# Applied Regression Analysis and Other Multivariable Methods

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1 THIRD EDITION Applied Regression Analysis and Other Multivariable Methods David G. Kleinbaum Emory University Lawrence L. Kupper University of North Carolina, Chapel Hill Keith E. Muller University of North Carolina, Chapel Hill Azhar Nizam Emory University An Alexander Kugushev Book ^P> Duxbury Press An Imprint of Brooks/Cole Publishing Company l P An International Thomson Publishing Company Pacific Grove Albany Belmont Bonn Boston Cincinnati Detroit Johannesburg London Madrid Melbourne Mexico City New York Paris Singapore Tokyo Toronto Washington

2 Contents 1 CONCEPTS AND EXAMPLES OF RESEARCH 1-1 Concepts Examples Concluding Remarks References 6 2 CLASSIFICATION OF VARIABLES AND THE CHOICE OF ANALYSIS 2-1 Classification of Variables Overlapping of Classification Schemes Choice of Analysis 11 References 13 3 BASIC STATISTICS: A REVIEW Preview Descriptive Statistics Random Variables and Distributions Sampling Distributions of /, # 2, and F Statistical Inference: Estimation Statistical Inference: Hypothesis Testing 24 xi

3 xii Contents 3-7 Error Rates, Power, and Sample Size 28 Problems 30 References 33 4 INTRODUCTIONTO REGRESSION ANALYSIS Preview Association versus Causality Statistical versus Deterministic Models Concluding Remarks 38 References 38 5 STRAIGHT-LINE REGRESSION ANALYSIS Preview Regression with a Single Independent Variable Mathematical Properties of a Straight Line Statistical Assumptions for a Straight-line Model Determining the Best-fitting Straight Line Measure of the Quality of the Straight-line Fit and Estimate of a Inferences About the Slope and Intercept Interpretations of Tests for Slope and Intercept Inferences About the Regression Line JU Y \X = ßo + ßi% Prediction of a New Value of FatX Assessing the Appropriateness of the Straight-line Model 60 Problems 60 References 87 THE CORRELATION COEFFICIENT AND STRAIGHT-LINE REGRESSION ANALYSIS Definition of r ras a Measure of Association The Bivariate Normal Distribution r and the Strength of the Straight-line Relationship What r Does Not Measure Tests of Hypotheses and Confidence Intervals for the Correlation Coefficient Testing for the Equality of Two Correlations 99 Problems 101 References 103

4 Contents xiii 7 THE ANALYSIS-OF-VARIANCE TABLE Preview The ANOVA Table for Straight-line Regression 104 Problems 108 ß ö MULTIPLE REGRESSION ANALYSIS: GENERAL CONSIDERATIONS Preview Multiple Regression Models Graphical Look at the Problem Assumptions of Multiple Regression Determining the Best Estimate of the Multiple Regression Equation The ANOVA Table for Multiple Regression Numerical Examples 121 Problems 123 References TESTING HYPOTHESES IN MULTIPLE REGRESSION Preview Test for Significant Overall Regression PartialFTest Multiple PartialFTest Strategies for Using Partial F Tests Tests Involving the Intercept 150 Problems 151 References 159 CORRELATIONS: MULTIPLE, PARTIAL, AND MULTIPLE PARTIAL Preview Correlation Matrix Multiple Correlation Coefficient Relationship of Ry\x h x 2,...,x k to the Multivariate Normal Distribution Partial Correlation Coefficient Alternative Representation of the Regression Model 172

5 xiv Contents 10-7 Multiple Partial Correlation Concluding Remarks 174 Problems 174 Reference CONFOUNDING AND INTERACTION IN REGRESSION 11-1 Preview Overview Interaction in Regression Confounding in Regression Summary and Conclusions 199 Problems 199 Reference REGRESSION DIAGNOSTICS Preview Simple Approaches to Diagnosing Problems in Data Residual Analysis Treating Outliers Collinearity Scaling Problems Treating Collinearity and Scaling Problems Alternate Strategies of Analysis An Important Caution 252 Problems 253 References POLYNOMIAL REGRESSION Preview Polynomial Models Least-squares Procedure for Fitting a Parabola ANOVA Table for Second-order Polynomial Regression Inferences Associated with Second-order Polynomial Regression 13-6 Example Requiring a Second-order Model Fitting and Testing Higher-order Models Lack-of-fit Tests Orthogonal Polynomials 292

