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CSSS 1 560 Lecture 1: What is Clustered Data and How Should it be Analyzed? Kevin Quinn University of Washington 2 Examples of Clustered Data Multilevel (contextual) data Longitudinal (panel) data Repeated measures data Data collected via cluster (multi-stage) sampling 3 Standard models (the basic linear model, logit, probit, etc.) assume that yi is conditionally independent of yj given measured covariates If this assumption is not met the standard estimators can be horribly misleading This assumption often does not hold in datasets that feature some type of clustering 4 Example 1: Repeated Measures of IQ We re interested in measuring the mean IQ of 6th graders in Seattle We have collected a random sample of 100 Seattle 6th graders Each child s IQ was then measured 3 times Dataset has 300 observations but these are not 300 independent observations 5 Ignoring this clustering will produce falsely precise estimates of mean IQ 6 Example 2: Relationship between Income and Outdoor Recreation Expenditures 10 households in Boise, Minneapolis, and New York City are sampled Plotting the raw data we see: log outdoor expenditures 4.8 5.0 5.2 5.4 5.6 9.0 9.5 10.0 10.5 11.0 log income 7 8 We could ignore the clustering and pool all the observations log outdoor expenditures 4.8 5.0 5.2 5.4 5.6 9.0 9.5 10.0 10.5 11.0 log income 9 10 However, because there are more opportunities for outdoor recreation in Idaho and to a lesser extent Minnesota than NYC we might expect the intercepts to vary by sampling unit log outdoor expenditures 4.8 5.0 5.2 5.4 5.6 9.0 9.5 10.0 10.5 11.0 log income 11 12 Key to both of these examples is that clustering (on student in example 1 and on city in example 2) produces dependent observations Problems only arise when we ignore this dependence 13 2 Ways to Think about Cluster-induced Dependence: Nuisance vs. Substance In some situations clustered data is simply a nuisance that has to be dealt with to make accurate inferences Collecting data via cluster sampling for convenience Being forced to use an existing data set collected via cluster-sampling 14 In these situations the dependence among observations may not be substantively interesting but still needs to be addressed 15 In many other situations the clustered nature of the data is of primary interest Contextual analysis of neighborhood characteristics on individual decisions to commit crimes Teacher-speci c e ects on individual student achievement Longitudinal analysis of HIV progression 16 macro units micro units schools teachers classrooms students neighborhoods families rms employees jawbones teeth families children litters animals doctors patients subjects measurements interviewers respondents judges suspects Table 1: Some examples of macro and micro units (from Snijders and Bosker) 17 Micro-level inference, macro-level inference, and cross-level inference Micro-level inference Example: High school students in Seattle are about 1.3 times more likely to drop out of school if they are employed Multi-stage (cluster) sampling not needed to make inferences models Hierarchical not needed unless sampling design 18 induces dependence 19 Macro-level inference Example: Increasing a school s budget by 5% increases graduation rates by 1.3% Multi-stage (cluster) sampling generally not necessary unless macro-variables not directly measured Once again, hierarchical models not needed unless sampling design induces dependence 20 Cross-level inference Example 1: students are more likely to graduate from private high schools than public high schools even after controlling for student-speci c factors Example 2: There is a strong positive relationship between student aptitude and student educational achievement in private schools while there is not a clear relationship between student aptitude and student educational achievement in public schools Multi-stage sampling necessary here Hierarchical modeling necessary here 21 What Exactly Are Hierarchical Models ? Hierarchical models go by many names and take many forms: Random e ects models, random coe cients models, hierarchical Bayes models, multi-level models, mixede ects models, etc. A very general way to think about a hierarchical model is as a mixed-e ects model 22 Mixed-E ects Models Mixed-e ects models get their name from the fact that they include both xed e ects (parameters associated with an entire population) and random e ects (parameters associated with particular micro units randomly sampled from the population) 23 The general mixed e ects model can be written as: yi = Xi + Zibi + i, i = 1, 2, . . . , M bi N (0, ), i N (0, 2I) Here is the vector of