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### Topic 03

Course: STAT 502, Fall 2011
School: Purdue University -...
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3 Topic Multiple Regression Analysis Regression on Several Predictor Variables (Chapter 8) 1 Topic Overview Systolic Blood Pressure Example Multiple Regression Models SAS Output for Regression Multicollinearity 2 Systolic Blood Pressure Data In this topic we will fully analyze the SBP dataset described in Problem #5.2 in the text. This dataset illustrates some excellent points regarding multiple...

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3 Topic Multiple Regression Analysis Regression on Several Predictor Variables (Chapter 8) 1 Topic Overview Systolic Blood Pressure Example Multiple Regression Models SAS Output for Regression Multicollinearity 2 Systolic Blood Pressure Data In this topic we will fully analyze the SBP dataset described in Problem #5.2 in the text. This dataset illustrates some excellent points regarding multiple regression. The file 03SBP.sas provides the data and all of the code that has been utilized to produce output shown in the CLG handout and lecture notes. 3 Dataset Overview (n = 32) Response Variable: systolic blood pressure for an individual Note: SBP is the maximum pressure exerted when the heart contracts (top number). Predictor Variables Age (measured in years) Body Size (measured using the quetelet index) Smoking Status (0 = nonsmoker, 1 = smoker) 4 Multiple Regression Analysis For the SBP data, our goal is to determine whether SBP may be reasonably well predicted by some combination of age, body size, and smoking status. Additionally we may want to try to describe relationships and answer questions such as: Does the SBP increase (on average) with an increase in size? 5 Multiple Regression Analysis (2) The first step in a multiple regression analysis is to consider the individual variables and their pairwise (SLR) relationships. Identify potential problems (e.g. outliers) Identify and assess the form, direction, and strength of pairwise relationships. 6 CLG Activity #1 Please discuss questions 3.1-3.6 from the handout. Additional slides will be made available on the website after we have done the activities in class. 7 Multiple Regression Analysis (3) The next step in multiple regression analysis is to consider using more than one predictor variable in the same model. You can think of adding variables to the model in a certain order. Each variable takes up a portion of the total sums of squares. 8 Graphical View of MLR Rectangle represents total SS; Ovals represent variables; Note OVERLAP! X1 X3 X2 Total SSY 9 Some Key Points Must take into account relationships (correlation) among the potential predictor variables these relationships are responsible for the overlap in explained SS. Interaction between predictors may also become a consideration. Interaction means that the effect of one predictor changes depending on the value of the other. More on this later... 10 Some Additional Concerns Dealing with multiple predictors is much more difficult than SLR: More difficult to choose the best model (we now have many more choices). Calculation of estimates can be problematic generally we always employ a computer. Interpretation of the parameter estimates is usually less clear, and can in fact be meaningless if highly correlated variables are used in the same model. Harder to visualize the multi-dimensional relationship expressed by the model. 11 Examples of Models Suppose we have two predictors X1 and X2. Then possible models include (but are not limited to): Model 1: Y = 0 + 2 X 2 + Model 2: Y = 0 + 1 X 1 + 2 X 2 + Model 3: Y = 0 + 1 X 1 + 2 X 2 + 3 X 1 X 2 + Model 4: Y = 0 + 1 X 1 + 2 X 2 + 3 X 1 X 2 + 4 X 12 + 5 X 22 + 12 Best Model The best model will contain significant variables and exclude those that are not significant. BUT: Because of possible overlaps, it is not always easy to determine which variables should be excluded! Use scatter plots and residual plots to check the FORM of the model. In scatter plots of Y-hat vs. individual predictors, non-linear patterns indicate the need to transform a given predictor. 13 Best Estimates The best estimates for a MLR model are still those that minimize the SSE (least squares) n ( SSE = Yi Yi i =1 ) 2 Deviations are measured as before by the difference between the observed response and the predicted response. The Xs come into play only in terms of estimating the predicted values. 