Generalized Linear. Mixed Models. Methods and Applications. Modern Concepts, Walter W. Stroup. Texts in Statistical Science.

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1 Texts in Statistical Science Generalized Linear Mixed Models Modern Concepts, Methods and Applications Walter W. Stroup CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup an informa business A CHAPMAN & HALL BOOK

2 Preface Acknowledgments xv xxv Part I The Big Picture 1. Modeling Basics What Is a Model? Two Model Forms: Model Equation and Probability Distribution Twist Illustrating the Weakness of the Model Equation Form Types of Model Effects Extension of the Linear Regression Example to Illustrate an Important Distinction between Types of Model Effects Writing Models in Matrix Form Fixed-Effects-Only Models Mixed Models: Models with Fixed and Random Effects Summary: Essential Elements for a Complete Statement of the Model 20 Exercises Design Matters Introductory Ideas for Translating Design and Objectives into Models Chapter Organization Describing "Data Architecture" to Facilitate Model Specification Every Data Set Has a "Plot Plan" Terminology for Treatment and Design Structure Nested and Cross-Classification: Alternative Ways of Organizing Treatment and Design Structure From Plot Plan to Linear Predictor Unit of Replication Approach "What Would Fisher Do?" Complication Linear Predictor for Nested Schools Matching the Objective Distribution Matters Model Effects: Fixed or Random Response Variable Distribution Fixed or Random?: Tough Calls More Complex Example: Multiple Factors with Different Units of Replication Variations on the Multifactor, Multisize Unit of Replication Theme 46

3 iii Exercises 49 b) Distributions 51 2.A Appendix A: Common Response Variable (y 2.B Appendix B: Communicating Your Model to Software or "How SAS PROC GLIMMIX 'Thinks'" 52 2.B.1 General Principles Setting the Stage Goals for Inference with Models: Overview Basic Tools of Inference Estimable Functions Linear Combinations of Fixed and Random Effects: Predictable Functions Three Issues for Inference Model Scale vs. Data Scale Inference Space Inference Based on Conditional and Marginal Models Issue I: Data Scale vs. Model Scale Model Scale Estimation Data Scale Issue II: Inference Space Broad Inference Narrow Inference GLIMMIX Implementation Issue III: Conditional and Marginal Models Normal Approximation Binomial GLMM Conditional and Marginal Distribution Visualizing the Marginal p.d.f What Do the Normal Approximation and the GLMM Estimate? Gaussian Conditional and Marginal Models Non-Gaussian Marginal vs. Conditional Model One Last Aspect of the Conditional Model: The Role of "Residual" in Gaussian LMM vs. One-Parameter Non-Gaussian GLMM Summary 115 Exercises 116 Part II Estimation and Inference Essentials 4. Estimation Introduction Essential Background Exponential Family Essential Terminology and Results Maximum Likelihood Estimation Newton-Raphson and Fisher Scoring Quasi-Likelihood 127

4 ix 4.3 Fixed Effects Only Relation to Least Squares Estimation Pseudo-Likelihood for GLM Gaussian Linear Models and Ordinary Least Squares Gaussian Mixed Models Mixed Model Equations for p and b Relation to Least Squares Unknown G and R: ML and REML Variance-Covariance Component Estimation ANOVA Estimator Maximum Likelihood Restricted Maximum Likelihood Generalized Linear Mixed Models Pseudo-Likelihood for GLMM Variance-Covariance Estimation with Pseudo-Likelihood Integral Approximation: Laplace and Quadrature Summary 145 Exercises Inference, Part I: Model Effects Introduction Essential Background Estimable and Predictable Functions Estimability and GLMMs Basics of Interval Estimates and Test Statistics Approximate Distribution of Estimable and Predictable Functions Distribution of (3 in the LM with Known V Distribution of the Quadratic Form Defined on p for the LM with Known V LM with Unknown V LM with Unknown V: Case 1 V = LM with Unknown V: Case 2 All Covariance Components c2z 153 Must Be Estimated GLM GLM: Case 1 No Scale Parameter to Estimate GLM: Case 2 Estimated Scale Parameter(s) Mixed Models Approaches to Testing Likelihood Ratio and Deviance Wald and Approximate F-statistics A Special Case: The Gaussian LM with V=Ici Multiple Effect Models and Order of Testing Inference Model-Based Statistics Using 165 of Freedom Naive Statistics and Degrees Satterthwaite Degrees of Freedom Approximation Bias Correction for Model-Based Standard Errors and Test Statistics 168

5 X 5.5 Inference Using Empirical Standard Error Sandwich (a.k.a Robust or Empirical) Estimator Bias Correction for Sandwich Estimators Summary of Main Ideas and General Guidelines for Implementation 173 Exercises Inference, Part II: Covariance Components Introduction Formal Testing of Covariance Components ANOVA-Based Tests for Variance-Component-Only LMMs Wald Statistics for Covariance Component Testing and Why They Should Not Be Used Likelihood Ratio Tests for Covariance Components 182 of Variance One-Way ANOVA: Test for Homogeneity Repeated Measures Example: Selecting a Parsimonious Covariance Model Consequences of PL versus Integral Approximation for GLMMs R-Side or Working Correlation Model "What Would Fisher Do?" The G-Side Approach R-Side versus G-Side: Consequences for Covariance Model Selection Fit Statistics to Compare Covariance Models AIC and AICC BIC Application to Comparison of Covariance Models Interval Estimation Wald Approach Based on the x Likelihood-Based Approach Summary 195 Exercises 196 Part III Working with GLMMs 7. Treatment and Explanatory Variable Structure Types of Treatment Structures Types of Estimable Functions Relation to Classical ANOVA Reduction Sums of Squares How Do We Know What We Are Testing? How to Decide What to Test Rather than Letting It Be Decided for Us Multiplicity Multiple Factor Models: Overview Multifactor Models with All Factors Qualitative Review of Options Tools for Qualitative Factorial Inference: "SLICE," "SLICEDIFF," and Other Tools Multiplicity Adjustments 216