xed e ects and bi is the vector of random e ects speci c to micro-unit i Note that the random e ects are assumed to be drawn from a population distribution Can think about these random e ects is as additional error terms that induce intra-cluster dependence 24 Example: E ect of Aptitude on Achievement We want to determine the e ect of student aptitude on educational achievement We randomly select M = 50 schools in the US Within in school i we randomly sample ni = 100 students We might want to t the following model: 25 achievementij = 0 + aptitudeij 1 + bi0 + i = 1, . . . , M j = 1, . . . , ni ij This is an example of a random intercept model (sometimes called a random e ects model) The e ect of aptitude on achievement ( 1) is assumed constant across schools However, each school has its own intercept ( 0 + bi0) 26 Might also want to t: achievementij = 0 + aptitudeij 1 + bi0 + aptitudeij bi1 + i = 1, . . . , M j = 1, . . . , ni ij This is an example of a random coe cients model Each school has its own intercept ( 0 + bi0) and slope ( 1 + bi1) 27 in both cases, the bis capture unmeasured, schoolspeci c factors that in uence achievement, and/or the relationship between aptitude and achievement. For example, bi0 might represent the unmeasured quality of school i If bi0 and bi1 are positively correlated this might suggest that relationship between aptitude and achievement is strongest in high quality schools
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Path: Washington >> STAT >> 567 Fall, 2008
Description: Earnings of Black Women 0.06 0.06 Earnings of Black Women 0.05 default=SheatherJones 0.05 h=2 h=10 0.04 Probability Probability 0 20 40 60 80 0.03 0.02 0.01 0.00 0.00 0 0.01 0.02 0.03 0.04 20 40 60 80 Earnings (thousands of doll...
lec2plots.pdf
Path: Washington >> STAT >> 567 Fall, 2008
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Path: Washington >> STAT >> 567 Fall, 2008
Description: Krackhardts Hightech Managers friends network q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Trace of var1 10 Density of var1 Density 0e+00 3e+05 Iterations 6e+05 5 10 15 20 0.00 20 0.02 0.04 0.06 0 0.08 ...
FigHW5.3.pdf
Path: Washington >> STAT >> 567 Fall, 2008
Description: Linear Quantile Regression 11.5 11.5 Spline Quantile Regression 11.0 10.5 Logearnings 10.0 Logearnings observed quantiles quantile regression lines 9.5 9.5 10.0 10.5 11.0 observed quantiles quantile regression splines 9.0 9.0 70 75 80...
570-Day-2.pdf
Path: Washington >> STAT >> 570 Fall, 2008
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Path: Washington >> STAT >> 573 Fall, 2008
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576-07s.pdf
Path: Washington >> STAT >> 576 Fall, 2008
Description: BIOSTAT/STAT 576 Statistical Methods for Survival Data MW 9:00am - 10:20am, Spring 2007; HST T530 Instructor: Ying Qing Chen Tel: 206.667.7051 Email: yqchen@u.washington.edu Oce hour: Th 2:30pm - 3:30pm; H657 Course Web Site: http:/www.scharp.org/u...
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Path: Washington >> STAT >> 576 Fall, 2008
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Path: Washington >> STAT >> 579 Fall, 2008
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Path: Washington >> STAT >> 579 Fall, 2008
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Path: Washington >> STAT >> 579 Fall, 2008
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581.day1.08.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: STATISTICS 581: Advanced Theory of Statistical Inference Fall, 2008 Time: Place: Professor: Oce: Phone: e-mail: Oce Hours: Texts: 10:30 - 11:20 MWF (lecture) MEB 245 Jon A. Wellner B320 Padelford 206-543-6207 jaw@stat.washington.edu 1:30 - 3:30 MWF...
ref.08.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: References for Statistics 581, Fall 2008 Analysis: Bartle, R. G., The Elements of Integration. Rudin, W., Principles of Mathematical Analysis. Royden, H. L., Real Analysis. Luenberger, D., Optimization by Vector Space Methods. Probability: Will...
ch2.figs-epsf.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: Statistics 581, Chapter 2 Empirical Distribution Function and Empirical Process Figures Wellner; 10/24/2008 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 Figure 1: Uniform Empirical Distribution Function, n = 50. 1 0.75 0.5 0.25 0.2 -0.25 0.4 0.6...
exam1.06.pdf
Path: Washington >> STAT >> 581 Fall, 2008
Description: Statistics 581, Midterm Exam Wellner; 11/06/2006 This exam is to be taken without any books or notes. 1. (24 points) Dene any three of the following ve terms. (a) A uniformly integrable sequence of random variables. (b) Convergence in rth mean of a...
