Click Continue. E Select Smoking as the column variable. E Click Define Range in the Correspondence Analysis dialog box. Figure 12-4 Define Column Range dialog box E Type 1 as the minimum value. E Type 4 as the maximum value. E Click Update. E Click Continue. E Click Statistics in the Correspondence Analysis dialog box. 279 Correspondence Analysis Figure 12-5 Statistics dialog box E Select Row profiles and Column profiles. E Select Permutations of the correspondence table. E Select condence statistics for Row points and Column points. E Click Continue. E Click OK in the Correspondence Analysis dialog box. Correspondence Table The correspondence table shows the distribution of smoking behavior for ve levels of job category. The rows of the correspondence table represent the job categories. The columns represent the smoking behavior. Figure 12-6 Correspondence table 280 Chapter 12 The marginal row totals show that the company has far more employees, both junior and senior, than managers and secretaries. However, the distribution of senior and junior positions for the managers is approximately the same as the distribution of senior and junior positions for the employees. Looking at the column totals, you see that there are similar numbers of nonsmokers and medium smokers. Furthermore, heavy smokers are outnumbered by each of the other three categories. But what, if anything, do any of these job categories have in common regarding smoking behavior? And what is the relationship between job category and smoking? Dimensionality Ideally, you want a correspondence analysis solution that represents the relationship between the row and column variables in as few dimensions as possible. But it is frequently useful to look at the maximum number of dimensions to see the relative contribution of each dimension. The maximum number of dimensions for a correspondence analysis solution equals the number of active rows minus 1 or the number of active columns minus 1, whichever is less. An active row or column is one for which a distinct set of scores is found. Supplementary rows or columns are not active. In the present example, the maximum number of dimensions is min(5,4) 1 = 3. The rst dimension displays as much of the inertia (a measure of the variation in the data) as possible, the second is orthogonal to the rst and displays as much of the remaining inertia as possible, and so on. It is possible to split the total inertia into components attributable to each dimension. You can then evaluate the inertia shown by a particular dimension by comparing it to the total inertia. For example, the rst dimension displays 87.8% (0.075/0.085) of the total inertia, whereas the second dimension displays only 11.8% (0.010/0.085). Figure 12-7 Inertia per dimension 281 Correspondence Analysis If you decide that the rst p dimensions of a q dimensional solution show enough of the total inertia, then you do not have to look at higher dimensions. In this example, the two-dimensional solution is sufcient, since the third dimension represents less than 1.0% of the total inertia. The singular values can be interpreted as the correlation between the row and column scores. They are analogous to the Pearson correlation coefcient (r) in correlation analysis. For each dimension, the singular value squared (eigenvalue) equals the inertia and thus is another measure of the importance of that dimension. Biplot Correspondence analysis generates a variety of plots that graphically illustrate the underlying relationships between categories and between variables. This is the scatterplot of the row and column scores for the two-dimensional solution. Figure 12-8 Plot of row and column scores (symmetrical normalization) The interpretation of the plot is fairly simplerow/column points that are close together are more alike than points that are far apart. The second dimension separates managers from other employees, while the rst separates senior from junior, with secretaries in between. 282 Chapter 12 The symmetrical normalization makes it easy to examine the relationship between job category and smoking. For example, managers are near the Heavy smoking category, while senior employees are closest to None. Junior employees seem to be associated with Medium or Light smoking, and secretaries are not strongly associated with any particular smoking behavior (but are far from Heavy). Profiles and Distances To determine the distance between categories, correspondence analysis considers the marginal distributions as well as the individual cell frequencies. It computes row and column proles, which give the row and column proportions for each cell, based on the marginal totals. Figure 12-9 Row profiles (symmetrical normalization) The row proles indicate the proportion of the row category in each column category. For example, among the senior employees, most are nonsmokers and very few are heavy smokers. In contrast, among the junior managers, most are medium smokers and very few are light smokers. The column proles indicate the proportion of the column in each row category. For example, most of the light smokers are junior employees. Similarly, most of the medium and heavy smokers are junior employees. Recall that the sample contains 283 Correspondence Analysis predominantly junior employees. It is not surprising that this staff category dominates the smoking categories. Figure 12-10 Column profiles Mass is a measure that indicates the inuence of an object based on its marginal frequency. Mass affects the centroid, which is the weighted mean row or column prole. The row centroid is the mean row prole. Points with a large mass, like junior employees, pull the centroid strongly to their location. A point with a small mass, like senior managers, pulls the row centroid only slightly to its location. If you prefer to think of difference in terms of distance, then the greater the difference between row proles, the greater the distance between points in a plot. For example, when using row principal normalization, the nal conguration is one in which Euclidean distances between row points in the full dimensional space equal the chi-square distances between rows of the correspondence table. In a reduced space, the Euclidean distances approximate the chi-square distances. In turn, the chi-square distances are weighted prole distances. These weighted distances are based on mass. Likewise, under column principal normalization, the Euclidean distances between column points in the full dimensional space equal the chi-square distances between columns of the correspondence table. Note, however, that under symmetric normalization, these quantities are not equal. The total inertia is dened as the weighted sum of all squared distances to the origin divided by the total over all cells, where the weights are the masses. Rows with a small mass inuence the inertia only when they are far from the centroid. Rows with a large mass inuence the total inertia, even when they are located close to the centroid. The same applies to columns. 284 Chapter 12 Row and Column Scores The row and column scores are the coordinates of the row and column points in the biplot. Figure 12-11 Row scores (symmetrical normalization) Figure 12-12 Column scores (symmetrical normalization) The column scores are related to the row scores via the proles and singular value (from the inertia per dimension table). Specically, the row scores are the matrix product of the row proles and column scores, scaled by the singular value for each dimension. For example, the score of 0.126 for senior managers on the rst dimension equals: For row principal normalization, the singular value does not gure into this equation. The row points are in the weighted centroid of the active column points, where the weights correspond to the entries in the row proles table. When the row points are the weighted average of the column points and the maximum dimensionality is used, the 285 Correspondence Analysis Euclidean distance between a row point and the origin equals the chi-square distance between the row and the average row, which in turn is equal to the inertia of a row. Because the chi-square statistic is equivalent to the total inertia times the sum of all cells of the correspondence table, you can think of the orientation of the row points as a pictorial representation of the chi-square statistic. A corresponding interpretation exists for column principal normalization but not for symmetrical. Contributions It is possible to compute the inertia displayed by a particular dimension. The scores on each dimension correspond to an orthogonal projection of the point onto that dimension. Thus, the inertia for a dimension equals the weighted sum of the squared distances from the scores on the dimension to the origin. However, whether this applies to row or column scores (or both) depends on the normalization method used. Each row and column point contributes to the inertia. Row and column points that contribute substantially to the inertia of a dimension are important to that dimension. The contribution of a point to the inertia of a dimension is the weighted squared distance from the projected point to the origin divided by the inertia for the dimension. The diagnostics that measure the contributions of points are an important aid in the interpretation of a correspondence analysis solution. Dominant points in the solution can easily be detected. For example, senior employees and junior employees are dominant points in the rst dimension, contributing 84% of the inertia. Among the column points, none contributes 65% of the inertia for the rst dimension alone. The contribution of a point to the inertia of the dimensions depends on both the mass and the distance from the origin. Points that are far from the origin and have a large mass contribute most to the inertia of the dimension. Because supplementary points do not play any part in dening the solution, they do not contribute to the inertia of the dimensions. In addition to examining the contribution of the points to the inertia per dimension, you can examine the contribution of the dimensions to the inertia per point. You can examine how the inertia of a point is spread over the dimensions by computing the percentage of the point inertia contributed by each dimension. Notice that the contributions of the dimensions to the point inertias do not all sum to one. In a reduced space, the inertia that is contributed by the higher dimensions is not represented. Using the maximum dimensionality would reveal the unaccounted inertia amounts. 286 Chapter 12 The rst two dimensions contribute all of the inertia for senior employees and junior employees and virtually all of the inertia for junior managers and secretaries. For senior managers, 11% of the inertia is not contributed by the rst two dimensions. Two dimensions contribute a very large proportion of the inertia of the row points. Similar results occur for the column points. For every active column point, two dimensions contribute at least 98% of the inertia. The third dimension contributes very little to these points. Permutations of the Correspondence Table Sometimes it is useful to order the categories of the rows and the columns. For example, you might have reason to believe that the categories of a variable correspond to a certain order, but you dont know the precise order. This ordination problem is found in various disciplinesthe seriation problem in archaeology, the ordination problem in phytosociology, and Guttmans scalogram problem in the social sciences. Ordering can be achieved by taking the row and column scores as ordering variables. If you have row and column scores in p dimensions, p permuted tables can be made. When the rst singular value is large, the rst table will show a particular structure, with larger-than-expected relative frequencies close to the diagonal. The following table shows the permutation of the correspondence table along the rst dimension. Looking at the row scores for dimension 1, you can see that the ranking from lowest to highest is senior employees (0.728), secretaries (0.385), senior managers (0.126), junior employees (0.446), and junior managers (0.495). Looking at the column scores for dimension 1, you see that the ranking is none, light, medium, and then heavy. These rankings are reected in the ordering of the rows and columns of the table. Figure 12-13 Permutation of the correspondence table 287 Correspondence Analysis Confidence Statistics Assuming that the table to be analyzed is a frequency table and that the data are a random sample from an unknown population, the cell frequencies follow a multinomial distribution. From this, it is possible to compute the standard deviations and correlations of the singular values, row scores, and column scores. In a one-dimensional correspondence analysis solution, you can compute a condence interval for each score in the population. If the standard deviation is large, correspondence analysis is very uncertain of the location of the point in the population. On the other hand, if the standard deviation is small, then the correspondence analysis is fairly certain that this point is located very close to the point given by the solution. In a multidimensional solution, if the correlation between dimensions is large, it may not be possible to locate a point in the correct dimension with much certainty. In such cases, multivariate condence intervals must be calculated using the variance/covariance matrix that can be written to a le. The condence statistics for the row and column scores are shown. The standard deviations for the two manager categories are larger than the others, likely due to their relatively small numbers. The standard deviation for heavy smokers is also larger for the same reason. If you look at the correlations between the dimensions for the scores, you see that the correlations are generally small for the row and column scores with the exception of junior employees, with a correlation of 0.611. Figure 12-14 Confidence statistics for row scores 288 Chapter 12 Figure 12-15 Confidence statistics for column scores Supplementary Profiles In correspondence analysis, additional categories can be represented in the space describing the relationships between the active categories. A supplementary prole denes a prole across categories of either the row or column variable and does not inuence the analysis in any way. The data le contains one supplementary row and two supplementary columns. The national average of people in each smoking category denes a supplementary row prole. The two supplementary columns dene two column proles across the categories of staff. The supplementary proles dene a point in either the row space or the column space. Because you will focus on both the rows and the columns separately, you will use principal normalization. 289 Correspondence Analysis Running the Analysis Figure 12-16 Define Row Range dialog box E To add the supplementary categories and obtain a principal normalization solution, recall the Correspondence Analysis dialog box. E Select staff and click Define Range. E Type 6 as the maximum value and click Update. E Select 6 in the Category Constraints list and select Category is supplemental. E Click Continue. E Select smoke and click Define Range in the Correspondence Analysis dialog box. 290 Chapter 12 Figure 12-17 Define Column Range dialog box E Type 6 as the maximum value and click Update. E Select 5 in the Category Constraints list and select Category is supplemental. E Select 6 in the Category Constraints list and select Category is supplemental. E Click Continue. E Click Model in the Correspondence Analysis dialog box. 291 Correspondence Analysis Figure 12-18 Model dialog box E Select Principal as the normalization method. E Click Continue. E Click Plots in the Correspondence Analysis dialog box. 292 Chapter 12 Figure 12-19 Plots dialog box E Select Row points and Column points in the Scatterplots group. E Click Continue. E Click OK in the Correspondence Analysis dialog box. The row points plot shows the rst two dimensions for the row points with the supplementary point for National Average. National Average lies far from the origin, indicating that the sample is not representative of the nation in terms of smoking behavior. Secretaries and senior employees are close to the national average, whereas 293 Correspondence Analysis junior managers are not. Thus, secretaries and senior employees have smoking behaviors similar to the national average, but junior managers do not. Figure 12-20 Row points (principal normalization) The column points plot displays the column space with the two supplementary points for alcohol consumption. Alcohol lies near the origin, indicating a close correspondence between the alcohol prole and the average column prole. However, No Alcohol differs from the average column prole, illustrated by the large distance from the origin. The closest point to No Alcohol is Light. The light smokers prole is most similar to the nondrinkers. Among the smokers, Medium is next closest and Heavy is farthest. Thus, there is a progression in similarity to nondrinking from light 294 Chapter 12 to heavy smoking. However, the relatively high proportion of secretaries in the No Alcohol group prevents any close correspondence to any of the smoking categories. Figure 12-21 Column points (principal normalization) Example: Perceptions of Coffee Brands The previous example involved a small table of hypothetical data. Actual applications often involve much larger tables. In this example, you will use data pertaining to perceived images of six iced-coffee brands (Kennedy, Riquier, and Sharp, 1996). This dataset can be found in coffee.sav. For more information, see Sample Files in Appendix A on p. 406. For each of 23 iced-coffee image attributes, people selected all brands that were described by the attribute. The six brands are denoted as AA, BB, CC, DD, EE, and FF to preserve condentiality. Table 12-1 Iced-coffee attributes Image attribute good hangover cure low fat/calorie brand Label cure low fat Image attribute fattening brand appeals to men Label fattening men 295 Correspondence Analysis Image attribute brand for children working class brand rich/sweet brand unpopular brand brand for fat/ugly people very fresh brand for yuppies nutritious brand brand for women minor brand Label children working sweet unpopular ugly fresh yuppies nutritious women minor Image attribute South Australian brand traditional/old fashioned brand premium quality brand healthy brand high caffeine brand new brand brand for attractive people tough brand popular brand Label South Australian traditional premium healthy caffeine new attractive tough popular Initially, you will focus on how the attributes are related to each other and how the brands are related to each other. Using principal normalization spreads the total inertia once over the rows and once over the columns. Although this prevents biplot interpretation, the distances between the categories for each variable can be examined. Running the Analysis E The setup of the data requires that the cases be weighted by the variable freq. To do this, from the menus choose: Data Weight Cases... 