14 Notation Similar to SLR Greek Letters represent parameters Lower Case English represent estimates Parameter estimates / standard error formulas are much more complicated due to the fact we have multiple predictors 15 Matrix Approach Matrix approach simplifies notation somewhat. Estimates are: b = ( X ' X) 1 ( X 'Y) This is discussed in Appendix B, however we will rely on SAS to do the work for us. Our goal is to learn how to interpret the information. 16 Slope Estimates Some important properties of the estimates include the following: Each slope estimate is a linear function of the Ys (the linear coefficients are based on the Xs as was the case in SLR). Thus they are random variables. Correlation between Y and Y is maximized (while the sum of squared error is minimized). 17 Assumptions on Errors These are the same as before as well! A simple statement of the assumptions is that i ~ N ( 0, 2 ) Normality Given a particular set of predictors, the errors (or equivalently the Y|Xs) have a normal distribution. This provides justification for the use of T and F tests. 18 Assumptions on Errors (2) Constancy of Variance Variance of errors is the same regardless of the values of the predictor variables. Equivalent to say that the variance of Y given X = (X1,X2,...,Xk) is constant. Transformations If needed, we can often transform Y to achieve normality and/or constancy of variance. 19 Assumptions on Errors (3) Independence Errors (and hence responses) are statistically independent of one another. One common violation is repeated measures (measuring the same individual over time). If the relationship is linear (or transformable) then it is possible to include time in the model as a predictor variable and avoid this issue. 20 Linear Model Assumption Linearity means: Mean of Y is a linear function of the Xs Y = 0 + 1 X 1 + 2 X 2 + ... + k X k + Can have weird terms from transformations e.g. X 5 = X 2 X 3 or X 2 = log ( X 2 ) 21 Checking Assumptions Also very much the same as SLR Epsilon is the error component and is estimated by the residuals: ei = Yi Yi Residual plots can be considered to check for violations of the assumptions. May plot residuals versus each single predictor to assess constant variance and linearity with respect to that predictor. 22 ANOVA Table Similar idea to simple linear regression; information is organized into an ANOVA table. Still have (Corrected, for the mean) Total Sums of Squares which is split into SS for the regression model and SS error. n ( Y Y ) i =1 i SSTOT 2 = ( = SS R n i =1 Yi Y ) + ( n 2 i =1 + Yi Yi ) 2 SS E 23 Degrees of Freedom Degrees of freedom change Always have n 1 total degrees of freedom. The degrees of freedom for the regression is the number of slope estimates k (you do not count the intercept). Degrees of freedom for error is therefore n 1 k. 24 Regression Sums of Squares The SSR can be divided up into parts according to the addition of variables into the model. When this is done, we have an extended ANOVA table the model line is broken down into one line for each variable and each variable has 1 DF associated to it. IMPORTANT: This breakdown is order dependent! In SAS we will call this the TYPE I sums of squares. 25 Graphical View of MLR Rectangle represents total SS; Ovals represent SS explained by the different variables. Note that there overlap, is and the order in which variables are added is important! SS(X1) SS(X3|X1,X2) Total SSY SS(X2|X1) 26 Coefficient of Determination R2 is still the coefficient of determination. But it has a slightly different meaning in the context of MLR. R2 can be thought of as the percentage of variation in Y (as represented by the total SS) that is explained by the group of predictor variables in the model. R2 gives no indication of the importance of any particular predictor variable. 27 MLR Hypothesis Testing: Basics There are similarities to SLR, but there are also some big differences. 28 Parameter Estimates We will have a slope estimate for each variable. Each slope estimate will also have a standard error. Note that in SAS, the variable name is used to identify each line. Confidence intervals may be requested as before by using the CLB option in PROC REG. 29 Parameter Estimates (2) These are joint estimates. They assume other variables in the model. They will all change some if you remove or add a variable. (Note: If we are interested in interpretation, how much the estimates change in such a case is very important!) A Bonferroni adjustment may be appropriate to adjust for the fact we are computing several intervals. 