mt03.pdf
Path: Washington >> STAT >> 582 Fall, 2008
Description: Stat 582 W03 Midterm exam Please give as complete solutions as possible. More paper is available if needed. 1. Let (X,Y) be bivariate normal, mean zero, variance 1, correlation r. Find a minimal sufficient statistic for r. Is it complete? 2. Suppo...
final2.89.pdf
Path: Washington >> STAT >> 582 Fall, 2008
Description: STAT 582 FINAL EXAM 1. Let T (F) = m 2 = ( xdF(x)2 . Find, using the asymptotic theory for statistical functionals, the limiting F distribution of T (F n ) when m F 0. 2. Let (X 1 , Y 1 ), . . . , (X n , Y n ) be independent random variables with ...
gamma.pdf
Path: Washington >> STAT >> 582 Fall, 2008
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hw1.sln.doc
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT583 Sp04 Homework1solution 1. (a)(c) E X i - m = 2 xdF(x) = d 2 2 s (1+ 2e) and (1- e)s + 3es ) = p p 0 E(X - m) 2 = s 2 (1+ 8e) .Standardasymptotictheoryhasthat n ( s2 - s 2 (1+ 8e) N(0,t 2 ) where 2 = E(X - m) 4 - E 2 (X - m) 2 = s 4 (3...
notes1.pdf
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT 583 SPRING 2008 Lecture Notes 1 Statistical Functionals The Gteux derivative of a statistical functional T(F) is the limit T (F + (G F ) T(F) . d1T(F;G F) = lim If Q( ) = T (F + (G F ) has a McLaurin expansion, we get an expansion (the von...
hw3.sln.pdf
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT 583 Sp08 Solutions, Homework 3 1. (a) T (F! ) = F + !(G \" F ) so IC(x) = x ! F . (b) By the same calculation as that for the median we get p ! 1(x < F !1 ( p) IC(x) = , x \" F !1 ( p) !1 f (F ( p) d 1 T (F + !(\" x # F) = (c) Using (b) d! 1 ...
HW2.pdf
Path: Washington >> STAT >> 583 Fall, 2008
Description: STAT 583 Sp08 Homework 2 Due April 16. 1. Let F be a cdf on [0,1], and define for ! > 1 T (F) = x\"[0,1] # ( F(x) ! F(x!) $ . (a) Compute the Gateux derivative of T at the uniform distribution U on [0,1]. (b) Show that nR1,n = n(T (Fn ) ! T (U ...
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Path: Washington >> STAT >> 592 Fall, 2008
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Path: Washington >> STAT >> 592 Fall, 2008
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Path: Washington >> STAT >> 593 Fall, 2008
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Path: Washington >> HUM >> 202 Fall, 2008
Description: Lecture 14: October 27, 2006 Questions Reading Lyrical Ballads (1798), published Anonymously; 3 poems by Coleridge; 20 poems by Wordsworth Preface added in 1800 2nd edition Lyrical Ballads: not just an experiment but a poeti...
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Path: Washington >> HUM >> 204 Fall, 2008
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Path: Washington >> HUM >> 207 Winter, 2008
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Path: Washington >> HUM >> 208 Fall, 2008
Description: HUMANITIES 208: VIOLENCE, MYTH, AND MEMORY (SPRING 2008) Tuesday and Thursday 1:00-2:20, Johnson Hall 102 Francisco Benitez (Comp Lit) and Laurie Sears (History) TAs: Cheryll Alipio, Katrina Hagen, William Mitchell \"Violence, Myth, and Memory\" is bui...
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Path: Washington >> SWED >> 101 Fall, 2008
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Path: Washington >> T C >> 402 Winter, 2008
Description: Appendices to JFQA, Vol. 40, No. 2, June 2005, \"Horses and Rabbits? Trade-Off Theory and Optimal Capital Structure,\" by Nengjiu Ju, Robert Parrino, Allen M. Poteshman, and Michael S. Weisbach Horses and Rabbits? Trade-Off Theory and Optimal Capital ...
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Path: Washington >> T C >> 403 Fall, 2008
Description: Kyle Kyros Starr TC 403 May 18, 2008 Do You Know the Way to San Jose? Exploring the Future During the course of this class, I have looked through a lens back at who I was in the past and evaluated that person against the person that I am now. In t...