296 Chapter 12 Figure 12-22 Weight Cases dialog box E Weight cases by freq. E Click OK. E To obtain an initial solution in ve dimensions with principal normalization, from the menus choose: Analyze Dimension Reduction Correspondence Analysis... Figure 12-23 Correspondence Analysis dialog box E Select image as the row variable. 297 Correspondence Analysis E Click Define Range. Figure 12-24 Define Row Range dialog box E Type 1 as the minimum value. E Type 23 as the maximum value. E Click Update. E Click Continue. E Select brand as the column variable. E Click Define Range in the Correspondence Analysis dialog box. 298 Chapter 12 Figure 12-25 Define Column Range dialog box E Type 1 as the minimum value. E Type 6 as the maximum value. E Click Update. E Click Continue. E Click Model in the Correspondence Analysis dialog box. 299 Correspondence Analysis Figure 12-26 Model dialog box E Select Principal as the normalization method. E Click Continue. E Click Plots in the Correspondence Analysis dialog box. 300 Chapter 12 Figure 12-27 Plots dialog box E Select Row points and Column points in the Scatterplots group. E Click Continue. E Click OK in the Correspondence Analysis dialog box. Dimensionality The inertia per dimension shows the decomposition of the total inertia along each dimension. Two dimensions account for 83% of the total inertia. Adding a third dimension adds only 8.6% to the accounted-for inertia. Thus, you elect to use a two-dimensional representation. 301 Correspondence Analysis Figure 12-28 Inertia per dimension Contributions The row points overview shows the contributions of the row points to the inertia of the dimensions and the contributions of the dimensions to the inertia of the row points. If all points contributed equally to the inertia, the contributions would be 0.043. Healthy and low fat both contribute a substantial portion to the inertia of the rst dimension. Men and tough contribute the largest amounts to the inertia of the second dimension. Both ugly and fresh contribute very little to either dimension. 302 Chapter 12 Figure 12-29 Attribute contributions Two dimensions contribute a large amount to the inertia for most row points. The large contributions of the rst dimension to healthy, new, attractive, low fat, nutritious, and women indicate that these points are very well represented in one dimension. Consequently, the higher dimensions contribute little to the inertia of these points, which will lie very near the horizontal axis. The second dimension contributes most to men, premium, and tough. Both dimensions contribute very little to the inertia for South Australian and ugly, so these points are poorly represented. 303 Correspondence Analysis The column points overview displays the contributions involving the column points. Brands CC and DD contribute the most to the rst dimension, whereas EE and FF explain a large amount of the inertia for the second dimension. AA and BB contribute very little to either dimension. Figure 12-30 Brand contributions In two dimensions, all brands but BB are well represented. CC and DD are represented well in one dimension. The second dimension contributes the largest amounts for EE and FF. Notice that AA is represented well in the rst dimension but does not have a very high contribution to that dimension. Plots The row points plot shows that fresh and ugly are both very close to the origin, indicating that they differ little from the average row prole. Three general classications emerge. Located in the upper left of the plot, tough, men, and working are all similar to each other. The lower left contains sweet, fattening, children, and premium. In contrast, healthy, low fat, nutritious, and new cluster on the right side of the plot. 304 Chapter 12 Figure 12-31 Plot of image attributes (principal normalization) 305 Correspondence Analysis Notice in the column points plot that all brands are far from the origin, so no brand is similar to the overall centroid. Brands CC and DD group together at the right, whereas brands BB and FF cluster in the lower half of the plot. Brands AA and EE are not similar to any other brand. Figure 12-32 Plot of brands (principal normalization) Symmetrical Normalization How are the brands related to the image attributes? Principal normalization cannot address these relationships. To focus on how the variables are related to each other, use symmetrical normalization. Rather than spread the inertia twice (as in principal normalization), symmetrical normalization divides the inertia equally over both the rows and columns. Distances between categories for a single variable cannot be interpreted, but distances between the categories for different variables are meaningful. 306 Chapter 12 Figure 12-33 Model dialog box E To produce the following solution with symmetrical normalization, recall the Correspondence Analysis dialog box and click Model. E Select Symmetrical as the normalization method. E Click Continue. E Click OK in the Correspondence Analysis dialog box. 307 Correspondence Analysis In the upper left of the resulting biplot, brand EE is the only tough, working brand and appeals to men. Brand AA is the most popular and also viewed as the most highly caffeinated. The sweet, fattening brands include BB and FF. Brands CC and DD, while perceived as new and healthy, are also the most unpopular. Figure 12-34 Biplot of the brands and the attributes (symmetrical normalization) For further interpretation, you can draw a line through the origin and the two image attributes men and yuppies, and project the brands onto this line. The two attributes are opposed to each other, indicating that the association pattern of brands for men is reversed compared to the pattern for yuppies. That is, men are most frequently associated with brand EE and least frequently with brand CC, whereas yuppies are most frequently associated with brand CC and least frequently with brand EE. Example: Flying Mileage between Cities Correspondence analysis is not restricted to frequency tables. The entries can be any positive measure of correspondence. In this example, you use the ying mileages between 10 American cities. This dataset can be found in ying.sav. For more information, see Sample Files in Appendix A on p. 406. 308 Chapter 12 Table 12-2 City labels City Atlanta Chicago Denver Houston Los Angeles Label Atl Chi Den Hou LA City Miami New York San Francisco Seattle Washington, DC Label Mia NY SF Sea DC E To view the ying mileages, rst weight the cases by the variable dist. From the menus choose: Data Weight Cases... Figure 12-35 Weight Cases dialog box E Weight cases by dist. E Click OK. E Now, to view the mileages as a crosstabulation, from the menus choose: Analyze Descriptive Statistics Crosstabs... 309 Correspondence Analysis Figure 12-36 Crosstabs dialog box E Select row as the row variable. E Select col as the column variable. E Click OK. The following table contains the ying mileages between the cities. Notice that there is only one variable for both rows and columns and that the table is symmetric; the distance from Los Angeles to Miami is the same as the distance from Miami to Los 310 Chapter 12 Angeles. Moreover, the distance between any city and itself is 0. The active margin reects the total ying mileage from each city to all other cities. Figure 12-37 Flying mileages between 10 American cities In general, distances are dissimilarities; large values indicate a large difference between the categories. However, correspondence analysis requires an association measure; thus, you need to convert dissimilarities into similarities. In other words, a large table entry must correspond to a small difference between the categories. Subtracting every table entry from the largest table entry converts the dissimilarities into similarities. E To create the similarities and store them in a new variable, sim, from the menus choose: Transform Compute Variable... 311 Correspondence Analysis Figure 12-38 Compute Variable dialog box E Type sim as the target variable. E Type 2734-dist as the numeric expression. E Click OK. 312 Chapter 12 Figure 12-39 Weight Cases dialog box Now reweight the cases by the similarity measure by recalling the Weight Cases dialog box: E Weight cases by sim. E Click OK. E Finally, to obtain a correspondence analysis for the similarities, from the menus choose: Analyze Dimension Reduction Correspondence Analysis... 313 Correspondence Analysis Figure 12-40 Correspondence Analysis dialog box E Select row as the row variable. E Click Define Range. Figure 12-41 Define Row Range dialog box E Type 1 as the minimum value. E Type 10 as the maximum value. 314 Chapter 12 E Click Update. E Click Continue. E Select col as the column variable. E Click Define Range in the Correspondence Analysis dialog box. Figure 12-42 Define Column Range dialog box E Type 1 as the minimum value. E Type 10 as the maximum value. E Click Update. E Click Continue. E Click Model in the Correspondence Analysis dialog box. 315 Correspondence Analysis Figure 12-43 Model dialog box E Select Principal as the normalization method. E Click Continue. E Click Plots in the Correspondence Analysis dialog box. 316 Chapter 12 Figure 12-44 Plots dialog box E Select Row points in the Scatterplots group. E Click Continue. E Click OK in the Correspondence Analysis dialog box. Correspondence Table The new distance of 0 between Seattle and Miami indicates that they are most distant (least similar), whereas the distance of 2529 between New York and Washington, D.C., indicates that they are the least distant (most similar) pair of cities. 317 Correspondence Analysis Figure 12-45 Correspondence table for similarities Row and Column Scores By using ying mileages instead of driving mileages, the terrain of the United States does not impact the distances. Consequently, all similarities should be representable in two dimensions. You center both the rows and columns and use principal normalization. Because of the symmetry of the correspondence table and the principal 318 Chapter 12 normalization, the row and column scores are equal and the total inertia is in both, so it does not matter whether you inspect the row or column scores. Figure 12-46 Points for 10 cities The locations of the cities are very similar to their actual geographical locations, rotated about the origin. Cities that are further south have larger values along the second dimension, whereas cities that are further west have larger values along the rst dimension. Recommended Readings See the following texts for more information on correspondence analysis: Fisher, R. A. 1938. Statistical methods for research workers. Edinburgh: Oliver and Boyd. Fisher, R. A. 1940. The precision of discriminant functions. Annals of Eugenics, 10, 422429. 319 Correspondence Analysis Gilula, Z., and S. J. Haberman. 1988. The analysis of multivariate contingency tables by restricted canonical and restricted association models. Journal of the American Statistical Association, 83, 760771. Multiple Correspondence Analysis 13 Chapter The purpose of multiple correspondence analysis, also known as homogeneity analysis, is to nd quantications that are optimal in the sense that the categories are separated from each other as much as possible. This implies that objects in the same category are plotted close to each other and objects in different categories are plotted as far apart as possible. The term homogeneity also refers to the fact that the analysis will be most successful when the variables are homogeneous; that is, when they partition the objects into clusters with the same or similar categories. Example: Characteristics of Hardware To explore how multiple correspondence analysis works, you will use data from Hartigan (Hartigan, 1975), which can be found in screws.sav. For more information, see Sample Files in Appendix A on p. 406. This dataset contains information on the characteristics of screws, bolts, nuts, and tacks. The following table shows the variables, along with their variable labels, and the value labels assigned to the categories of each variable in the Hartigan hardware dataset. Table 13-1 Hartigan hardware dataset Variable name thread head indhead bottom length Variable label Thread Head form Indentation of head Bottom shape Length in half inches Value label Yes_Thread, No_Thread Flat, Cup, Cone, Round, Cylinder None, Star, Slit sharp, at 1/2_in, 1_in, 1_1/2_ in, 2_in, 2_1/2_in 320 321 Multiple Correspondence Analysis Variable name brass object Variable label Brass Object Value label Yes_Br, Not_Br tack, nail1, nail2, nail3, nail4, nail5, nail6, nail7, nail8, screw1, screw2, screw3, screw4, screw5, bolt1, bolt2, bolt3, bolt4, bolt5, bolt6, tack1, tack2, nailb, screwb Running the Analysis E To obtain a Multiple Correspondence Analysis, from the menus choose: Analyze Dimension Reduction Optimal Scaling... Figure 13-1 Optimal Scaling dialog box E Make sure All variables multiple nominal and One set are selected, and click Define. 322 Chapter 13 Figure 13-2 Multiple Correspondence Analysis dialog box E Select Thread through Length in half-inches as analysis variables. E Select object as a labeling variable. E Click Object in the Plots group. 323 Multiple Correspondence Analysis Figure 13-3 Object Plots dialog box E Choose to label objects by Variable. E Select thread through object as labeling variables. E Click Continue, and then click Variable in the Plots group of the Multiple Correspondence Analysis dialog box. 324 Chapter 13 Figure 13-4 Variable Plots dialog box E Choose to produce a joint category plot for thread through length. E Click Continue. E Click OK in the Multiple Correspondence Analysis dialog box. 325 Multiple Correspondence Analysis Model Summary Homogeneity analysis can compute a solution for several dimensions. The maximum number of dimensions equals either the number of categories minus the number of variables with no missing data or the number of observations minus one, whichever is smaller. However, you should rarely use the maximum number of dimensions. A smaller number of dimensions is easier to interpret, and after a certain number of dimensions, the amount of additional association accounted for becomes negligible. A one-, two-, or three-dimensional solution in homogeneity analysis is very common. Figure 13-5 Model summary Nearly all of the variance in the data is accounted for by the solution, 62.1% by the rst dimension and 36.8% by the second. The two dimensions together provide an interpretation in terms of distances. If a variable discriminates well, the objects will be close to the categories to which they belong. Ideally, objects in the same category will be close to each other (that is, they should have similar scores), and categories of different variables will be close if they belong to the same objects (that is, two objects that have similar scores for one variable should also score close to each other for the other variables in the solution). Object Scores After examining the model summary, you should look at the object scores. You can specify one or more variables to label the object scores plot. Each labeling variable produces a separate plot labeled with the values of that variable. Well take a look at the plot of object scores labeled by the variable object. This is just a case-identication variable and was not used in any computations. 326 Chapter 13 The distance from an object to the origin reects variation from the average response pattern. This average response pattern corresponds to the most frequent category for each variable. Objects with many characteristics corresponding to the most frequent categories lie near the origin. In contrast, objects with unique characteristics are located far from the origin. Figure 13-6 Object scores plot labeled by object Examining the plot, you see that the rst dimension (the horizontal axis) discriminates the screws and bolts (which have threads) from the nails and tacks (which dont have threads). This is easily seen on the plot since screws and bolts are on one end of the horizontal axis and tacks and nails are on the other. To a lesser extent, the rst dimension also separates the bolts (which have at bottoms) from all the others (which have sharp bottoms). The second dimension (the vertical axis) seems to separate SCREW1 and NAIL6 from all other objects. What SCREW1 and NAIL6 have in common are their values on variable lengththey are the longest objects in the data. Moreover, SCREW1 lies much farther from the origin than the other objects, suggesting that, taken as a whole, many of the characteristics of this object are not shared by the other objects. The object scores plot is particularly useful for spotting outliers. SCREW1 might be considered an outlier. Later, well consider what happens if you drop this object. 