30 Some Questions of Interest 1. Does the set of independent variables contribute anything worthwhile to the prediction of the response variable? 2. Given the present model, which variable(s) add significantly to the prediction of the response (over and above other predictors already included)? 3. Given the present model, is there any group of variables which will add significantly to the prediction of the response? 31 Answers? We find the answers to these questions by considering: F-test from the ANOVA table T-tests from the parameter estimates table Partial F tests for comparing different models; these are based on Type I and II SS which we will discuss in Topic 4 32 ANOVA F-test Null hypothesis is that ALL of the slope parameters are zero. Written: H 0 : 1 = 2 = L = k = 0 Alternative hypothesis is that at least one of the slope parameters is non-zero: H a : i 0 for some i Rejecting the null hypothesis says There is something in the model that is important in explaining the variation in the response. 33 ANOVA F-test (2) MSR The test statistic is F = MSE 2. MSE estimates the variance MSR estimates the variance under H0. Under HA, MSR will be larger than MSE; hence we reject H0 if F >> 1. Compare to F statistic with DF associated to the MSR and MSE; p-value computed by SAS. 34 ANOVA F-test (3) Failing to reject means that there is no evidence of association between the response variable Y and the group of predictors that are currently in the model. Failing to reject doesnt guarantee that no variables in the model are important. We might be lacking in power; we might also be missing an important variable that would reduce the total SS. Rejecting means that at least one predictor variable is important in predicting Y. Reject doesnt indicate which predictor(s) are important. 35 T-tests Null hypothesis for each t-test is that the slope parameter associated with the corresponding variable is zero. The alternative is that the associated slope parameter is not zero. H 0 : i = 0 vs. H a : i 0 KEY POINT: Other predictors are considered to be already in the model. So these tests are variable added last tests. 36 T-tests (2) Rejection means: Even with all of the other variables in the model and taking up as much of the SS as they can, this variable Xi is still important! Failing to reject means: This variable does not do anything useful when added to the other variables in the model. Key Point: This may just be due to overlap. The variable Xi if considered by itself may be associated to Y. 37 T-tests General Comments If only one of the variables has an insignificant t-test, and if the sample size is reasonable, then probably that variable is not important. Any variable that has a significant t-test is important and should remain in the model. If multiple variables have insignificant ttests, it is possible (and even likely) that some of them may still be important!!!! 38 Collinearity Issues If predictors are highly associated to each other, then they will likely be redundant when we consider them in the same model. 39 Collinearity Collinearity occurs when there are strong relationships (high intercorrelations) among the predictor variables. Some examples: X1 highly correlated to X2 X1 not highly correlated to X2, X3, or X4 individually, but together they explain most of the variation in X1. Note that when squares or interactions are used, e.g., X3 = X1*X2 will by default be correlated to X1 and X2. Centering can help in this case to decrease correlation. 40 Consequences Whenever we have collinearity, the predictor variables will fight to explain the same portion of the variation in the response (SS). If X1 and X2 are redundant, then the variable added last T-test will be insignificant for both even though we probably do want one of them in the model. When collinearity is present, standard errors will be large (inflated) and interpretation of parameter estimates is adversely affected. 41 Pictorial Representation X3 is highly correlated to X1 and X2. So in a regression model we are likely to use either X3 or (X1,X2), but not both. SS(X3|X1,X2) SS(X1) SS(X2|X1) Total SSY 42 Finding Intercollinearity Plots (X1 vs X2, etc.) Use PROC CORR to determine pairwise correlations. High values are problematic. (>0.9 is considered most serious, but even values between 0.5 and 0.9 can cause trouble). Variance Inflation Factors (>10 indicates multicollinearity) catch intercorrelation of the form X1 = X2 X3 X4. 