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Path: Washington >> T C >> 493 Fall, 2008
Description: Excerpts from Volume 1 of Capital (text from http:/www.marxists.org/archive/marx/works/1867-c1/index.htm) Introductory Note: Well discuss this in class, but just so you have it in front of you, Marx develops an analysis of capitalist production stres...
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Path: Washington >> T C >> 505 Fall, 2008
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Path: Washington >> T C >> 505 Fall, 2008
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intro.pdf
Path: Washington >> T C >> 509 Fall, 2008
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Path: Washington >> T C >> 509 Fall, 2008
Description: GFD I: Winter 2007 Midterm Solutions Professor: Chris Bretherton TA: Tom Connolly \\) As A.\' ( ynOt/e g iawardCg- n\"a ir\",{ ina f r{ r,\\6rI V c\\.,r P *e*^p *ropi cc,t {cto nP- r {n 3ce t,( tt 17o r* lo t% scr rq- Av-ops {r\" ^ l o \\ o ,*b ...
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Path: Washington >> T C >> 509 Fall, 2008
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Path: Washington >> T C >> 516 Fall, 2008
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ps4.pdf
Path: Washington >> T C >> 521 Winter, 2008
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ps7.pdf
Path: Washington >> T C >> 521 Winter, 2008
Description: ...
sol.ps3.pdf
Path: Washington >> T C >> 521 Winter, 2008
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Path: Washington >> T C >> 598 Fall, 2008
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Path: Washington >> T C >> 598 Fall, 2008
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Path: Washington >> THAI >> 302 Fall, 2008
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mt2_sol.pdf
Path: Washington >> IND E >> 101 Fall, 2008
Description: Name: _Section/TA Name: _ Atmospheric Science 101, Spring 2003 Midterm 2 Thursday 22 May 2003 Version A Multiple Choice (2 points each) Choose the best answer and mark it on a Scantron sheet. 1. Choose the words which best complete the following sent...
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Path: Washington >> IND E >> 101 Fall, 2008
Description: ABET Course Description IND E 101 Introduction to Industrial Engineering Elective Course Catalog course description: Examines the basic concepts and methods of industrial engineering through team-based hands-on activities. Explores the profession of ...
250.doc
Path: Washington >> IND E >> 250 Fall, 2008
Description: ABET Course Description IND E 250 Fundamentals of Engineering Economy Required Course Catalog course description: Basics of industrial cost analysis and accounting. Application of engineering economics to decision making. Analysis of engineering alte...
316.doc
Path: Washington >> IND E >> 316 Fall, 2008
Description: ABET Course Description IND E 316 Design of Experiments and Regression Analysis Required Course Catalog course description: Introduction to the analysis of data from planned experiments. Analysis of variance for multiple factors and applications of o...
351.doc
Path: Washington >> IND E >> 351 Spring, 2008
Description: ABET Course Description IND E 351 Human Factors in Design Elective Course Catalog Course Description: Engineering considerations of the abilities and limitations of the human aspect in the design of operational systems and components. Functional, psy...
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Path: Washington >> IND E >> 410 Fall, 2008
Description: UNIVERSITY OF OREGON Department of History Spring, 2006 Professor Quintard Taylor Office: 364 McKenzie Hall Email: qtaylor@uoregon.edu Phone: 346-6160 Web: http:/faculty.washington.edu/qtaylor/ Office Hours: 2:00-3:30 Tuesday History 410/510 Urban Ar...
412.doc
Path: Washington >> IND E >> 412 Fall, 2008
Description: ABET Course Description IND E 412 Integer and Dynamic Programming Elective Course Catalog course description: Modeling and optimization of problems and dynamic programming approach to optimization. Topics include: integer programming formulation tech...
424.doc
Path: Washington >> IND E >> 424 Winter, 2008
Description: ABET Course Description IND E 424 Simulation Elective Course Catalog course description: In this course, students will learn the processes, tools, and techniques for performing effective simulation analyses, specifically: i) (the basic underlying pri...
426.doc
Path: Washington >> IND E >> 426 Spring, 2008
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Path: Washington >> IND E >> 439 Fall, 2008
Description: ABET Course Description IND E 439 Plant Layout and Material Handling Elective Course Catalog course description: In this course, students will learn how to apply the concept of material flow in: 1) facilities design (designing high-performance interi...