327 Multiple Correspondence Analysis Discrimination Measures Before examining the rest of the object scores plots, lets see if the discrimination measures agree with what weve said so far. For each variable, a discrimination measure, which can be regarded as a squared component loading, is computed for each dimension. This measure is also the variance of the quantied variable in that dimension. It has a maximum value of 1, which is achieved if the object scores fall into mutually exclusive groups and all object scores within a category are identical. (Note: This measure may have a value greater than 1 if there are missing data.) Large discrimination measures correspond to a large spread among the categories of the variable and, consequently, indicate a high degree of discrimination between the categories of a variable along that dimension. The average of the discrimination measures for any dimension equals the percentage of variance accounted for that dimension. Consequently, the dimensions are ordered according to average discrimination. The rst dimension has the largest average discrimination, the second dimension has the second largest average discrimination, and so on, for all dimensions in the solution. Figure 13-7 Plot of discrimination measures 328 Chapter 13 As noted on the object scores plot, the discrimination measures plot shows that the rst dimension is related to variables Thread and Bottom shape. These variables have large discrimination measures on the rst dimension and small discrimination measures on the second dimension. Thus, for both of these variables, the categories are spread far apart along the rst dimension only. Length in half-inches has a large value on the second dimension but a small value on the rst dimension. As a result, length is closest to the second dimension, agreeing with the observation from the object scores plot that the second dimension seems to separate the longest objects from the rest. Indentation of head and Head form have relatively large values on both dimensions, indicating discrimination in both the rst and second dimensions. The variable Brass, located very close to the origin, does not discriminate at all in the rst two dimensions. This makes sense, since all of the objects can be made of brass or not made of brass. Category Quantifications Recall that a discrimination measure is the variance of the quantied variable along a particular dimension. The discrimination measures plot contains these variances, indicating which variables discriminate along which dimension. However, the same variance could correspond to all of the categories being spread moderately far apart or to most of the categories being close together, with a few categories differing from this group. The discrimination plot cannot differentiate between these two conditions. Category quantication plots provide an alternative method of displaying discrimination of variables that can identify category relationships. In this plot, the coordinates of each category on each dimension are displayed. Thus, you can determine which categories are similar for each variable. 329 Multiple Correspondence Analysis Figure 13-8 Category quantifications Length in half-inches has ve categories, three of which group together near the top of the plot. The remaining two categories are in the lower half of the plot, with the 2_1/2_in category very far from the group. The large discrimination for length along dimension 2 is a result of this one category being very different from the other categories of length. Similarly, for Head form, the category STAR is very far from the other categories and yields a large discrimination measure along the second dimension. These patterns cannot be illustrated in a plot of discrimination measures. The spread of the category quantications for a variable reects the variance and thus indicates how well that variable is discriminated in each dimension. Focusing on dimension 1, the categories for Thread are far apart. However, along dimension 2, the categories for this variable are very close. Thus, Thread discriminates better in dimension 1 than in dimension 2. In contrast, the categories for Head form are spread far apart along both dimensions, suggesting that this variable discriminates well in both dimensions. In addition to determining the dimensions along which a variable discriminates and how that variable discriminates, the category quantication plot also compares variable discrimination. A variable with categories that are far apart discriminates better than a variable with categories that are close together. For example, along dimension 1, 330 Chapter 13 the two categories of Brass are much closer to each other than the two categories of Thread, indicating that Thread discriminates better than Brass along this dimension. However, along dimension 2, the distances are very similar, suggesting that these variables discriminate to the same degree along this dimension. The discrimination measures plot discussed previously identies these same relationships by using variances to reect the spread of the categories. A More Detailed Look at Object Scores A greater insight into the data can be gained by examining the object scores plots labeled by each variable. Ideally, similar objects should form exclusive groups, and these groups should be far from each other. Figure 13-9 Object scores labeled with Thread The plot labeled with Thread shows that the rst dimension separates Yes_Thread and No_Thread perfectly. All of the objects with threads have negative object scores, whereas all of the nonthreaded objects have positive scores. Although the two categories do not form compact groups, the perfect differentiation between the categories is generally considered a good result. 331 Multiple Correspondence Analysis Figure 13-10 Object scores labeled with Head form The plot labeled with Head form shows that this variable discriminates in both dimensions. The FLAT objects group together in the lower right corner of the plot, whereas the CUP objects group together in the upper right. CONE objects all lie in the upper left. However, these objects are more spread out than the other groups and, thus, are not as homogeneous. Finally, CYLINDER objects cannot be separated from ROUND objects, both of which lie in the lower left corner of the plot. 332 Chapter 13 Figure 13-11 Object scores labeled with Length in half-inches The plot labeled with Length in half-inches shows that this variable does not discriminate in the rst dimension. Its categories display no grouping when projected onto a horizontal line. However, Length in half-inches does discriminate in the second dimension. The shorter objects correspond to positive scores, and the longer objects correspond to large negative scores. 333 Multiple Correspondence Analysis Figure 13-12 Object scores labeled with Brass The plot labeled with Brass shows that this variable has categories that cannot be separated very well in the rst or second dimensions. The object scores are widely spread throughout the space. The brass objects cannot be differentiated from the nonbrass objects. Omission of Outliers In homogeneity analysis, outliers are objects that have too many unique features. As noted earlier, SCREW1 might be considered an outlier. To delete this object and run the analysis again, from the menus choose: Data Select Cases... 334 Chapter 13 Figure 13-13 Select Cases dialog box E Select If condition is satisfied. E Click If. 335 Multiple Correspondence Analysis Figure 13-14 If dialog box E Type object ~= 16 as the condition. E Click Continue. E Click OK in the Select Cases dialog box. E Finally, recall the Multiple Correspondence Analysis dialog box, and click OK. Figure 13-15 Model summary (outlier removed) The eigenvalues shift slightly. The rst dimension now accounts for a little more of the variance. 336 Chapter 13 Figure 13-16 Discrimination measures As shown in the discrimination plot, Indentation of head no longer discriminates in the second dimension, whereas Brass changes from no discrimination in either dimension to discrimination in the second dimension. Discrimination for the other variables is largely unchanged. 337 Multiple Correspondence Analysis Figure 13-17 Object scores labeled with Brass (outlier removed) The object scores plot labeled by Brass shows that the four brass objects all appear near the bottom of the plot (three objects occupy identical locations), indicating high discrimination along the second dimension. As was the case for Thread in the previous analysis, the objects do not form compact groups, but the differentiation of objects by categories is perfect. 338 Chapter 13 Figure 13-18 Object scores labeled with Indentation of head (outlier removed) The object scores plot labeled by Indentation of head shows that the rst dimension discriminates perfectly between the non-indented objects and the indented objects, as in the previous analysis. In contrast to the previous analysis, however, the second dimension cannot now distinguish the two categories. Thus, the omission of SCREW1, which is the only object with a star-shaped head, dramatically affects the interpretation of the second dimension. This dimension now differentiates objects based on Brass, Head form, and Length in half-inches. Recommended Readings See the following texts for more information on multiple correspondence analysis: Benzcri, J. P. 1992. Correspondence analysis handbook. New York: Marcel Dekker. Guttman, L. 1941. The quantication of a class of attributes: A theory and method of scale construction. In: The Prediction of Personal Adjustment, P. Horst, ed. New York: Social Science Research Council, 319348. Meulman, J. J. 1982. Homogeneity analysis of incomplete data. Leiden: DSWO Press. 339 Multiple Correspondence Analysis Meulman, J. J. 1996. Fitting a distance model to homogeneous subsets of variables: Points of view analysis of categorical data. Journal of Classication, 13, 249266. Meulman, J. J., and W. J. Heiser. 1997. Graphical display of interaction in multiway contingency tables by use of homogeneity analysis. In: Visual Display of Categorical Data, M. Greenacre, and J. Blasius, eds. New York: Academic Press, 277296. Nishisato, S. 1984. Forced classication: A simple application of a quantication method. Psychometrika, 49, 2536. Tenenhaus, M., and F. W. Young. 1985. An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis, and other methods for quantifying categorical multivariate data. Psychometrika, 50, 91119. Van Rijckevorsel, J. 1987. The application of fuzzy coding and horseshoes in multiple correspondence analysis. Leiden: DSWO Press. Multidimensional Scaling 14 Chapter Given a set of objects, the goal of multidimensional scaling is to nd a representation of the objects in a low-dimensional space. This solution is found by using the proximities between the objects. The procedure minimizes the squared deviations between the original, possibly transformed, object proximities and their Euclidean distances in the low-dimensional space. The purpose of the low-dimensional space is to uncover relationships between the objects. By restricting the solution to be a linear combination of independent variables, you may be able to interpret the dimensions of the solution in terms of these variables. In the following example, you will see how 15 different kinship terms can be represented in three dimensions and how that space can be interpreted with respect to the gender, generation, and degree of separation of each of the terms. Example: An Examination of Kinship Terms Rosenberg and Kim (Rosenberg and Kim, 1975) set out to analyze 15 kinship terms (aunt, brother, cousin, daughter, father, granddaughter, grandfather, grandmother, grandson, mother, nephew, niece, sister, son, uncle). They asked four groups of college students (two female, two male) to sort these terms on the basis of similarities. Two groups (one female, one male) were asked to sort twice, with the second sorting based on a different criteria from the rst sort. Thus, a total of six sources were obtained, as outlined in the following table. Table 14-1 Source structure of kinship data Source 1 2 3 Gender Female Male Female Condition Single sort Single sort First sort Sample size 85 85 80 340 341 Multidimensional Scaling Source 4 5 6 Gender Female Male Male Condition Sample size 80 Second sort 80 First sort 80 Second sort Each source corresponds to a proximity matrix, whose cells are equal to the number of people in a source minus the number of times that the objects were partitioned together in that source. This dataset can be found in kinship_dat.sav. For more information, see Sample Files in Appendix A on p. 406. Choosing the Number of Dimensions It is up to you to decide how many dimensions the solution should have. The scree plot can help you make this decision. E To create a scree plot, from the menus choose: Analyze Scale Multidimensional Scaling (PROXSCAL)... 342 Chapter 14 Figure 14-1 Data Format dialog box E Select Multiple matrix sources in the Number of Sources group. E Click Define. 343 Multidimensional Scaling Figure 14-2 Multidimensional Scaling dialog box E Select Aunt through Uncle as proximities variables. E Select sourceid as the variable identifying the source. E Click Model. 344 Chapter 14 Figure 14-3 Model dialog box E Type 10 as the maximum number of dimensions. E Click Continue. E Click Restrictions in the Multidimensional Scaling dialog box. 345 Multidimensional Scaling Figure 14-4 Restrictions dialog box E Select Linear combination of independent variables. E Click File to select the source of the independent variables. E Select kinship_var.sav. 346 Chapter 14 Figure 14-5 Restrictions dialog box E Select gender, gener, and degree as restriction variables. Note that the variable gender has a user-missing value9 = missing (for cousin). The procedure treats this as a valid category. Thus, the default linear transformation is unlikely to be appropriate. Use a nominal transformation instead. 347 Multidimensional Scaling Figure 14-6 Restrictions dialog box E Select gender. E Select Nominal from the Independent Variable Transformations drop-down list. E Click Change. E Click Continue. E Click Plots in the Multidimensional Scaling dialog box. 348 Chapter 14 Figure 14-7 Plots dialog box E Select Stress in the Plots group. E Click Continue. E Click OK in the Multidimensional Scaling dialog box. 349 Multidimensional Scaling Figure 14-8 Scree plot The procedure begins with a 10-dimensional solution and works down to a 2-dimensional solution. The scree plot shows the normalized raw stress of the solution at each dimension. You can see from the plot that increasing the dimensionality from 2 to 3 and from 3 to 4 offers large improvements in the stress. After 4, the improvements are rather small. You will choose to analyze the data by using a 3-dimensional solution, because the results are easier to interpret. 350 Chapter 14 A Three-Dimensional Solution The independent variables gender, gener (generation), and degree (of separation) were constructed with the intention of using them to interpret the dimensions of the solution. The independent variables were constructed as follows: gender gener 1 = male, 2 = female, 9 = missing (for cousin) The number of generations from you if the term refers to your kin, with lower numbers corresponding to older generations. Thus, grandparents are 2, grandchildren are 2, and siblings are 0. The number of degrees of separation along your family tree. Thus, your parents are up 1 node, while your children are down 1 node. Your siblings are up 1 node to your parents and then down 1 node to them, for 2 degrees of separation. Your cousin is 4 degrees away2 up to your grandparents and then 2 down through your aunt/uncle to them. degree The external variables can be found in kinship_var.sav. Additionally, an initial conguration from an earlier analysis is supplied in kinship_ini.sav. For more information, see Sample Files in Appendix A on p. 406. 351 Multidimensional Scaling Running the Analysis Figure 14-9 Model dialog box E To obtain a three-dimensional solution, recall the Multidimensional Scaling dialog box and click Model. E Type 3 as the minimum and maximum number of dimensions. E Click Continue. E Click Options in the Multidimensional Scaling dialog box. 352 Chapter 14 Figure 14-10 Options dialog box E Select Custom as the initial conguration. E Select kinship_ini.sav as the le to read variables from. E Select dim01, dim02, and dim03 as variables. E Click Continue. E Click Plots in the Multidimensional Scaling dialog box. 353 Multidimensional Scaling Figure 14-11 Plots dialog box E Select Original vs. transformed proximities and Transformed independent variables. E Click Continue. E Click Output in the Multidimensional Scaling dialog box. 354 Chapter 14 Figure 14-12 Output dialog box E Select Input data, Stress decomposition, and Variable and dimension correlations. E Click Continue. E Click OK in the Multidimensional Scaling dialog box. Stress Measures The stress and t measures give an indication of how well the distances in the solution approximate the original distances. 355 Multidimensional Scaling Figure 14-13 Stress and fit measures Each of the four stress statistics measures the mist of the data, while the dispersion accounted for and Tuckers coefcient of congruence measure the t. Lower stress measures (to a minimum of 0) and higher t measures (to a maximum of 1) indicate better solutions. Figure 14-14 Decomposition of normalized raw stress The decomposition of stress helps you identify which sources and objects contribute the most to the overall stress of the solution. In this case, most of the stress among the sources is attributable to sources 1 and 2, while among the objects, most of the stress is attributable to Brother, Granddaughter, Grandfather, Grandmother, Grandson, and Sister. 