43 Pairwise Scatterplots Can look at plots to find relationships among the predictors. Can use GPLOT for individual plots; for a first look, SAS Interactive Data Analysis provides a convenient scatter plot matrix: Solutions Analysis Interactive Data Analysis Select WORK library and the dataset for which you wish to obtain plots. Select columns you want to plot (can use control key to select multiple columns). Analyze Scatterplot Clicking a point in a plot will highlight that point in every plot. 44 PROC CORR Output contains r and a p-value. Significant relationships among the Xs can be problematic. Note that for both PROC CORR and scatterplots we are considering only pairwise correlations. It is possible to have more complicated correlations (e.g. maybe X2 and X3 explain 98% of X1 together, but only 50% individually). 45 Variance Inflation Factors VIF is related to the variance of the estimated regression coefficients (you can think of the SEs being inflated by having intercorrelation among the predictors) 1 V IFi = 1 - R i2 2 i R is the coefficient of determination obtained in regression of Xi on all other predictors. 46 VIFs in SAS Obtained by using VIF option in the model statement of PROC REG. VIF > 10 suggests multicollinearity. If this happens, a simplistic strategy is to remove the variable with the highest VIF and rerun the analysis. 47 CLG Activity #2 We now look at the SBP data from the MLR model viewpoint. Please discuss questions 3.7-3.9 from the handout. 48 Questions? 49 Upcoming in Topic 4... Partial F Tests in MLR Related Reading: Chapter 9 50
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Purdue University - Main Campus - STAT - 502
Topic 4 CLG Addendum1CLG Activity #1In your groups, please attempt thefirst set of questions. These willinvolve computing various extrasums of squares.2Question 4.1 &amp; 4.2Univariate and bivariate issues that mayaffect modeling include:For HSM, H
Purdue University - Main Campus - STAT - 502
Topic 4 Extra Sums of Squaresand the General Linear TestUsing Partial F Tests in MultipleRegression Analysis(Chapter 9)1Recall: Types of Tests1.ANOVA F Test: Does the group of predictorvariables explain a significant percentage of thevariation i
Purdue University - Main Campus - STAT - 502
CLG ActivityIn CLGs, please attempt Activity#1 from the handout for Topic 5.Note that this activity builds on theone we used for Topic #4.15.1 (a) &amp; (b)2GPA, HSM | HSS , HSE , SATM , SATVr2GPA , HSE | HSMr6.835== 0.061105.65 + 6.8352.48=
Purdue University - Main Campus - STAT - 502
Topic 5 Partial Correlations; Diagnostics &amp;Remedial MeasuresChapters 10 &amp; 141OverviewReview: MLR Tests &amp; Extra SSPartial Correlations Think Extra SS beingused to compute Extra R2Of the variation left to explain.How much isexplained by adding anot
Purdue University - Main Campus - STAT - 502
Topic 6 Model SelectionSelecting the BestRegression Model(Chapters 9 &amp; 16)1OverviewWe already have many of the pieces inplace. Well use those to developalgorithmic procedures.There are some additional statistics that canbe used for comparison of
Purdue University - Main Campus - STAT - 502
Topic 7 Other Regression IssuesReading: Some parts ofChapters 11 and 15OverviewConfounding (Chapter 11)Interaction (Chapter 11)Using Polynomial Terms (Chapter 15)Regression: Primary GoalsWe usually are focused on one of thefollowing goals:Predic
Purdue University - Main Campus - STAT - 502
Topic 8 One-Way ANOVASingle Factor Analysis of VarianceReading: 17.1, 17.2, &amp; 17.5Skim: 12.3, 17.3, 17.41OverviewNote: Entire topic constitutes some reviewas this would be the last thing you coveredin 501. We will cover it perhaps insomewhat more
Purdue University - Main Campus - STAT - 502
Topic 8 One-Way ANOVASingle Factor Analysis of VarianceReading: 17.1, 17.2, &amp; 17.5Skim: 12.3, 17.3, 17.41OverviewNote: Entire topic constitutes some reviewas this would be the last thing you coveredin 501. We will cover it perhaps insomewhat more
Purdue University - Main Campus - STAT - 502
Topic 9 Multiple ComparisonsMultiple Comparisonsof Treatment MeansReading: 17.7-17.81OverviewBrief Review of One-Way ANOVAPairwise Comparisons of Treatment MeansMultiplicity of TestingLinear Combinations &amp; Contrasts ofTreatment Means2Review: O
Purdue University - Main Campus - STAT - 502