6 Contents xv Strategies for Choosing a Polynomial Model 301 Problems DUMMY VARIABLES IN REGRESSION Preview Definitions Rule for Defming Dummy Variables Comparing Two Straight-line Regression Equations: An Example Questions for Comparing Two Straight Lines Methods of Comparing Two Straight Lines Method I: Using Separate Regression Fits to Compare Two Straight Lines Method II: Using a Single Regression Equation to Compare Two Straight Lines Comparison of Methods I and II Testing Strategies and Interpretation: Comparing Two Straight Lines Other Dummy Variable Models Comparing Four Regression Equations Comparing Several Regression Equations Involving Two Nominal Variables 336 Problems 338 References 360 -, c ANALYSIS OF COVARIANCE AND OTHER 1 ^ METHODS FOR ADJUSTING CONTINUOUS DATA Preview Adjustment Problem Analysis of Covariance Assumption of Parallelism: A Potential Drawback Analysis of Covariance: Several Groups and Several Covariates Comments and Cautions Summary 371 Problems 371 Reference SELECTING THE BEST REGRESSION EQUATION Preview Steps in Selecting the Best Regression Equation Step 1: Specifying the Maximum Model Step 2: Specifying a Criterion for Selecting a Model Step 3: Specifying a Strategy for Selecting Variables 392

7 xvi Contents 16-6 Step 4: Conducting the Analysis Step 5: Evaluating Reliability with Split Samples Example Analysis ofactual Data Issues in Selecting the Most Valid Model 409 Problems 409 References ONE-WAY ANALYSIS OFVARIANCE Preview One-way ANOVA: The Problem, Assumptions, and Data Configuration Methodology for One-way Fixed-effects ANOVA Regression Model for Fixed-effects One-way ANOVA Fixed-effects Model for One-way ANOVA Random-effects Model for One-way ANOVA Multiple-comparison Procedures for Fixed-effects One-way ANOVA Choosing a Multiple-comparison Technique Orthogonal Contrasts and Partitioning an ANOVA Sum of Squares 457 Problems 463 References RANDOMIZED BLOCKS: SPECIAL CASE OF TWO-WAY ANOVA 18-1 Preview Equivalent Analysis of a Matched Pairs Experiment PrincipleofBlocking Analysis of a Randomized-blocks Experiment ANOVA Table for a Randomized-blocks Experiment Regression Models for a Randomized-blocks Experiment Fixed-effects ANOVA Model for a Randomized-blocks Experiment 502 Problems 503 References TWO-WAY ANOVA WITH EQUAL CELL NUMBERS Preview Usinga Table of Cell Means General Methodology F Tests for Two-way ANOVA Regression Model for Fixed-effects Two-way ANOVA Interactions in Two-way ANOVA Random- and Mixed-effects Two-way ANOVA Models 541 Problems 544 References 560

8 Contents xvii 20 TWO-WAY ANOVA WITH UNEQUAL CELL NUMBERS Preview Problems with Unequal Cell Numbers: Nonorthogonality Regression Approach for Unequal Cell Sample Sizes Higher-way ANOVA 571 Problems 572 References ANALYSIS OF REPEATED MEASURES DATA Preview Examples General Approach for Repeated Measures ANOVA Overview of Selected Repeated Measures Designs and ANOVA-based Analyses Repeated Measures ANOVA for Unbalanced Data Other Approaches to Analyzing Repeated Measures Data 612 Appendix 21-A Examples of SAS's GLM and MIXED Procedures 613 Problems 616 References THE METHOD OF MAXIMUM LIKELIHOOD Preview The Principle of Maximum Likelihood Statistical Inference via Maximum Likelihood Summary 652 Problems 653 References LOGISTIC REGRESSION ANALYSIS Preview The Logistic Model Estimating the Odds Ratio Using Logistic Regression A Numerical Example of Logistic Regression Theoretical Considerations An Example of Conditional ML Estimation Involving Pair-matched Data with Unmatched Covariates Summary 681 Problems 682 References 686

9 xviii Contents 24 POISSON REGRESSION ANALYSIS Preview The Poisson Distribution An Example of Poisson Regression Poisson Regression: General Considerations Measures of Goodness of Fit Continuation of Skin Cancer Data Example A Second Illustration of Poisson Regression Analysis Summary 704 Problems 705 References 709 A APPENDIX A TABLES 711 A-l Standard Normal Cumulative Probabilities 712 A-2 Percentiles of the t Distribution 715 A-3 Percentiles of the Chi-square Distribution 716 A-4 Percentiles of the F Distribution 717 A-5 Values off In- 1 +r 1 -r 724 A-6 Upper a Point of Studentized Range 726 A-7 Orthogonal Polynomial Coefficients 728 A-8 Bonferroni Corrected Jackknife and Studentized Residual Critical Values 729 A-9 Critical Values for Leverages 730 A-10 Critical Values for the Maximum of N Values of Cook's d(i) times (n-k-\) 731 B APPENDIX B MATRICES AND THEIR RELATIONSHIP TO REGRESSION ANALYSIS 732 APPENDIX C ANOVA INFORMATION FOR FOUR COMMON BALANCED REPEATED MEASURES DESIGNS 744 C-1 Balanced Repeated Measures Design with One Crossover Factor (Treatments) 744 C-2 Balanced Repeated Measures Design with Two Crossover Factors 746 C-3 Balanced Repeated Measures Design with One Nest Factor (Treatments) 750 C-4 Balanced Repeated Measures Design with One Crossover Factor and One Nest Factor 752 C-5 Balanced Two-group Pre/Posttest Design 755 References 757 ) SOLUTIONS TO EXERCISES 758 INDEX 787

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