356 Chapter 14 The two sources that are accountable for most of the stress are the two groups that sorted the terms only once. This information suggests that the students considered multiple factors when sorting the terms, and those students who were allowed to sort twice focused on a portion of those factors for the rst sort and then considered the remaining factors during the second sort. The objects that account for most of the stress are those objects with a degree of 2. These people are relations who are not part of the nuclear family (Mother, Father, Daughter, Son) but are nonetheless closer than other relations. This middle position could easily cause some differential sorting of these terms. Final Coordinates of the Common Space The common space plot gives a visual representation of the relationships between the objects. Figure 14-15 Common space coordinates Look at the nal coordinates for the objects in dimensions 1 and 3; this is the plot in the lower left corner of the scatterplot matrix. This plot shows that dimension 1 (on the x axis) is correlated with the variable gender, and dimension 3 (on the y axis) is correlated with gener. From left to right, you see that dimension 1 separates the female 357 Multidimensional Scaling and male terms, with the genderless term Cousin in the middle. From the bottom of the plot to the top, increasing values along the axis correspond to terms that are older. Now look at the nal coordinates for the objects in dimensions 2 and 3; this plot is the plot on the middle right side of the scatterplot matrix. From this plot, you can see that the second dimension (along the y axis) corresponds to the variable degree, with larger values along the axis corresponding to terms that are further from the nuclear family. A Three-Dimensional Solution with Nondefault Transformations The previous solution was computed by using the default ratio transformation for proximities and interval transformations for the independent variables gener and degree. The results are pretty good, but you may be able to do better by using other transformations. For example, the proximities, gener, and degree all have natural orderings, but they may be better modeled by an ordinal transformation than a linear transformation. 358 Chapter 14 Figure 14-16 Model dialog box E To rerun the analysis, scaling the proximities, gener, and degree at the ordinal level (keeping ties), recall the Multidimensional Scaling dialog box and click Model. E Select Ordinal as the proximity transformation. E Click Continue. E Click Restrictions in the Multidimensional Scaling dialog box. 359 Multidimensional Scaling Figure 14-17 Restrictions dialog box E Select gener and degree. E Select Ordinal (keep ties) from the Independent Variable Transformations drop-down list. E Click Change. E Click Continue. E Click OK in the Multidimensional Scaling dialog box. 360 Chapter 14 Transformation Plots The transformation plots are a good rst check to see whether the original transformations were appropriate. If the plots are approximately linear, the linear assumption is appropriate. If not, check the stress measures to see whether there is an improvement in t and check the common space plot to see whether the interpretation is more useful. The independent variables each obtain approximately linear transformations, so it may be appropriate to interpret them as numerical. However, the proximities do not obtain a linear transformation, so it is possible that the ordinal transformation is more appropriate for the proximities. Figure 14-18 Transformed proximities Stress Measures The stress for the current solution supports the argument for scaling the proximities at the ordinal level. 361 Multidimensional Scaling Figure 14-19 Stress and fit measures The normalized raw stress for the previous solution is 0.06234. Scaling the variables by using nondefault transformations halves the stress to 0.03137. Final Coordinates of the Common Space The common space plots offer essentially the same interpretation of the dimensions as the previous solution. Figure 14-20 Common space coordinates 362 Chapter 14 Discussion It is best to treat the proximities as ordinal variables, because there is great improvement in the stress measures. As a next step, you may want to untie the ordinal variablesthat is, allow equivalent values of the original variables to obtain different transformed values. For example, in the rst source, the proximities between Aunt and Son, and Aunt and Grandson, are 85. The tied approach to ordinal variables forces the transformed values of these proximities to be equivalent, but there is no particular reason for you to assume that they should be. In this case, allowing the proximities to become untied frees you from an unnecessary restriction. Recommended Readings See the following texts for more information on multidimensional scaling: Commandeur, J. J. F., and W. J. Heiser. 1993. Mathematical derivations in the proximity scaling (PROXSCAL) of symmetric data matrices. Leiden: Department of Data Theory, University of Leiden. De Leeuw, J., and W. J. Heiser. 1980. Multidimensional scaling with restrictions on the conguration. In: Multivariate Analysis, Vol. V, P. R. Krishnaiah, ed. Amsterdam: North-Holland, 501522. Heiser, W. J. 1981. Unfolding analysis of proximity data. Leiden: Department of Data Theory, University of Leiden. Heiser, W. J., and F. M. T. A. Busing. 2004. Multidimensional scaling and unfolding of symmetric and asymmetric proximity relations. In: Handbook of Quantitative Methodology for the Social Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: SagePublications, Inc., 2548. Kruskal, J. B. 1964. Multidimensional scaling by optimizing goodness of t to a nonmetric hypothesis. Psychometrika, 29, 128. Kruskal, J. B. 1964. Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115129. Shepard, R. N. 1962. The analysis of proximities: Multidimensional scaling with an unknown distance function I. Psychometrika, 27, 125140. 363 Multidimensional Scaling Shepard, R. N. 1962. The analysis of proximities: Multidimensional scaling with an unknown distance function II. Psychometrika, 27, 219246. Multidimensional Unfolding 15 Chapter The Multidimensional Unfolding procedure attempts to nd a common quantitative scale that allows you to visually examine the relationships between two sets of objects. Example: Breakfast Item Preferences In a classic study (Green and Rao, 1972), 21 Wharton School MBA students and their spouses were asked to rank 15 breakfast items in order of preference, from 1 = most preferred to 15 = least preferred. This information is collected in breakfast_overall.sav. For more information, see Sample Files in Appendix A on p. 406. The results of the study provide a typical example of the degeneracy problem inherent in most multidimensional unfolding algorithms that is solved by penalizing the coefcient of variation of the transformed proximities (Busing, Groenen, and Heiser, 2005). You will see a degenerate solution and will see how to solve the problem using Multidimensional Unfolding, allowing you to determine how individuals discriminate between breakfast items. Syntax for reproducing these analyses can be found in prefscal_breakfast-overall.sps. Producing a Degenerate Solution E To run a Multidimensional Unfolding analysis, from the menus choose: Analyze Scale Multidimensional Unfolding (PREFSCAL)... 364 365 Multidimensional Unfolding Figure 15-1 Multidimensional Unfolding main dialog box E Select Toast pop-up through Corn mufn and butter as proximities variables. E Click Options. 366 Chapter 15 Figure 15-2 Options dialog box E Select Spearman as the imputation method for the Classical start. E In the Penalty Term group, type 1.0 as the value of the Strength parameter and 0.0 as the value of the Range parameter. This turns off the penalty term. E Click Continue. E Click OK in the Multidimensional Unfolding dialog box. 367 Multidimensional Unfolding Following is the command syntax generated by these selections: PREFSCAL VARIABLES=TP BT EMM JD CT BMM HRB TMd BTJ TMn CB DP GD CC CMB /INITIAL=CLASSICAL (SPEARMAN) /TRANSFORMATION=NONE /PROXIMITIES=DISSIMILARITIES /CRITERIA=DIMENSIONS(2,2) DIFFSTRESS(.000001) MINSTRESS(.0001) MAXITER(5000) /PENALTY=LAMBDA(1.0) OMEGA(0.0) /PRINT=MEASURES COMMON /PLOT=COMMON . This syntax species an analysis on variables tp (Toast pop-up) through cmb (Corn mufn and butter). The INITIAL subcommand species that the starting values be imputed using Spearman distances. The specied values on the PENALTY subcommand essentially turn off the penalty term, and as a result, the procedure will minimize Kruskals Stress-I. This will result in a degenerate solution. The PLOT subcommand requests plots of the common space. All other parameters fall back to their default values. 368 Chapter 15 Measures Figure 15-3 Measures for degenerate solution The algorithm converges to a solution after 154 iterations, with a penalized stress (marked nal function value) of 0.0000990. Since the penalty term has been turned off, penalized stress is equal to Kruskals Stress-I (the stress part of the function value is equivalent to Kruskals badness-of-t measure). Low stress values generally indicate that the solution ts the data well, but there are several warning signs of a degenerate solution: The coefcient of variation for the transformed proximities is very small relative to the coefcient of variation for the original proximities. This suggests that the transformed proximities for each row are near-constant, and thus the solution will not provide any discrimination between objects. The sum-of-squares of DeSarbos intermixedness indices are a measure of how well the points of the different sets are intermixed. If they are not intermixed, this is a warning sign that the solution may be degenerate. The closer to 0, the 369 Multidimensional Unfolding more intermixed the solution. The reported value is very large, indicating that the solution is not intermixed. Shepards rough nondegeneracy index, which is reported as a percentage of different distances, is equal to 0. This is a clear numerical indication that there are insufciently different distances and that the solution is probably degenerate. Common Space Figure 15-4 Joint plot of common space for degenerate solution Visual conrmation that the solution is degenerate is found in the joint plot of the common space of row and column objects. The row objects (individuals) are situated on the circumference of a circle centered on the column objects (breakfast items), whose coordinates have collapsed to a single point. 370 Chapter 15 Running a Nondegenerate Analysis Figure 15-5 Options dialog box E To produce a nondegenerate solution, click the Dialog Recall tool and select Multidimensional Unfolding. E Click Options in the Multidimensional Unfolding dialog box. E In the Penalty Term group, type 0.5 as the value of the Strength parameter and 1.0 as the value of the Range parameter. This turns off the penalty term. 371 Multidimensional Unfolding E Click Continue. E Click OK in the Multidimensional Unfolding dialog box. Following is the command syntax generated by these selections: PREFSCAL VARIABLES=TP BT EMM JD CT BMM HRB TMd BTJ TMn CB DP GD CC CMB /INITIAL=CLASSICAL (SPEARMAN) /TRANSFORMATION=NONE /PROXIMITIES=DISSIMILARITIES /CRITERIA=DIMENSIONS(2,2) DIFFSTRESS(.000001) MINSTRESS(.0001) MAXITER(5000) /PENALTY=LAMBDA(0.5) OMEGA(1.0) /PRINT=MEASURES COMMON /PLOT=COMMON . The only change is on the PENALTY subcommand. LAMBDA has been set to 0.5, and OMEGA has been set to 1.0, their default values. 372 Chapter 15 Measures Figure 15-6 Measures for nondegenerate solution The problems noted in the measures for the degenerate solution have been corrected here. The normalized stress is no longer 0. The coefcient of variation for the transformed proximities now has a similar value to the coefcient of variation for the original proximities. DeSarbos intermixedness indices are much closer to 0, indicating that the solution is much better intermixed. Shepards rough nondegeneracy index, which is reported as a percentage of different distances, is now nearly 80%. There are sufciently different distances, and the solution is probably nondegenerate. 373 Multidimensional Unfolding Common Space Figure 15-7 Joint plot of common space for nondegenerate solution The joint plot of the common space allows for an interpretation of the dimensions. The horizontal dimension appears to discriminate between soft and hard bread or toast, with softer items as you move right along the axis. The vertical dimension does not have a clear interpretation, although it perhaps discriminates based on convenience, with more formal items as you move down along the axis. This creates several clusters of breakfast items. For example, the donuts, cinnamon buns, and Danish pastry form a cluster of soft and somewhat informal items. The mufns and cinnamon toast form a cluster of harder but more formal items. The other toasts and hard rolls form a cluster of hard and somewhat informal items. The toast pop-up is a hard item that is extremely informal. The individuals represented by the row objects are clearly split into clusters according to preference for hard or soft items, with considerable within-cluster variation along the vertical dimension. 374 Chapter 15 Example: Three-Way Unfolding of Breakfast Item Preferences In a classic study (Green et al., 1972), 21 Wharton School MBA students and their spouses were asked to rank 15 breakfast items in order of preference, from 1 = most preferred to 15 = least preferred. Their preferences were recorded under six different scenarios, from Overall preference to Snack, with beverage only. This information is collected in breakfast.sav. For more information, see Sample Files in Appendix A on p. 406. The six scenarios can be treated as separate sources. Use PREFSCAL to perform a three-way unfolding of the rows, columns, and sources. Syntax for reproducing these analyses can be found in prefscal_breakfast.sps. Running the Analysis E To run a Multidimensional Unfolding analysis, from the menus choose: Analyze Scale Multidimensional Unfolding (PREFSCAL)... 375 Multidimensional Unfolding Figure 15-8 Multidimensional Unfolding main dialog box E Select Toast pop-up through Corn mufn and butter as proximities variables. E Select Menu scenarios as the source variable. E Click Model. 376 Chapter 15 Figure 15-9 Model dialog box E Select Weighted Euclidean as the scaling model. E Click Continue. E Click Options in the Multidimensional Unfolding dialog box. 377 Multidimensional Unfolding Figure 15-10 Options dialog box E Select Spearman as the imputation method for the Classical start. E Click Continue. E Click Plots in the Multidimensional Unfolding dialog box. 378 Chapter 15 Figure 15-11 Plots dialog box E Select Individual spaces in the Plots group. E Click Continue. E Click OK in the Multidimensional Unfolding dialog box. 379 Multidimensional Unfolding Following is the command syntax generated by these selections: PREFSCAL VARIABLES=TP BT EMM JD CT BMM HRB TMd BTJ TMn CB DP GD CC CMB /INPUT=SOURCES(srcid ) /INITIAL=CLASSICAL (SPEARMAN) /CONDITION=ROW /TRANSFORMATION=NONE /PROXIMITIES=DISSIMILARITIES /MODEL=WEIGHTED /CRITERIA=DIMENSIONS(2,2) DIFFSTRESS(.000001) MINSTRESS(.0001) MAXITER(5000) /PENALTY=LAMBDA(0.5) OMEGA(1.0) /PRINT=MEASURES COMMON /PLOT=COMMON WEIGHTS INDIVIDUAL ( ALL ) . This syntax species an analysis on variables tp (Toast pop-up) through cmb (Corn mufn and butter). The variable srcid is used to identify the sources. The INITIAL subcommand species that the starting values be imputed using Spearman distances. The MODEL subcommand species a weighted Euclidean model, which allows each individual space to weight the dimensions of the common space differently. The PLOT subcommand requests plots of the common space, individual spaces, and individual space weights. All other parameters fall back to their default values. 380 Chapter 15 Measures Figure 15-12 Measures The algorithm converges after 481 iterations, with a nal penalized stress of 0.8199642. The variation coefcients and Shepards index are sufciently large, and DeSarbos indices are sufciently low, to suggest that there are no problems with degeneracy. 381 Multidimensional Unfolding Common Space Figure 15-13 Joint plot of common space The joint plot of the common space shows a nal conguration that is very similar to the two-way analysis on the overall preferences, with the solution ipped over the 45-degree line. Thus, the vertical dimension now appears to discriminate between soft and hard bread or toast, with softer items as you move up along the axis. The horizontal dimension now does not have a clear interpretation, though perhaps it discriminates based on convenience, with more formal items as you move left along the axis. The individuals represented by the row objects are still clearly split into clusters according to preference for hard or soft items, with considerable within-cluster variation along the horizontal dimension. 382 Chapter 15 Individual Spaces Figure 15-14 Dimension weights An individual space is computed for each source. The dimension weights show how the individual spaces load on the dimensions of the common space. A larger weight indicates a larger distance in the individual space and thus greater discrimination between the objects on that dimension for that individual space. Specicity is a measure of how different an individual space is from the common space. An individual space that was identical to the common space would have identical dimension weights and a specicity of 0, while an individual space that was specic to a particular dimension would have a single large dimension weight and a specicity of 1. In this case, the most divergent sources are Breakfast, with juice, bacon and eggs, and beverage, and Snack, with beverage only. Importance is a measure of the relative contribution of each dimension to the solution. In this case, the dimensions are equally important. 