Topic 10 ANCOVA &amp; RCBDAnalysis of Covariance (Ch. 13)Randomized Complete BlockDesigns (Ch. 18)1ReviewRecall the idea of confounding. Suppose I want todraw inference about a certain predictor.If meaningfully different interpretations would bemade
Purdue University - Main Campus - STAT - 502
Topic 11 ANOVA IIBalanced Two-Way ANOVA(Chapter 19)1Two Way ANOVAWe are now interested the combined effectsof two factors, A and B, on a response (note:text refers to these as R = rows andC = columns well call them A, B, and laterC for a 3-way AN
Purdue University - Main Campus - STAT - 502
Topic 12 Further Topics in ANOVAUnequal Cell Sizes(Chapter 20)1OverviewWell start with the Learning Activity.More practice in interpreting ANOVA results; anda baby-step into 3-way ANOVA.An illustration of the problems that anunbalanced design wil
Purdue University - Main Campus - STAT - 502
Topic 13 Random EffectsBackground Reading:Parts of Chapters 17 and 191Reading SummarySection 17.1 (particularly page 422)Section 17.6 (pages 438-440)Section 19.7 (pages 538-541)2Random EffectsSo far we have really only dealt with fixedeffects w
Purdue University - Main Campus - STAT - 502
Topic 14 Experimental DesignCrossoverNested FactorsRepeated Measures1OverviewWe will conclude the course by consideringsome different topics that can arise in amulti-way ANOVA, as well as some othermiscellaneous topics.Some of these are discusse
Purdue University - Main Campus - STAT - 506
STAT 506Homework 1You should first go the the website and download the Data folder for the Programming 1course, it is available in a zip file called prg1.zip. I suggest you use your H: drive for storagebut its really up to you. You will need to unzip i
Purdue University - Main Campus - STAT - 506
STAT 506Homework 2For most problems you will still need to access the data in the PRG1 folder. Use the libname statement we learned toload this each time you work on your assignments. You may now call it orion to be consistent with the SASmaterials. Th
Purdue University - Main Campus - STAT - 506
STAT 506Homework 3For most problems you will still need to access the data in the PRG1 folder. Use the libname statement we learned toload this each time you work on your assignments calling the library orion. I tried to bold the parts where I expectyo
Purdue University - Main Campus - STAT - 506
STAT 506Homework 4For most problems you will still need to access the data in the PRG1 folder. Use the libname statement we learned toload this each time you work on your assignments calling the library orion. I tried to bold the parts where I expectyo
Purdue University - Main Campus - STAT - 506
STAT 506Homework 5For most problems you will still need to access the data in the PRG1 folder. Use the libname statement we learned toload this each time you work on your assignments calling the library orion. I tried to bold the parts where I expectyo
Purdue University - Main Campus - STAT - 506
STAT 506Homework 6For most problems you will need to access the data in the PRG2 folder. Use the libname statement we learned to loadthis each time you work on your assignments calling the library orion. I tried to bold the parts where I expect you toa
Purdue University - Main Campus - STAT - 506
STAT 506Homework 7For most problems you will need to access the data in the PRG2 folder. Use the libname statement we learned to loadthis each time you work on your assignments calling the library orion. I tried to bold the parts where I expect you toa
Purdue University - Main Campus - STAT - 506
Chapter 1: Introduction1.1 Course Logistics1.2 Purpose of the Macro Facility1.3 Program Flow1Chapter 1: Introduction1.1 Course Logistics1.2 Purpose of the Macro Facility1.3 Program Flow2Objectives3Explain the naming convention that is used for
Purdue University - Main Campus - STAT - 506
Chapter 2: Macro Variables2.1 Introduction to Macro Variables2.2 Automatic Macro Variables2.3 Macro Variable References2.4 User-Defined Macro Variables2.5 Delimiting Macro Variable References2.6 Macro Functions1Chapter 2: Macro Variables2.1 Intro
Purdue University - Main Campus - STAT - 506
Chapter 3: Macro Definitions3.1 Defining and Calling a Macro3.2 Macro Parameters3.3 Macro Storage (Self-Study)1Chapter 3: Macro Definitions3.1 Defining and Calling a Macro3.2 Macro Parameters3.3 Macro Storage (Self-Study)2Objectives3Define and
Purdue University - Main Campus - STAT - 506