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Path: Washington >> IND E >> 455 Fall, 2008
Description: ABET Course Description IND E 455 User Interface Design Elective Course Catalog Course Description: Design oriented to cover fundamentals of user interface design; models on human computer interaction, software psychology, input devices, usability, c...
494.doc
Path: Washington >> IND E >> 494 Winter, 2008
Description: ABET Course Description IND E 494 Design in the Manufacturing Firm Required Course Catalog course description: Engineering design in manufacturing firms is presented. Topics include design methodology, concurrent engineering, and project management. ...
Aut07-537-Storch.pdf
Path: Washington >> IND E >> 537 Fall, 2008
Description: Date 09/27 10/02 10/04 10/09 10/11 10/16 10/18 10/23 10/25 10/30 11/01 11/06 11/08 11/13 11/15 11/20 11/22 11/27 11/29 12/04 12/06 12/10 IND E 337A/537A INTRODUCTION TO MANUFACTURING SYSTEMS AUTUMN 2007 Topic Assignment _ Introduction _ Manu. Issues...
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Path: Washington >> INDN >> 401 Fall, 2008
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Path: Washington >> INDON >> 111 Fall, 2008
Description: ATM S 111 Global Warming: Understanding the Debate Fall 2008 Syllabus Course overview Human-induced climate change - popularly known as \"global warming\" - is emerging as one of the great challenges facing society in the 21st century. If we ignore ...
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Path: Washington >> INDON >> 111 Fall, 2008
Description: Land Clearing and the Biofuel Carbon Debt Joseph Fargione, et al. Science 319, 1235 (2008); DOI: 10.1126/science.1152747 The following resources related to this article are available online at www.sciencemag.org (this information is current as of Ap...
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Path: Washington >> INDON >> 111 Fall, 2008
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Path: Washington >> INDON >> 111 Fall, 2008
Description: Climate Change 2007: Impacts, Adaptation and Vulnerability The Intergovernmental Panel on Climate Change (IPCC) was set up jointly by the World Meteorological Organization and the United Nations Environment Programme to provide an authoritative inter...
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Path: Washington >> INDON >> 211 Fall, 2008
Description: Atmospheric Sciences 211: Climate and Climate Change Homework Set 3. Due February 14, 2000 February 7, 2000 1 Background: The purpose of this problem set is to use some of the concepts we have studied on atmospheric circulations and the hydrological...
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Path: Washington >> INDON >> 211 Fall, 2008
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homework4.pdf
Path: Washington >> INDON >> 211 Fall, 2008
Description: ATMOS SCI 211 HOMEWORK 4 Due: MONDAY, MAY 3 Part A About 11,000 years ago, when Europe and North America were recovering from the last ice age, the climate of Europe suddenly became colder again for several hundred years, before warming resumed. Wall...
hw5_sq2008_solns.pdf
Path: Washington >> INDON >> 211 Fall, 2008
Description: ATMS 211: Climate and Climate Change SQ2008 Assignment #5 Due Wed 28 May, 2008, in class or online via collect it. Name SOLNS Quiz Section: A (10:30) | B (11:30) Work with others, but write up answers independently. Points in bold after each questi...
UCONJ 450.doc
Path: Washington >> UCONJ >> 450 Winter, 2008
Description: UCONJ 450: Healthcare in Underserved Communities Offered Winter Quarter Elective course 1 credit with credit/no-credit grading Course description: Healthcare in the Underserved Community is a course organized by the Students in the Community (SITC) ...
UCONJ 540 - Mercury in Dent 2.ppt
Path: Washington >> UCONJ >> 540 Winter, 2008
Description: Mercury (Hg) in the Dental Setting Mercury Toxicity Elemental Mercury Little or no toxic effect when ingested Mercury vapors Readily absorb in tissue Significant toxicity if inhaled Targets: lungs, CNS, kidneys Mercury is a Common, Everyday ...
HISTORY.txt?limit=100&rev=198&mode=stop_on_copy&format=changelog
Path: Washington >> INFO >> 198 Fall, 2008
Description: # # ChangeLog for /ethnomed.contenttypes/trunk/docs/HISTORY.txt # # Generated by Trac 0.10.5 # 01/18/09 07:24:25 # 10/20/08 15:41:10 dunlapm [198] * ethnomed.contenttypes/trunk/docs/HISTORY.txt (modified) forgot to make notes in History 09/10/08 16:1...