383 Multidimensional Unfolding Figure 15-15 Dimension weights The dimension weights chart provides a visualization of the weights table. Breakfast, with juice, bacon and eggs, and beverage and Snack, with beverage only are the nearest to the dimension axes, but neither are strongly specic to a particular dimension. 384 Chapter 15 Figure 15-16 Joint plot of individual space Breakfast, with juice, bacon and eggs, and beverage The joint plot of the individual space Breakfast, with juice, bacon and eggs, and beverage shows the effect of this scenario on the preferences. This source loads more heavily on the rst dimension, so the differentiation between items is mostly due to the rst dimension. 385 Multidimensional Unfolding Figure 15-17 Joint plot of individual space Snack, with beverage only The joint plot of the individual space Snack, with beverage only shows the effect of this scenario on the preferences. This source loads more heavily on the second dimension, so the differentiation between items is mostly due to the second dimension. However, there is still quite a bit of differentiation along the rst dimension because of the fairly low specicity of this source. Using a Different Initial Configuration The nal conguration can depend on the starting points given to the algorithm. Ideally, the general structure of the solution should remain the same; otherwise, it can be difcult to ascertain which is correct. However, details may come into sharper focus as you try different initial congurations, such as using a correspondence start on the three-way analysis of the breakfast data. E To produce a solution with a correspondence start, click the Dialog Recall tool and select Multidimensional Unfolding. 386 Chapter 15 E Click Options in the Multidimensional Unfolding dialog box. Figure 15-18 Options dialog box E Select Correspondence in the Initial Conguration group. E Click Continue. E Click OK in the Multidimensional Unfolding dialog box. 387 Multidimensional Unfolding Following is the command syntax generated by these selections: PREFSCAL VARIABLES=TP BT EMM JD CT BMM HRB TMd BTJ TMn CB DP GD CC CMB /INPUT=SOURCES(srcid ) /INITIAL=CORRESPONDENCE /TRANSFORMATION=NONE /PROXIMITIES=DISSIMILARITIES /CRITERIA=DIMENSIONS(2,2) DIFFSTRESS(.000001) MINSTRESS(.0001) MAXITER(5000) /PENALTY=LAMBDA(0.5) OMEGA(1.0) /PRINT=MEASURES COMMON /PLOT=COMMON WEIGHTS INDIVIDUAL ( ALL ) . The only change is on the INITIAL subcommand. The starting conguration has been set to CORRESPONDENCE, which uses the results of a correspondence analysis on the reversed data (similarities instead of dissimilarities), with a symmetric normalization of row and column scores. 388 Chapter 15 Measures Figure 15-19 Measures for correspondence initial configuration The algorithm converges after 385 iterations, with a nal penalized stress of 0.8140741. This statistic, the badness of t, the goodness of t, variation coefcients, and Shepards index are all very similar to those for the solution using the classical Spearman start. DeSarbos indices is somewhat different, with a value of 1.7571887 instead of 0.2199287, which suggests that the solution using the correspondence start is not as well mixed. To see how this affects the solution, look at the joint plot of the common space. 389 Multidimensional Unfolding Common Space Figure 15-20 Joint plot of common space for correspondence initial configuration The joint plot of the common space shows a nal conguration that is similar to the analysis with the classical Spearman initial conguration; however, the column objects (breakfast items) are situated around the row objects (individuals) rather than intermixed with them. 390 Chapter 15 Individual Spaces Figure 15-21 Dimension weights for correspondence initial configuration Under the correspondence initial conguration, each of the individual spaces has a higher specicity; that is, each situation under which the participants ranked the breakfast items is more strongly associated with a specic dimension. The most divergent sources are still Breakfast, with juice, bacon and eggs, and beverage, and Snack, with beverage only. 391 Multidimensional Unfolding Figure 15-22 Joint plot of individual space Breakfast, with juice, bacon and eggs, and beverage for correspondence initial configuration The higher specicity is evident in the joint plot of the individual space Breakfast, with juice, bacon and eggs, and beverage. The source loads even more heavily on the rst dimension than under the classical Spearman start, so the row and column objects show a little less variation on the vertical axis and a little more variation on the horizontal axis. 392 Chapter 15 Figure 15-23 Joint plot of individual space Snack, with beverage only for correspondence initial configuration The joint plot of the individual space Snack, with beverage only shows that the row and column objects lie more closely to a vertical line than under the classical Spearman start. Example: Examining Behavior-Situation Appropriateness In a classic example (Price and Bouffard, 1974), 52 students were asked to rate the combinations of 15 situations and 15 behaviors on a 10-point scale ranging from 0 = extremely appropriate to 9 = extremely inappropriate. Averaged over individuals, the values are taken as dissimilarities. This information is collected in behavior.sav. For more information, see Sample Files in Appendix A on p. 406. Use Multidimensional Unfolding to nd clusterings of similar situations and the behaviors with which they are most closely associated. Syntax for reproducing these analyses can be found in prefscal_behavior.sps. 393 Multidimensional Unfolding Running the Analysis E To run a Multidimensional Unfolding analysis, from the menus choose: Analyze Scale Multidimensional Unfolding (PREFSCAL)... Figure 15-24 Multidimensional Unfolding main dialog box E Select Run through Shout as proximities variables. E Select ROWID as the row variable. E Click Model. 394 Chapter 15 Figure 15-25 Model dialog box E Select Linear as the proximity transformation, and choose to Include intercept. E Choose to apply transformations Across all sources simultaneously. E Click Continue. E Click Options in the Multidimensional Unfolding dialog box. 395 Multidimensional Unfolding Figure 15-26 Options dialog box E Select Custom in the Initial Conguration group. E Browse to and choose behavior_ini.sav as the le containing the custom initial conguration. For more information, see Sample Files in Appendix A on p. 406. E Select dim1 and dim2 as the variables specifying the initial conguration. E Click Continue. E Click Plots in the Multidimensional Unfolding dialog box. 396 Chapter 15 Figure 15-27 Plots dialog box E Select Transformation plots in the Plots group. E Click Continue. E Click OK in the Multidimensional Unfolding dialog box. 397 Multidimensional Unfolding Following is the command syntax generated by these selections: PREFSCAL VARIABLES=Run Talk Kiss Write Eat Sleep Mumble Read Fight Belch Argue Jump Cry Laugh Shout /INPUT=ROWS(ROWID ) /INITIAL=( 'samplesDirectory/behavior_ini.sav' ) dim1 dim2 /CONDITION=UNCONDITIONAL /TRANSFORMATION=LINEAR (INTERCEPT) /PROXIMITIES=DISSIMILARITIES /MODEL=IDENTITY /CRITERIA=DIMENSIONS(2,2) DIFFSTRESS(.000001) MINSTRESS(.0001) MAXITER(5000) /PENALTY=LAMBDA(0.5) OMEGA(1.0) /PRINT=MEASURES COMMON /PLOT=COMMON TRANSFORMATIONS . This syntax species an analysis on variables run through shout. The variable rowid is used to identify the rows. The INITIAL subcommand species that the starting values be taken from the le behavior_ini.sav. The row and column coordinates are stacked, with the column coordinates following the row coordinates. The CONDITION subcommand species that all proximities can be compared with each other. This is true in this analysis, since you should be able to compare the proxmities for running in a park and running in church and see that one behavior is considered less appropriate than the other. The TRANSFORMATION subcommand species a linear transformation of the proximities, with intercept. This is appropriate if a 1-point difference in proximities is equivalent across the range of the 10-point scale. That is, if the students have assigned their scores so that the difference between 0 and 1 is the same as the difference between 5 and 6, then a linear transformation is appropriate. The PLOT subcommand requests plots of the common space and transformation plots. All other parameters fall back to their default values. 398 Chapter 15 Measures Figure 15-28 Measures The algorithm converges after 169 iterations, with a nal penalized stress of 0.6427725. The variation coefcients and Shepards index are sufciently large, and DeSarbos indices are sufciently low, to suggest that there are no problems with degeneracy. 399 Multidimensional Unfolding Common Space Figure 15-29 Joint plot of common space The horizontal dimension appears to be more strongly associated with the column objects (behaviors) and discriminates between inappropriate behaviors (ghting, belching) and more appropriate behaviors. The vertical dimension appears to be more strongly associated with the row objects (situations) and denes different situational-behavior restrictions. Toward the bottom of the vertical dimension are situations (church, class) that restrict behavior to the quieter/introspective types of behaviors (read, write). Thus, these behaviors are pulled down the vertical axis. Toward the top of the vertical dimension are situations (movies, game, date) that restrict behavior to the social/extroverted types of behaviors (eat, kiss, laugh). Thus, these behaviors are pulled up the vertical axis. At the center of the vertical dimension, situations are separated on the horizontal dimension based on the general restrictiveness of the situation. Those further from the behaviors (interview) are the most restricted, while those closer to the behaviors (room, park) are generally less restricted. 400 Chapter 15 Proximity Transformations Figure 15-30 Transformation plot The proximities were treated as linear in this analysis, so the plot of the transformed values versus the original proximities forms a straight line. The t of this solution is good, but perhaps a better t can be achieved with a different transformation of the proximities. Changing the Proximities Transformation (Ordinal) E To produce a solution with an ordinal transformation of the proximities, click the Dialog Recall tool and select Multidimensional Unfolding. 401 Multidimensional Unfolding E Click Model in the Multidimensional Unfolding dialog box. Figure 15-31 Model dialog box E Select Ordinal as the proximity transformation. E Click Continue. E Click OK in the Multidimensional Unfolding dialog box. Following is the command syntax generated by these selections: PREFSCAL VARIABLES=Run Talk Kiss Write Eat Sleep Mumble Read Fight Belch Argue Jump Cry Laugh Shout /INPUT=ROWS(ROWID ) /INITIAL=( 'samplesDirectory/behavior_ini.sav' ) dim1 dim2 /CONDITION=UNCONDITIONAL /TRANSFORMATION=ORDINAL (KEEPTIES) 402 Chapter 15 /PROXIMITIES=DISSIMILARITIES /MODEL=IDENTITY /CRITERIA=DIMENSIONS(2,2) DIFFSTRESS(.000001) MINSTRESS(.0001) MAXITER(5000) /PENALTY=LAMBDA(0.5) OMEGA(1.0) /PRINT=MEASURES COMMON /PLOT=COMMON TRANSFORMATIONS . The only change is on the TRANSFORMATION subcommand. The transformation has been set to ORDINAL, which preserves the order of proximities but does not require that the transformed values be proportional to the original values. Measures Figure 15-32 Measures for solution with ordinal transformation The algorithm converges after 268 iterations, with a nal penalized stress of 0.6044671. This statistic and the other measures are slightly better for this solution than the one with a linear transformation of the proximities. 403 Multidimensional Unfolding Common Space Figure 15-33 Joint plot of common space for solution with ordinal transformation The interpretation of the common space is the same under both solutions. Perhaps this solution (with the ordinal transformation) has relatively less variation on the vertical dimension than on the horizontal dimension than is evident in the solution with the linear transformation. 404 Chapter 15 Proximity Transformations Figure 15-34 Transformation plot for solution with ordinal transformation Aside from the values with the largest proximities, which bend up from the rest of the values, the ordinal transformation of proximities is fairly linear. These proximities likely account for most of the differences between the ordinal and linear solutions; however, there isnt enough information here to determine whether this nonlinear trend in the higher values is a true trend or an anomaly. Recommended Readings See the following texts for more information: Busing, F. M. T. A., P. J. F. Groenen, and W. J. Heiser. 2005. Avoiding degeneracy in multidimensional unfolding by penalizing on the coefcient of variation. Psychometrika, 70, 7198. 405 Multidimensional Unfolding Green, P. E., and V. Rao. 1972. Applied multidimensional scaling. Hinsdale, Ill.: Dryden Press. Price, R. H., and D. L. Bouffard. 1974. Behavioral appropriateness and situational constraints as dimensions of social behavior. Journal of Personality and Social Psychology, 30, 579586. Appendix Sample Files A The sample les installed with the product can be found in the Samples subdirectory of the installation directory. There is a separate folder within the Samples subdirectory for each of the following languages: English, French, German, Italian, Japanese, Korean, Polish, Russian, Simplied Chinese, Spanish, and Traditional Chinese. Not all sample les are available in all languages. If a sample le is not available in a language, that language folder contains an English version of the sample le. Descriptions Following are brief descriptions of the sample les used in various examples throughout the documentation. accidents.sav. This is a hypothetical data le that concerns an insurance company that is studying age and gender risk factors for automobile accidents in a given region. Each case corresponds to a cross-classication of age category and gender. adl.sav. This is a hypothetical data le that concerns efforts to determine the benets of a proposed type of therapy for stroke patients. Physicians randomly assigned female stroke patients to one of two groups. The rst received the standard physical therapy, and the second received an additional emotional therapy. Three months following the treatments, each patients abilities to perform common activities of daily life were scored as ordinal variables. advert.sav. This is a hypothetical data le that concerns a retailers efforts to examine the relationship between money spent on advertising and the resulting sales. To this end, they have collected past sales gures and the associated advertising costs.. 406 407 Sample Files aflatoxin.sav. This is a hypothetical data le that concerns the testing of corn crops for aatoxin, a poison whose concentration varies widely between and within crop yields. A grain processor has received 16 samples from each of 8 crop yields and measured the alfatoxin levels in parts per billion (PPB). aflatoxin20.sav. This data le contains the aatoxin measurements from each of the 16 samples from yields 4 and 8 from the aatoxin.sav data le. anorectic.sav. While working toward a standardized symptomatology of anorectic/bulimic behavior, researchers (Van der Ham, Meulman, Van Strien, and Van Engeland, 1997) made a study of 55 adolescents with known eating disorders. Each patient was seen four times over four years, for a total of 220 observations. At each observation, the patients were scored for each of 16 symptoms. Symptom scores are missing for patient 71 at time 2, patient 76 at time 2, and patient 47 at time 3, leaving 217 valid observations. autoaccidents.sav. This is a hypothetical data le that concerns the efforts of an insurance analyst to model the number of automobile accidents per driver while also accounting for driver age and gender. Each case represents a separate driver and records the drivers gender, age in years, and number of automobile accidents in the last ve years. band.sav. This data le contains hypothetical weekly sales gures of music CDs for a band. Data for three possible predictor variables are also included. bankloan.sav. This is a hypothetical data le that concerns a banks efforts to reduce the rate of loan defaults. The le contains nancial and demographic information on 850 past and prospective customers. The rst 700 cases are customers who were previously given loans. The last 150 cases are prospective customers that the bank needs to classify as good or bad credit risks. bankloan_binning.sav. This is a hypothetical data le containing nancial and demographic information on 5,000 past customers. behavior.sav. In a classic example (Price and Bouffard, 1974), 52 students were asked to rate the combinations of 15 situations and 15 behaviors on a 10-point scale ranging from 0=extremely appropriate to 9=extremely inappropriate. Averaged over individuals, the values are taken as dissimilarities. behavior_ini.sav. This data le contains an initial conguration for a two-dimensional solution for behavior.sav. 408 Appendix A brakes.sav. This is a hypothetical data le that concerns quality control at a factory that produces disc brakes for high-performance automobiles. The data le contains diameter measurements of 16 discs from each of 8 production machines. The target diameter for the brakes is 322 millimeters. breakfast.sav. In a classic study (Green and Rao, 1972), 21 Wharton School MBA students and their spouses were asked to rank 15 breakfast items in order of preference with 1=most preferred to 15=least preferred. Their preferences were recorded under six different scenarios, from Overall preference to Snack, with beverage only. breakfast-overall.sav. This data le contains the breakfast item preferences for the rst scenario, Overall preference, only. broadband_1.sav. This is a hypothetical data le containing the number of subscribers, by region, to a national broadband service. The data le contains monthly subscriber numbers for 85 regions over a four-year period. broadband_2.sav. This data le is identical to broadband_1.sav but contains data for three additional months. car_insurance_claims.sav. A dataset presented and analyzed elsewhere (McCullagh and Nelder, 1989) concerns damage claims for cars. The average claim amount can be modeled as having a gamma distribution, using an inverse link function to relate the mean of the dependent variable to a linear combination of the policyholder age, vehicle type, and vehicle age. The number of claims led can be used as a scaling weight. car_sales.sav. This data le contains hypothetical sales estimates, list prices, and physical specications for various makes and models of vehicles. The list prices and physical specications were obtained alternately from edmunds.com and manufacturer sites. carpet.sav. In a popular example (Green and Wind, 1973), a company interested in marketing a new carpet cleaner wants to examine the inuence of ve factors on consumer preferencepackage design, brand name, price, a Good Housekeeping seal, and a money-back guarantee. There are three factor levels for package design, each one differing in the location of the applicator brush; three brand names (K2R, Glory, and Bissell); three price levels; and two levels (either no or yes) for each of the last two factors. Ten consumers rank 22 proles dened by these factors. The variable Preference contains the rank of the average rankings for each prole. Low rankings correspond to high preference. This variable reects an overall measure of preference for each prole. 