Chapter 4: DATA Step and SQL Interfaces4.1 Creating Macro Variables in the DATA Step4.2 Indirect References to Macro Variables4.3 Retrieving Macro Variables in the DATA Step(Self-Study)4.4 Creating Macro Variables in SQL1Chapter 4: DATA Step and SQ
Purdue University - Main Campus - STAT - 506
Chapter 5: Macro Programs5.1 Conditional Processing5.2 Parameter Validation5.3 Iterative Processing5.4 Global and Local Symbol Tables1Chapter 5: Macro Programs5.1 Conditional Processing5.2 Parameter Validation5.3 Iterative Processing5.4 Global a
Purdue University - Main Campus - STAT - 506
Chapter 6: Learning More6.1: SAS Resources6.2: Beyond This Course1Chapter 6: Learning More6.1: SAS Resources6.2: Beyond This Course2Objectives3Identify areas of support that SAS offers.EducationComprehensive training to deliver greater valuet
Purdue University - Main Campus - STAT - 506
Chapter 1: Introduction1.1 Course Logistics1.2 An Overview of Foundation SAS1Chapter 1: Introduction1.1 Course Logistics1.2 An Overview of Foundation SAS2Objectives3Explain the naming convention that is used for thecourse files.Compare the thr
Purdue University - Main Campus - STAT - 506
Chapter 2: Getting Started with SAS2.1 Introduction to SAS Programs2.2 Submitting a SAS Program1Chapter 2: Getting Started with SAS2.1 Introduction to SAS Programs2.2 Submitting a SAS Program2Objectives3List the components of a SAS program.Stat
Purdue University - Main Campus - STAT - 506
Chapter 3: Working with SAS Syntax3.1 Mastering Fundamental Concepts3.2 Diagnosing and Correcting Syntax Errors1Chapter 3: Working with SAS Syntax3.1 Mastering Fundamental Concepts3.2 Diagnosing and Correcting Syntax Errors2Objectives3Identify t
Purdue University - Main Campus - STAT - 506
Chapter 4: Getting Familiar withSAS Data Sets4.1 Examining Descriptor and Data Portions4.2 Accessing SAS Data Libraries4.3 Accessing Relational Databases (Self-Study)1Chapter 4: Getting Familiar withSAS Data Sets4.1 Examining Descriptor and Data P
Purdue University - Main Campus - STAT - 506
Chapter 5: Reading SAS Data Sets5.1 Introduction to Reading Data5.2 Using SAS Data as Input5.3 Subsetting Observations and Variables5.4 Adding Permanent Attributes1Chapter 5: Reading SAS Data Sets5.1 Introduction to Reading Data5.2 Using SAS Data
Purdue University - Main Campus - STAT - 506
Chapter 6: Reading Excel Worksheets6.1 Using Excel Data as Input6.2 Doing More with Excel Worksheets (Self-Study)1Chapter 6: Reading Excel Worksheets6.1 Using Excel Data as Input6.2 Doing More with Excel Worksheets (Self-Study)2Objectives3Use th
Purdue University - Main Campus - STAT - 506
Chapter 7: Reading Delimited Raw Data Files7.1 Using Standard Delimited Data as Input7.2 Using Nonstandard Delimited Data as Input1Chapter 7: Reading Delimited Raw Data Files7.1 Using Standard Delimited Data as Input7.2 Using Nonstandard Delimited D
Purdue University - Main Campus - STAT - 506
Chapter 8: Validating and Cleaning Data8.1 Introduction to Validating and Cleaning Data8.2 Examining Data Errors When Reading Raw Data Files8.3 Validating Data with the PRINT and FREQ Procedures8.4 Validating Data with the MEANS andUNIVARIATE Procedu
Purdue University - Main Campus - STAT - 506
Chapter 9: Manipulating Data9.1 Creating Variables9.2 Creating Variables Conditionally9.3 Subsetting Observations1Chapter 9: Manipulating Data9.1 Creating Variables9.2 Creating Variables Conditionally9.3 Subsetting Observations2Objectives3Crea
Purdue University - Main Campus - STAT - 511
Statistical Methods (STAT 511)Midterm #1, Spring 20118:00-10:00pm, LWSN B155Thursday, February 24, 2011There are totally 27 points in the exam. The students with score higher than or equal to 25points will receive 25 points. Please write down your na
Purdue University - Main Campus - STAT - 511
Statistical Methods (STAT 511)Midterm #1, Spring 20118:00-10:00pm, LWSN B155Thursday, February 24, 2011There are totally 27 points in the exam. The students with score higher than or equal to 25points will receive 25 points. Please write down your na
Purdue University - Main Campus - STAT - 511
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Purdue University - Main Campus - STAT - 511
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Purdue University - Main Campus - STAT - 511
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Purdue University - Main Campus - STAT - 511
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Purdue University - Main Campus - STAT - 511
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