409 Sample Files carpet_prefs.sav. This data le is based on the same example as described for carpet.sav, but it contains the actual rankings collected from each of the 10 consumers. The consumers were asked to rank the 22 product proles from the most to the least preferred. The variables PREF1 through PREF22 contain the identiers of the associated proles, as dened in carpet_plan.sav. catalog.sav. This data le contains hypothetical monthly sales gures for three products sold by a catalog company. Data for ve possible predictor variables are also included. catalog_seasfac.sav. This data le is the same as catalog.sav except for the addition of a set of seasonal factors calculated from the Seasonal Decomposition procedure along with the accompanying date variables. cellular.sav. This is a hypothetical data le that concerns a cellular phone companys efforts to reduce churn. Churn propensity scores are applied to accounts, ranging from 0 to 100. Accounts scoring 50 or above may be looking to change providers. ceramics.sav. This is a hypothetical data le that concerns a manufacturers efforts to determine whether a new premium alloy has a greater heat resistance than a standard alloy. Each case represents a separate test of one of the alloys; the heat at which the bearing failed is recorded. cereal.sav. This is a hypothetical data le that concerns a poll of 880 people about their breakfast preferences, also noting their age, gender, marital status, and whether or not they have an active lifestyle (based on whether they exercise at least twice a week). Each case represents a separate respondent. clothing_defects.sav. This is a hypothetical data le that concerns the quality control process at a clothing factory. From each lot produced at the factory, the inspectors take a sample of clothes and count the number of clothes that are unacceptable. coffee.sav. This data le pertains to perceived images of six iced-coffee brands (Kennedy, Riquier, and Sharp, 1996) . For each of 23 iced-coffee image attributes, people selected all brands that were described by the attribute. The six brands are denoted AA, BB, CC, DD, EE, and FF to preserve condentiality. contacts.sav. This is a hypothetical data le that concerns the contact lists for a group of corporate computer sales representatives. Each contact is categorized by the department of the company in which they work and their company ranks. Also recorded are the amount of the last sale made, the time since the last sale, and the size of the contacts company. 410 Appendix A creditpromo.sav. This is a hypothetical data le that concerns a department stores efforts to evaluate the effectiveness of a recent credit card promotion. To this end, 500 cardholders were randomly selected. Half received an ad promoting a reduced interest rate on purchases made over the next three months. Half received a standard seasonal ad. customer_dbase.sav. This is a hypothetical data le that concerns a companys efforts to use the information in its data warehouse to make special offers to customers who are most likely to reply. A subset of the customer base was selected at random and given the special offers, and their responses were recorded. customer_information.sav. A hypothetical data le containing customer mailing information, such as name and address. customers_model.sav. This le contains hypothetical data on individuals targeted by a marketing campaign. These data include demographic information, a summary of purchasing history, and whether or not each individual responded to the campaign. Each case represents a separate individual. customers_new.sav. This le contains hypothetical data on individuals who are potential candidates for a marketing campaign. These data include demographic information and a summary of purchasing history for each individual. Each case represents a separate individual. debate.sav. This is a hypothetical data le that concerns paired responses to a survey from attendees of a political debate before and after the debate. Each case corresponds to a separate respondent. debate_aggregate.sav. This is a hypothetical data le that aggregates the responses in debate.sav. Each case corresponds to a cross-classication of preference before and after the debate. demo.sav. This is a hypothetical data le that concerns a purchased customer database, for the purpose of mailing monthly offers. Whether or not the customer responded to the offer is recorded, along with various demographic information. demo_cs_1.sav. This is a hypothetical data le that concerns the rst step of a companys efforts to compile a database of survey information. Each case corresponds to a different city, and the region, province, district, and city identication are recorded. demo_cs_2.sav. This is a hypothetical data le that concerns the second step of a companys efforts to compile a database of survey information. Each case corresponds to a different household unit from cities selected in the rst step, and 411 Sample Files the region, province, district, city, subdivision, and unit identication are recorded. The sampling information from the rst two stages of the design is also included. demo_cs.sav. This is a hypothetical data le that contains survey information collected using a complex sampling design. Each case corresponds to a different household unit, and various demographic and sampling information is recorded. dietstudy.sav. This hypothetical data le contains the results of a study of the Stillman diet (Rickman, Mitchell, Dingman, and Dalen, 1974). Each case corresponds to a separate subject and records his or her pre- and post-diet weights in pounds and triglyceride levels in mg/100 ml. dischargedata.sav. This is a data le concerning Seasonal Patterns of Winnipeg Hospital Use, (Menec , Roos, Nowicki, MacWilliam, Finlayson , and Black, 1999) from the Manitoba Centre for Health Policy. dvdplayer.sav. This is a hypothetical data le that concerns the development of a new DVD player. Using a prototype, the marketing team has collected focus group data. Each case corresponds to a separate surveyed user and records some demographic information about them and their responses to questions about the prototype. flying.sav. This data le contains the ying mileages between 10 American cities. german_credit.sav. This data le is taken from the German credit dataset in the Repository of Machine Learning Databases (Blake and Merz, 1998) at the University of California, Irvine. grocery_1month.sav. This hypothetical data le is the grocery_coupons.sav data le with the weekly purchases rolled-up so that each case corresponds to a separate customer. Some of the variables that changed weekly disappear as a result, and the amount spent recorded is now the sum of the amounts spent during the four weeks of the study. grocery_coupons.sav. This is a hypothetical data le that contains survey data collected by a grocery store chain interested in the purchasing habits of their customers. Each customer is followed for four weeks, and each case corresponds to a separate customer-week and records information about where and how the customer shops, including how much was spent on groceries during that week. guttman.sav. Bell (Bell, 1961) presented a table to illustrate possible social groups. Guttman (Guttman, 1968) used a portion of this table, in which ve variables describing such things as social interaction, feelings of belonging to a group, physical proximity of members, and formality of the relationship were crossed with seven theoretical social groups, including crowds (for example, people at a 412 Appendix A football game), audiences (for example, people at a theater or classroom lecture), public (for example, newspaper or television audiences), mobs (like a crowd but with much more intense interaction), primary groups (intimate), secondary groups (voluntary), and the modern community (loose confederation resulting from close physical proximity and a need for specialized services). healthplans.sav. This is a hypothetical data le that concerns an insurance groups efforts to evaluate four different health care plans for small employers. Twelve employers are recruited to rank the plans by how much they would prefer to offer them to their employees. Each case corresponds to a separate employer and records the reactions to each plan. health_funding.sav. This is a hypothetical data le that contains data on health care funding (amount per 100 population), disease rates (rate per 10,000 population), and visits to health care providers (rate per 10,000 population). Each case represents a different city. hivassay.sav. This is a hypothetical data le that concerns the efforts of a pharmaceutical lab to develop a rapid assay for detecting HIV infection. The results of the assay are eight deepening shades of red, with deeper shades indicating greater likelihood of infection. A laboratory trial was conducted on 2,000 blood samples, half of which were infected with HIV and half of which were clean. hourlywagedata.sav. This is a hypothetical data le that concerns the hourly wages of nurses from ofce and hospital positions and with varying levels of experience. insure.sav. This is a hypothetical data le that concerns an insurance company that is studying the risk factors that indicate whether a client will have to make a claim on a 10-year term life insurance contract. Each case in the data le represents a pair of contracts, one of which recorded a claim and the other didnt, matched on age and gender. judges.sav. This is a hypothetical data le that concerns the scores given by trained judges (plus one enthusiast) to 300 gymnastics performances. Each row represents a separate performance; the judges viewed the same performances. kinship_dat.sav. Rosenberg and Kim (Rosenberg and Kim, 1975) set out to analyze 15 kinship terms (aunt, brother, cousin, daughter, father, granddaughter, grandfather, grandmother, grandson, mother, nephew, niece, sister, son, uncle). They asked four groups of college students (two female, two male) to sort these terms on the basis of similarities. Two groups (one female, one male) were asked to sort twice, with the second sorting based on a different criterion from the rst sort. Thus, a total of six sources were obtained. Each source corresponds to a 413 Sample Files proximity matrix, whose cells are equal to the number of people in a source minus the number of times the objects were partitioned together in that source. kinship_ini.sav. This data le contains an initial conguration for a three-dimensional solution for kinship_dat.sav. kinship_var.sav. This data le contains independent variables gender, gener(ation), and degree (of separation) that can be used to interpret the dimensions of a solution for kinship_dat.sav. Specically, they can be used to restrict the space of the solution to a linear combination of these variables. mailresponse.sav. This is a hypothetical data le that concerns the efforts of a clothing manufacturer to determine whether using rst class postage for direct mailings results in faster responses than bulk mail. Order-takers record how many weeks after the mailing each order is taken. marketvalues.sav. This data le concerns home sales in a new housing development in Algonquin, Ill., during the years from 19992000. These sales are a matter of public record. mutualfund.sav. This data le concerns stock market information for various tech stocks listed on the S&P 500. Each case corresponds to a separate company. nhis2000_subset.sav. The National Health Interview Survey (NHIS) is a large, population-based survey of the U.S. civilian population. Interviews are carried out face-to-face in a nationally representative sample of households. Demographic information and observations about health behaviors and status are obtained for members of each household. This data le contains a subset of information from the 2000 survey. National Center for Health Statistics. National Health Interview Survey, 2000. Public-use data le and documentation. ftp://ftp.cdc.gov/pub/Health_Statistics/NCHS/Datasets/NHIS/2000/. Accessed 2003. ozone.sav. The data include 330 observations on six meteorological variables for predicting ozone concentration from the remaining variables. Previous researchers (Breiman and Friedman, 1985), (Hastie and Tibshirani, 1990), among others found nonlinearities among these variables, which hinder standard regression approaches. pain_medication.sav. This hypothetical data le contains the results of a clinical trial for anti-inammatory medication for treating chronic arthritic pain. Of particular interest is the time it takes for the drug to take effect and how it compares to an existing medication. 414 Appendix A patient_los.sav. This hypothetical data le contains the treatment records of patients who were admitted to the hospital for suspected myocardial infarction (MI, or heart attack). Each case corresponds to a separate patient and records many variables related to their hospital stay. patlos_sample.sav. This hypothetical data le contains the treatment records of a sample of patients who received thrombolytics during treatment for myocardial infarction (MI, or heart attack). Each case corresponds to a separate patient and records many variables related to their hospital stay. polishing.sav. This is the Nambeware Polishing Times data le from the Data and Story Library. It concerns the efforts of a metal tableware manufacturer (Nambe Mills, Santa Fe, N. M.) to plan its production schedule. Each case represents a different item in the product line. The diameter, polishing time, price, and product type are recorded for each item. poll_cs.sav. This is a hypothetical data le that concerns pollsters efforts to determine the level of public support for a bill before the legislature. The cases correspond to registered voters. Each case records the county, township, and neighborhood in which the voter lives. poll_cs_sample.sav. This hypothetical data le contains a sample of the voters listed in poll_cs.sav. The sample was taken according to the design specied in the poll.csplan plan le, and this data le records the inclusion probabilities and sample weights. Note, however, that because the sampling plan makes use of a probability-proportional-to-size (PPS) method, there is also a le containing the joint selection probabilities (poll_jointprob.sav). The additional variables corresponding to voter demographics and their opinion on the proposed bill were collected and added the data le after the sample as taken. property_assess.sav. This is a hypothetical data le that concerns a county assessors efforts to keep property value assessments up to date on limited resources. The cases correspond to properties sold in the county in the past year. Each case in the data le records the township in which the property lies, the assessor who last visited the property, the time since that assessment, the valuation made at that time, and the sale value of the property. property_assess_cs.sav. This is a hypothetical data le that concerns a state assessors efforts to keep property value assessments up to date on limited resources. The cases correspond to properties in the state. Each case in the data le records the county, township, and neighborhood in which the property lies, the time since the last assessment, and the valuation made at that time. 415 Sample Files property_assess_cs_sample.sav. This hypothetical data le contains a sample of the properties listed in property_assess_cs.sav. The sample was taken according to the design specied in the property_assess.csplan plan le, and this data le records the inclusion probabilities and sample weights. The additional variable Current value was collected and added to the data le after the sample was taken. recidivism.sav. This is a hypothetical data le that concerns a government law enforcement agencys efforts to understand recidivism rates in their area of jurisdiction. Each case corresponds to a previous offender and records their demographic information, some details of their rst crime, and then the time until their second arrest, if it occurred within two years of the rst arrest. recidivism_cs_sample.sav. This is a hypothetical data le that concerns a government law enforcement agencys efforts to understand recidivism rates in their area of jurisdiction. Each case corresponds to a previous offender, released from their rst arrest during the month of June, 2003, and records their demographic information, some details of their rst crime, and the data of their second arrest, if it occurred by the end of June, 2006. Offenders were selected from sampled departments according to the sampling plan specied in recidivism_cs.csplan; because it makes use of a probability-proportional-to-size (PPS) method, there is also a le containing the joint selection probabilities (recidivism_cs_jointprob.sav). rfm_transactions.sav. A hypothetical data le containing purchase transaction data, including date of purchase, item(s) purchased, and monetary amount of each transaction. salesperformance.sav. This is a hypothetical data le that concerns the evaluation of two new sales training courses. Sixty employees, divided into three groups, all receive standard training. In addition, group 2 gets technical training; group 3, a hands-on tutorial. Each employee was tested at the end of the training course and their score recorded. Each case in the data le represents a separate trainee and records the group to which they were assigned and the score they received on the exam. satisf.sav. This is a hypothetical data le that concerns a satisfaction survey conducted by a retail company at 4 store locations. 582 customers were surveyed in all, and each case represents the responses from a single customer. screws.sav. This data le contains information on the characteristics of screws, bolts, nuts, and tacks (Hartigan, 1975). 416 Appendix A shampoo_ph.sav. This is a hypothetical data le that concerns the quality control at a factory for hair products. At regular time intervals, six separate output batches are measured and their pH recorded. The target range is 4.55.5. ships.sav. A dataset presented and analyzed elsewhere (McCullagh et al., 1989) that concerns damage to cargo ships caused by waves. The incident counts can be modeled as occurring at a Poisson rate given the ship type, construction period, and service period. The aggregate months of service for each cell of the table formed by the cross-classication of factors provides values for the exposure to risk. site.sav. This is a hypothetical data le that concerns a companys efforts to choose new sites for their expanding business. They have hired two consultants to separately evaluate the sites, who, in addition to an extended report, summarized each site as a good, fair, or poor prospect. siteratings.sav. This is a hypothetical data le that concerns the beta testing of an e-commerce rms new Web site. Each case represents a separate beta tester, who scored the usability of the site on a scale from 020. smokers.sav. This data le is abstracted from the 1998 National Household Survey of Drug Abuse and is a probability sample of American households. Thus, the rst step in an analysis of this data le should be to weight the data to reect population trends. smoking.sav. This is a hypothetical table introduced by Greenacre (Greenacre, 1984). The table of interest is formed by the crosstabulation of smoking behavior by job category. The variable Staff Group contains the job categories Sr Managers, Jr Managers, Sr Employees, Jr Employees, and Secretaries, plus the category National Average, which can be used as supplementary to an analysis. The variable Smoking contains the behaviors None, Light, Medium, and Heavy, plus the categories No Alcohol and Alcohol, which can be used as supplementary to an analysis. storebrand.sav. This is a hypothetical data le that concerns a grocery store managers efforts to increase sales of the store brand detergent relative to other brands. She puts together an in-store promotion and talks with customers at check-out. Each case represents a separate customer. stores.sav. This data le contains hypothetical monthly market share data for two competing grocery stores. Each case represents the market share data for a given month. stroke_clean.sav. This hypothetical data le contains the state of a medical database after it has been cleaned using procedures in the Data Preparation option. 417 Sample Files stroke_invalid.sav. This hypothetical data le contains the initial state of a medical database and contains several data entry errors. stroke_survival. This hypothetical data le concerns survival times for patients exiting a rehabilitation program post-ischemic stroke face a number of challenges. Post-stroke, the occurrence of myocardial infarction, ischemic stroke, or hemorrhagic stroke is noted and the time of the event recorded. The sample is left-truncated because it only includes patients who survived through the end of the rehabilitation program administered post-stroke. stroke_valid.sav. This hypothetical data le contains the state of a medical database after the values have been checked using the Validate Data procedure. It still contains potentially anomalous cases. survey_sample.sav. This hypothetical data le contains survey data, including demographic data and various attitude measures. tastetest.sav. This is a hypothetical data le that concerns the effect of mulch color on the taste of crops. Strawberries grown in red, blue, and black mulch were rated by taste-testers on an ordinal scale of 1 to 5 (far below to far above average). Each case represents a separate taste-tester. telco.sav. This is a hypothetical data le that concerns a telecommunications companys efforts to reduce churn in their customer base. Each case corresponds to a separate customer and records various demographic and service usage information. telco_extra.sav. This data le is similar to the telco.sav data le, but the tenure and log-transformed customer spending variables have been removed and replaced by standardized log-transformed customer spending variables. telco_missing.sav. This data le is a subset of the telco.sav data le, but some of the demographic data values have been replaced with missing values. testmarket.sav. This hypothetical data le concerns a fast food chains plans to add a new item to its menu. There are three possible campaigns for promoting the new product, so the new item is introduced at locations in several randomly selected markets. A different promotion is used at each location, and the weekly sales of the new item are recorded for the rst four weeks. Each case corresponds to a separate location-week. testmarket_1month.sav. This hypothetical data le is the testmarket.sav data le with the weekly sales rolled-up so that each case corresponds to a separate location. Some of the variables that changed weekly disappear as a result, and the sales recorded is now the sum of the sales during the four weeks of the study. 418 Appendix A tree_car.sav. This is a hypothetical data le containing demographic and vehicle purchase price data. tree_credit.sav. This is a hypothetical data le containing demographic and bank loan history data. tree_missing_data.sav This is a hypothetical data le containing demographic and bank loan history data with a large number of missing values. tree_score_car.sav. This is a hypothetical data le containing demographic and vehicle purchase price data. tree_textdata.sav. A simple data le with only two variables intended primarily to show the default state of variables prior to assignment of measurement level and value labels. tv-survey.sav. This is a hypothetical data le that concerns a survey conducted by a TV studio that is considering whether to extend the run of a successful program. 906 respondents were asked whether they would watch the program under various conditions. Each row represents a separate respondent; each column is a separate condition. ulcer_recurrence.sav. This le contains partial information from a study designed to compare the efcacy of two therapies for preventing the recurrence of ulcers. It provides a good example of interval-censored data and has been presented and analyzed elsewhere (Collett, 2003). ulcer_recurrence_recoded.sav. This le reorganizes the information in ulcer_recurrence.sav to allow you model the event probability for each interval of the study rather than simply the end-of-study event probability. It has been presented and analyzed elsewhere (Collett et al., 2003). verd1985.sav. This data le concerns a survey (Verdegaal, 1985). The responses of 15 subjects to 8 variables were recorded. The variables of interest are divided into three sets. Set 1 includes age and marital, set 2 includes pet and news, and set 3 includes music and live. Pet is scaled as multiple nominal and age is scaled as ordinal; all of the other variables are scaled as single nominal. virus.sav. This is a hypothetical data le that concerns the efforts of an Internet service provider (ISP) to determine the effects of a virus on its networks. They have tracked the (approximate) percentage of infected e-mail trafc on its networks over time, from the moment of discovery until the threat was contained. 419 Sample Files waittimes.sav. This is a hypothetical data le that concerns customer waiting times for service at three different branches of a local bank. Each case corresponds to a separate customer and records the time spent waiting and the branch at which they were conducting their business. webusability.sav. This is a hypothetical data le that concerns usability testing of a new e-store. Each case corresponds to one of ve usability testers and records whether or not the tester succeeded at each of six separate tasks. wheeze_steubenville.sav. This is a subset from a longitudinal study of the health effects of air pollution on children (Ware, Dockery, Spiro III, Speizer, and Ferris Jr., 1984). The data contain repeated binary measures of the wheezing status for children from Steubenville, Ohio, at ages 7, 8, 9 and 10 years, along with a xed recording of whether or not the mother was a smoker during the rst year of the study. workprog.sav. This is a hypothetical data le that concerns a government works program that tries to place disadvantaged people into better jobs. A sample of potential program participants were followed, some of whom were randomly selected for enrollment in the program, while others were not. Each case represents a separate program participant. Bibliography Barlow, R. E., D. J. Bartholomew, D. J. Bremner, and H. D. Brunk. 1972. Statistical inference under order restrictions. New York: John Wiley and Sons. Bell, E. H. 1961. Social foundations of human behavior: Introduction to the study of sociology. New York: Harper & Row. Benzcri, J. P. 1969. Statistical analysis as a tool to make patterns emerge from data. In: Methodologies of Pattern Recognition, S. Watanabe, ed. New York: Academic Press, 3574. Benzcri, J. P. 1992. Correspondence analysis handbook. New York: Marcel Dekker. Bishop, Y. M., S. E. Feinberg, and P. W. Holland. 1975. Discrete multivariate analysis: Theory and practice. Cambridge, Mass.: MIT Press. Blake, C. L., and C. J. Merz. 1998. "UCI Repository of machine learning databases." Available at http://www.ics.uci.edu/~mlearn/MLRepository.html. Breiman, L., and J. H. Friedman. 1985. Estimating optimal transformations for multiple regression and correlation. Journal of the American Statistical Association, 80, 580598. Buja, A. 1990. Remarks on functional canonical variates, alternating least squares methods and ACE. Annals of Statistics, 18, 10321069. Busing, F. M. T. A., P. J. F. Groenen, and W. J. Heiser. 2005. Avoiding degeneracy in multidimensional unfolding by penalizing on the coefcient of variation. Psychometrika, 70, 7198. Carroll, J. D. 1968. Generalization of canonical correlation analysis to three or more sets of variables. In: Proceedings of the 76th Annual Convention of the American Psychological Association, 3, Washington, D.C.: American Psychological Association, 227228. Collett, D. 2003. Modelling survival data in medical research, 2 ed. Boca Raton: Chapman & Hall/CRC. Commandeur, J. J. F., and W. J. Heiser. 1993. Mathematical derivations in the proximity scaling (PROXSCAL) of symmetric data matrices. Leiden: Department of Data Theory, University of Leiden. 420 421 Bibliography De Haas, M., J. A. Algera, H. F. J. M. Van Tuijl, and J. J. Meulman. 2000. Macro and micro goal setting: In search of coherence. Applied Psychology, 49, 579595. De Leeuw, J. 1982. Nonlinear principal components analysis. In: COMPSTAT Proceedings in Computational Statistics, Vienna: Physica Verlag, 7789. De Leeuw, J. 1984. Canonical analysis of categorical data, 2nd ed. Leiden: DSWO Press. De Leeuw, J. 1984. The Gi system of nonlinear multivariate analysis. In: Data Analysis and Informatics III, E. Diday, et al., ed., 415424. De Leeuw, J., and W. J. Heiser. 1980. Multidimensional scaling with restrictions on the conguration. In: Multivariate Analysis, Vol. V, P. R. Krishnaiah, ed. Amsterdam: North-Holland, 501522. De Leeuw, J., and J. Van Rijckevorsel. 1980. HOMALS and PRINCALSSome generalizations of principal components analysis. In: Data Analysis and Informatics, E. Diday,et al., ed. Amsterdam: North-Holland, 231242. De Leeuw, J., F. W. Young, and Y. Takane. 1976. Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika, 41, 471503. De Leeuw, J. 1990. Multivariate analysis with optimal scaling. In: Progress in Multivariate Analysis, S. Das Gupta, and J. Sethuraman, eds. Calcutta: Indian Statistical Institute. Eckart, C., and G. Young. 1936. The approximation of one matrix by another one of lower rank. Psychometrika, 1, 211218. Fisher, R. A. 1938. Statistical methods for research workers. Edinburgh: Oliver and Boyd. Fisher, R. A. 1940. The precision of discriminant functions. Annals of Eugenics, 10, 422429. Gabriel, K. R. 1971. The biplot graphic display of matrices with application to principal components analysis. Biometrika, 58, 453467. Gi, A. 1985. PRINCALS. Research Report UG-85-02. Leiden: Department of Data Theory, University of Leiden. Gi, A. 1990. Nonlinear multivariate analysis. Chichester: John Wiley and Sons. Gilula, Z., and S. J. Haberman. 1988. The analysis of multivariate contingency tables by restricted canonical and restricted association models. Journal of the American Statistical Association, 83, 760771. 422 Bibliography Gower, J. C., and J. J. Meulman. 1993. The treatment of categorical information in physical anthropology. International Journal of Anthropology, 8, 4351. Green, P. E., and V. Rao. 1972. Applied multidimensional scaling. Hinsdale, Ill.: Dryden Press. Green, P. E., and Y. Wind. 1973. Multiattribute decisions in marketing: A measurement approach. Hinsdale, Ill.: Dryden Press. Greenacre, M. J. 1984. Theory and applications of correspondence analysis. London: Academic Press. Guttman, L. 1941. The quantication of a class of attributes: A theory and method of scale construction. In: The Prediction of Personal Adjustment, P. Horst, ed. New York: Social Science Research Council, 319348. Guttman, L. 1968. A general nonmetric technique for nding the smallest coordinate space for congurations of points. Psychometrika, 33, 469506. Hartigan, J. A. 1975. Clustering algorithms. New York: John Wiley and Sons. Hastie, T., and R. Tibshirani. 1990. Generalized additive models. London: Chapman and Hall. Hastie, T., R. Tibshirani, and A. Buja. 1994. Flexible discriminant analysis. Journal of the American Statistical Association, 89, 12551270. Hayashi, C. 1952. On the prediction of phenomena from qualitative data and the quantication of qualitative data from the mathematico-statistical point of view. Annals of the Institute of Statitical Mathematics, 2, 9396. Heiser, W. J. 1981. Unfolding analysis of proximity data. Leiden: Department of Data Theory, University of Leiden. Heiser, W. J., and F. M. T. A. Busing. 2004. Multidimensional scaling and unfolding of symmetric and asymmetric proximity relations. In: Handbook of Quantitative Methodology for the Social Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: SagePublications, Inc., 2548. Heiser, W. J., and J. J. Meulman. 1994. Homogeneity analysis: Exploring the distribution of variables and their nonlinear relationships. In: Correspondence Analysis in the Social Sciences: Recent Developments and Applications, M. Greenacre, and J. Blasius, eds. New York: Academic Press, 179209. Heiser, W. J., and J. J. Meulman. 1995. Nonlinear methods for the analysis of homogeneity and heterogeneity. In: Recent Advances in Descriptive Multivariate Analysis, W. J. Krzanowski, ed. Oxford: Oxford UniversityPress, 5189. 423 Bibliography Horst, P. 1961. Generalized canonical correlations and their applications to experimental data. Journal of Clinical Psychology, 17, 331347. Horst, P. 1961. Relations among m sets of measures. Psychometrika, 26, 129149. Israls, A. 1987. Eigenvalue techniques for qualitative data. Leiden: DSWO Press. Kennedy, R., C. Riquier, and B. Sharp. 1996. Practical applications of correspondence analysis to categorical data in market research. Journal of Targeting, Measurement, and Analysis for Marketing, 5, 5670. Kettenring, J. R. 1971. Canonical analysis of several sets of variables. Biometrika, 58, 433460. Kruskal, J. B. 1964. Multidimensional scaling by optimizing goodness of t to a nonmetric hypothesis. Psychometrika, 29, 128. Kruskal, J. B. 1964. Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115129. Kruskal, J. B. 1965. Analysis of factorial experiments by estimating monotone transformations of the data. Journal of the Royal Statistical Society Series B, 27, 251263. Kruskal, J. B. 1978. Factor analysis and principal components analysis: Bilinear methods. In: International Encyclopedia of Statistics, W. H. Kruskal, and J. M. Tanur, eds. New York: The Free Press, 307330. Kruskal, J. B., and R. N. Shepard. 1974. A nonmetric variety of linear factor analysis. Psychometrika, 39, 123157. Krzanowski, W. J., and F. H. C. Marriott. 1994. Multivariate analysis: Part I, distributions, ordination and inference. London: Edward Arnold. Lebart, L., A. Morineau, and K. M. Warwick. 1984. Multivariate descriptive statistical analysis. New York: John Wiley and Sons. Lingoes, J. C. 1968. The multivariate analysis of qualitative data. Multivariate Behavioral Research, 3, 6194. Max, J. 1960. Quantizing for minimum distortion. Proceedings IEEE (Information Theory), 6, 712. McCullagh, P., and J. A. Nelder. 1989. Generalized Linear Models, 2nd ed. London: Chapman & Hall. Menec , V., N. Roos, D. Nowicki, L. MacWilliam, G. Finlayson , and C. Black. 1999. Seasonal Patterns of Winnipeg Hospital Use. : Manitoba Centre for Health Policy. Meulman, J. J. 1982. Homogeneity analysis of incomplete data. Leiden: DSWO Press. 424 Bibliography Meulman, J. J. 1986. A distance approach to nonlinear multivariate analysis. Leiden: DSWO Press. Meulman, J. J. 1992. The integration of multidimensional scaling and multivariate analysis with optimal transformations of the variables. Psychometrika, 57, 539565. Meulman, J. J. 1993. Principal coordinates analysis with optimal transformations of the variables: Minimizing the sum of squares of the smallest eigenvalues. British Journal of Mathematical and Statistical Psychology, 46, 287300. Meulman, J. J. 1996. Fitting a distance model to homogeneous subsets of variables: Points of view analysis of categorical data. Journal of Classication, 13, 249266. Meulman, J. J. 2003. Prediction and classication in nonlinear data analysis: Something old, something new, something borrowed, something blue. Psychometrika, 4, 493517. Meulman, J. J., and W. J. Heiser. 1997. Graphical display of interaction in multiway contingency tables by use of homogeneity analysis. In: Visual Display of Categorical Data, M. Greenacre, and J. Blasius, eds. New York: Academic Press, 277296. Meulman, J. J., and P. Verboon. 1993. Points of view analysis revisited: Fitting multidimensional structures to optimal distance components with cluster restrictions on the variables. Psychometrika, 58, 735. Meulman, J. J., A. J. Van der Kooij, and A. Babinec. 2000. New features of categorical principal components analysis for complicated data sets, including data mining. In: Classication, Automation and New Media, W. Gaul, and G. Ritter, eds. Berlin: Springer-Verlag, 207217. Meulman, J. J., A. J. Van der Kooij, and W. J. Heiser. 2004. Principal components analysis with nonlinear optimal scaling transformations for ordinal and nominal data. In: Handbook of Quantitative Methodology for the Social Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: Sage Publications, Inc., 4970. Nishisato, S. 1980. Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press. Nishisato, S. 1984. Forced classication: A simple application of a quantication method. Psychometrika, 49, 2536. Nishisato, S. 1994. Elements of dual scaling: An introduction to practical data analysis. Hillsdale, N.J.: Lawrence Erlbaum Associates, Inc. Pratt, J. W. 1987. Dividing the indivisible: Using simple symmetry to partition variance explained. In: Proceedings of the Second International Conference in Statistics, T. Pukkila, and S. Puntanen, eds. Tampere, Finland: Universityof Tampere, 245260. 425 Bibliography Price, R. H., and D. L. Bouffard. 1974. Behavioral appropriateness and situational constraints as dimensions of social behavior. Journal of Personality and Social Psychology, 30, 579586. Ramsay, J. O. 1989. Monotone regression splines in action. Statistical Science, 4, 425441. Rao, C. R. 1973. Linear statistical inference and its applications, 2nd ed. New York: John Wiley and Sons. Rao, C. R. 1980. Matrix approximations and reduction of dimensionality in multivariate statistical analysis. In: Multivariate Analysis, Vol. 5, P. R. Krishnaiah, ed. Amsterdam: North-Holland, 322. Rickman, R., N. Mitchell, J. Dingman, and J. E. Dalen. 1974. Changes in serum cholesterol during the Stillman Diet. Journal of the American Medical Association, 228, 5458. Rosenberg, S., and M. P. Kim. 1975. The method of sorting as a data-gathering procedure in multivariate research. Multivariate Behavioral Research, 10, 489502. Roskam, E. E. 1968. Metric analysis of ordinal data in psychology. Voorschoten: VAM. Shepard, R. N. 1962. The analysis of proximities: Multidimensional scaling with an unknown distance function I. Psychometrika, 27, 125140. Shepard, R. N. 1962. The analysis of proximities: Multidimensional scaling with an unknown distance function II. Psychometrika, 27, 219246. Shepard, R. N. 1966. Metric structures in ordinal data. Journal of Mathematical Psychology, 3, 287315. Tenenhaus, M., and F. W. Young. 1985. An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis, and other methods for quantifying categorical multivariate data. Psychometrika, 50, 91119. Theunissen, N. C. M., J. J. Meulman, A. L. Den Ouden, H. M. Koopman, G. H. Verrips, S. P. Verloove-Vanhorick, and J. M. Wit. 2003. Changes can be studied when the measurement instrument is different at different time points. Health Services and Outcomes Research Methodology, 4, 109126. Tucker, L. R. 1960. Intra-individual and inter-individual multidimensionality. In: Psychological Scaling: Theory & Applications, H. Gulliksen, and S. Messick, eds. NewYork: John Wiley and Sons, 155167. 426 Bibliography Van der Burg, E. 1988. Nonlinear canonical correlation and some related techniques. Leiden: DSWO Press. Van der Burg, E., and J. De Leeuw. 1983. Nonlinear canonical correlation. British Journal of Mathematical and Statistical Psychology, 36, 5480. Van der Burg, E., J. De Leeuw, and R. Verdegaal. 1988. Homogeneity analysis with k sets of variables: An alternating least squares method with optimal scaling features. Psychometrika, 53, 177197. Van der Ham, T., J. J. Meulman, D. C. Van Strien, and H. Van Engeland. 1997. Empirically based subgrouping of eating disorders in adolescents: A longitudinal perspective. British Journal of Psychiatry, 170, 363368. Van der Kooij, A. J., and J. J. Meulman. 1997. MURALS: Multiple regression and optimal scaling using alternating least squares. In: Softstat 97, F. Faulbaum, and W. Bandilla, eds. Stuttgart: Gustav Fisher, 99106. Van Rijckevorsel, J. 1987. The application of fuzzy coding and horseshoes in multiple correspondence analysis. Leiden: DSWO Press. Verboon, P., and I. A. Van der Lans. 1994. Robust canonical discriminant analysis. Psychometrika, 59, 485507. Verdegaal, R. 1985. Meer sets analyse voor kwalitatieve gegevens (in Dutch). Leiden: Department of Data Theory, University of Leiden. Vlek, C., and P. J. Stallen. 1981. Judging risks and benets in the small and in the large. Organizational Behavior and Human Performance, 28, 235271. Wagenaar, W. A. 1988. Paradoxes of gambling behaviour. London: Lawrence Erlbaum Associates, Inc. Ware, J. H., D. W. Dockery, A. Spiro III, F. E. Speizer, and B. G. Ferris Jr.. 1984. Passive smoking, gas cooking, and respiratory health of children living in six cities. American Review of Respiratory Diseases, 129, 366374. Winsberg, S., and J. O. Ramsay. 1980. Monotonic transformations to additivity using splines. Biometrika, 67, 669674. Winsberg, S., and J. O. Ramsay. 1983. Monotone spline transformations for dimension reduction. Psychometrika, 48, 575595. Wolter, K. M. 1985. Introduction to variance estimation. Berlin: Springer-Verlag. Young, F. W. 1981. Quantitative analysis of qualitative data. Psychometrika, 46, 357387. 427 Bibliography Young, F. W., J. De Leeuw, and Y. Takane. 1976. Regression with qualitative and quantitative variables: An alternating least squares method with optimal scaling features. Psychometrika, 41, 505528. Young, F. W., Y. Takane, and J. De Leeuw. 1978. The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features. Psychometrika, 43, 279281. Zeijl, E., Y. te Poel, M. du Bois-Reymond, J. Ravesloot, and J. J. Meulman. 2000. The role of parents and peers in the leisure activities of young adolescents. Journal of Leisure Research, 32, 281302. Index ANOVA in Categorical Regression, 29 biplots in Categorical Principal Components Analysis, 49 in Correspondence Analysis, 72, 281 in Multiple Correspondence Analysis, 88 Categorical Principal Components Analysis, 35, 43, 179, 196 category points, 220 command additional features, 54 component loadings, 191, 196, 216 iteration history, 187 model summary, 187, 194, 216 object scores, 190, 194, 218 optimal scaling level, 38 quantications, 188, 212 save variables, 48 Categorical Regression, 19, 125 command additional features, 34 correlations, 140, 142 importance, 142 intercorrelations, 139 model t, 140 optimal scaling level, 21 plots, 19 regularization, 28 residuals, 146 save, 32 statistics, 19 transformation plots, 144 category coordinates in Nonlinear Canonical Correlation Analysis, 259 category plots in Categorical Principal Components Analysis, 51 in Multiple Correspondence Analysis, 90 category points in Categorical Principal Components Analysis, 220 category quantications in Categorical Principal Components Analysis, 46 in Categorical Regression, 29 in Multiple Correspondence Analysis, 85, 328 in Nonlinear Canonical Correlation Analysis, 60 centroids in Nonlinear Canonical Correlation Analysis, 60, 260 coefcient of variation in Multidimensional Unfolding, 368, 372, 380, 388, 398 coefcients in Categorical Regression, 140 column principal normalization in Correspondence Analysis, 274 column scores in Correspondence Analysis, 284 column scores plots in Correspondence Analysis, 303 common space in Multidimensional Scaling, 356, 361 in Multidimensional Unfolding, 369, 373, 381, 389, 399, 403 common space coordinates in Multidimensional Scaling, 108 in Multidimensional Unfolding, 121 common space plots in Multidimensional Scaling, 105 in Multidimensional Unfolding, 118 component loadings in Categorical Principal Components Analysis, 46, 191, 196, 216 in Nonlinear Canonical Correlation Analysis, 60, 254 component loadings plots in Categorical Principal Components Analysis, 52 condence statistics in Correspondence Analysis, 71, 287 contributions in Correspondence Analysis, 284, 301 428 429 Index correlation matrix in Categorical Principal Components Analysis, 46 in Multiple Correspondence Analysis, 85 correlations in Multidimensional Scaling, 108 correlations plots in Multidimensional Scaling, 105 Correspondence Analysis, 64, 6668, 7172, 273, 275, 294 biplots, 281 column scores plots, 303 command additional features, 74 condence statistics, 287 contributions, 284, 301 correspondence tables, 279, 316 dimensions, 300 inertia per dimension, 280 normalization, 274 permutations, 286 plots, 64 proles, 282 row and column scores, 284 row scores plots, 303, 317 statistics, 64 correspondence tables in Correspondence Analysis, 279, 316 Cronbachs alpha in Categorical Principal Components Analysis, 187 DeSarbos intermixedness indices in Multidimensional Unfolding, 368, 372, 380, 388, 398 descriptive statistics in Categorical Regression, 29 dimension weights in Multidimensional Unfolding, 382, 390 dimensions in Correspondence Analysis, 68, 300 discretization in Categorical Principal Components Analysis, 40 in Categorical Regression, 23 in Multiple Correspondence Analysis, 79 discrimination measures in Multiple Correspondence Analysis, 85, 327 discrimination measures plots in Multiple Correspondence Analysis, 90 distance measures in Correspondence Analysis, 68 distances in Multidimensional Scaling, 108 in Multidimensional Unfolding, 121 eigenvalues in Categorical Principal Components Analysis, 187, 194, 216 in Nonlinear Canonical Correlation Analysis, 250 elastic net in Categorical Regression, 28 nal common space plots in Multidimensional Unfolding, 118 t in Nonlinear Canonical Correlation Analysis, 60 t values in Nonlinear Canonical Correlation Analysis, 250 generalized Euclidean model in Multidimensional Unfolding, 112 identity model in Multidimensional Unfolding, 112 importance in Categorical Regression, 142 individual space coordinates in Multidimensional Unfolding, 121 individual space weights in Multidimensional Scaling, 108 in Multidimensional Unfolding, 121 individual space weights plots in Multidimensional Scaling, 105 in Multidimensional Unfolding, 118 individual spaces in Multidimensional Unfolding, 382, 390 individual spaces plots in Multidimensional Scaling, 105 in Multidimensional Unfolding, 118 430 Index inertia in Correspondence Analysis, 71, 280, 284 initial common space plots in Multidimensional Unfolding, 118 initial conguration in Categorical Regression, 26 in Multidimensional Scaling, 103 in Multidimensional Unfolding, 115 in Nonlinear Canonical Correlation Analysis, 60 intercorrelations in Categorical Regression, 139 iteration criteria in Multidimensional Scaling, 103 in Multidimensional Unfolding, 115 iteration history in Categorical Principal Components Analysis, 46, 187 in Multidimensional Scaling, 108 in Multidimensional Unfolding, 121 in Multiple Correspondence Analysis, 85 joint category plots in Categorical Principal Components Analysis, 51 in Multiple Correspondence Analysis, 90 joint plot of common space in Multidimensional Unfolding, 369, 373, 381, 389, 399, 403 joint plot of individual spaces in Multidimensional Unfolding, 382, 390 lasso in Categorical Regression, 28 loss values in Nonlinear Canonical Correlation Analysis, 250 missing values in Categorical Principal Components Analysis, 42 in Categorical Regression, 25 in Multiple Correspondence Analysis, 81 model summary in Multiple Correspondence Analysis, 325 Multidimensional Scaling, 92, 9599, 340 command additional features, 109 common space, 356, 361 model, 100 options, 103 output, 108 plots, 92, 105, 107 restrictions, 102 statistics, 92 stress measures, 354, 360 transformation plots, 360 Multidimensional Unfolding, 110, 364, 392 command additional features, 122 common space, 369, 373, 381, 389, 399, 403 degenerate solutions, 364 individual spaces, 382, 390 measures, 368, 372, 380, 388, 398, 402 model, 112 options, 115 output, 121 plots, 110, 118 proximity transformations, 400, 404 restrictions on common space, 114 statistics, 110 three-way unfolding , 374 Multiple Correspondence Analysis, 76, 82, 320 category quantications, 328 command additional features, 91 discrimination measures, 327 model summary, 325 object scores, 325, 330 optimal scaling level, 78 outliers, 333 save variables, 87 multiple R in Categorical Regression, 29 multiple starts plots in Multidimensional Unfolding, 118 Nonlinear Canonical Correlation Analysis, 55, 5960, 239 category coordinates, 259 centroids, 260 command additional features, 62 component loadings, 251, 254 plots, 55 quantications, 255 statistics, 55 summary of analysis, 250 weights, 251 431 Index normalization in Correspondence Analysis, 68, 274 object points plots in Categorical Principal Components Analysis, 49 in Multiple Correspondence Analysis, 88 object scores in Categorical Principal Components Analysis, 46, 190, 194, 218 in Multiple Correspondence Analysis, 85, 325, 330 in Nonlinear Canonical Correlation Analysis, 60 optimal scaling level in Categorical Principal Components Analysis, 38 in Multiple Correspondence Analysis, 78 outliers in Multiple Correspondence Analysis, 333 part correlations in Categorical Regression, 142 partial correlations in Categorical Regression, 142 penalized stress in Multidimensional Unfolding, 368, 380, 388, 398, 402 penalty term in Multidimensional Unfolding, 115 permutations in Correspondence Analysis, 286 plots in Categorical Regression, 33 in Correspondence Analysis, 72 in Multidimensional Scaling, 105, 107 in Nonlinear Canonical Correlation Analysis, 60 PREFSCAL, 110 principal normalization in Correspondence Analysis, 274 proles in Correspondence Analysis, 282 projected centroids in Nonlinear Canonical Correlation Analysis, 260 projected centroids plots in Categorical Principal Components Analysis, 51 proximity transformations in Multidimensional Unfolding, 112 quantications in Categorical Principal Components Analysis, 188, 212 in Nonlinear Canonical Correlation Analysis, 255 R2 in Categorical Regression, 140 regression coefcients in Categorical Regression, 29 relaxed updates in Multidimensional Scaling, 103 residuals in Categorical Regression, 146 residuals plots in Multidimensional Unfolding, 118 restrictions in Multidimensional Scaling, 102 restrictions on common space in Multidimensional Unfolding, 114 ridge regression in Categorical Regression, 28 row principal normalization in Correspondence Analysis, 274 row scores in Correspondence Analysis, 284 row scores plots in Correspondence Analysis, 303, 317 sample les location, 406 scaling model in Multidimensional Unfolding, 112 scatterplot of t in Multidimensional Unfolding, 118 Shepard plots in Multidimensional Unfolding, 118 Shepards rough nondegeneracy index in Multidimensional Unfolding, 368, 372, 380, 388, 398 space weights plots in Multidimensional Unfolding, 118 standardization in Correspondence Analysis, 68 stress measures in Multidimensional Scaling, 108, 354, 360 432 Index in Multidimensional Unfolding, 121 stress plots in Multidimensional Scaling, 105 in Multidimensional Unfolding, 118 supplementary objects in Categorical Regression, 26 supplementary points in Correspondence Analysis, 288 symmetrical normalization in Correspondence Analysis, 274 three-way unfolding in Multidimensional Unfolding, 374 transformation plots in Categorical Principal Components Analysis, 51 in Categorical Regression, 144 in Multidimensional Scaling, 105, 360 in Multidimensional Unfolding, 118, 400, 404 in Multiple Correspondence Analysis, 90 transformed independent variables in Multidimensional Scaling, 108 transformed proximities in Multidimensional Scaling, 108 in Multidimensional Unfolding, 121 triplots in Categorical Principal Components Analysis, 49 variable weight in Categorical Principal Components Analysis, 38 in Multiple Correspondence Analysis, 78 variance accounted for in Categorical Principal Components Analysis, 46, 187, 216 weighted Euclidean model in Multidimensional Unfolding, 112 weights in Nonlinear Canonical Correlation Analysis, 60, 251 zero-order correlations in Categorical Regression, 142 ... View Full Document