6 xi 7.5 Multifactor: Some Factors Qualitative, Some Factors Quantitative Generic Form of the Linear Predictor Many Uses of the Generic Linear Predictor Latent Growth Curve Models Analysis of Covariance Factorial Treatment Design Multifactor: All Factors Quantitative Second-Order Polynomial, a.k.a. Classical Response Surface Linear Predictors Other Quantitative-by-Quantitative Models Nonlinear Mean Models Spline or Segmented Regression Summary Multilevel Models Types of Design Structure: Single- and Multilevel Models Defined Types of Multilevel Models and How They Arise Units of Replication: Not Just in Designed Experiments "What Would Fisher Do?" Revisited: Topographical and Treatment Component Role of Blocking in Multilevel Models "Block Effects Fixed vs. Block Effects Random" Revisited Fixed Blocks, Multilevel Designs, and Spurious Nonestimability Working with Multilevel Designs Examples of Multilevel Structures Multifactor Treatment and Multilevel Design Structures: How They Fit Together Marginal and Conditional Multilevel Models Gaussian Data Non-Gaussian Models Summary 267 Exercises Best Linear Unbiased Prediction Review of Estimable and Predictable Functions BLUP in Random-Effects-Only Models One-Way Random Effects Model Two-Way Random Effects Nested Model Analysis: Balanced Case Unbalanced Case Gaussian Data with Fixed and Random Effects Mixed-Model Analysis with BLUP to Modify the Inference Space Relationship 9.4 Advanced Applications with Complex between BLUP and Fixed Effect Estimators 288 Z Matrices Summary Rates and Proportions Types of Rate and Proportion Data 299

7 xii 10.2 Discrete Proportions: Binary and Binomial Data Pseudo-Likelihood or Integral Approximation Example of Explanatory-Response Models Models for Contingency Tables Alternative Link Functions for Binomial Data Role of "Residual" in Binomial Models Continuous Proportions Beta Distribution Continuous Proportion Example Using the Beta Distribution Summary 330 Exercises Counts Introduction Count Data and the Poisson Distribution Example Comparing Pre-GLM ANOVA-Based Analysis to Poisson GLM Overdispersion in Count Data Overdispersion Defined Detecting Overdispersion Strategies Scale Parameter "What Would Fisher Do?" Revisited Alternative Distributions More on Alternative Distributions Negative Binomial Generalized Poisson Conditional and Marginal Too Many Zeroes Formal Description of Zero-Inflated and Hurdle Models GLMM for Poisson and Negative Binomial Zero-Inflated and Hurdle Models Summary 369 Exercises Time-to-Event Data Introduction: Probability Concepts for Time-to-Event Data Gamma GLMMs Hierarchical (Split-Plot) Gamma GLMM What Happens If We Fit This Model Using a Gaussian LMM? Gamma Generalized Linear Model Response Surface for Time-to-Event: An Example Using the Box-Behnken Design Gaussian LMM Gamma GLMM GLMMs and Survival Analysis Basic Concepts and Terminology 387

8 xiii Exponential Survival GLMM for Uncensored Data Exponential Survival GLMM for Censored Data Summary Multinomial Data Overview Multinomial Data with Ordered Categories Nominal Categories: Generalized Logit Models Model Comparison Summary 410 Exercises Correlated Errors, Part I: Repeated Measures Overview What Are Repeated Measures/Longitudinal Data Pre-GLMM Methods Gaussian Data: Correlation and Covariance Models for LMMs Covariance Model Selection, Why Does It Matter? Covariance Model Selection Methods Non-Gaussian Case GEE-Type Models GLMMs Issues for Non-Gaussian Repeated Measures How Do Correlated Errors Arise? Deciding What We Are Modeling Covariance Model Selection and Non-Gaussian Repeated Measures Inference Space, Standard Errors, and Test Statistics Summary 437 Exercises Correlated Errors, Part II: Spatial Variability Overview Types of Spatial Variability Pre-GLMM Methods Nearest-Neighbor Adjustment Blocking Gaussian Case with Covariance Model Covariance Model Selection Impact of Spatial Variability on Inference Spatial Covariance Modeling by Smoothing Spline Non-Gaussian Case Randomized Complete Block Model Incomplete Block Model GLIMMIX Statements RCB Lattice Incomplete Blocks 457

9 xiv GEE-Type "R-Side" Spatial Correlation Model "G-Side" Spatial Correlation Model G-Side Spatial Radial Smoothing Model Relevant Output Summary 464 Exercise Power, Sample Size, and Planning Basics of GLMM-Based Power and Precision Analysis Essential GLMM Theory for Power and Precision Analysis Using SAS PROC GLIMMIX to Implement a Power Analysis Gaussian Example Power for Binomial GLMMs GLMM-Based Power Analysis for Count Data Power and Planning for Repeated Measures Straightforward Cases: Gaussian and One-Parameter Exponential Family On the Frontier: The Two-Parameter Exponential Family Summary 492 Exercises 494 Appendices: Essential Matrix Operations and Results 499 Appendix A: Matrix Operations 501 Appendix B: Distribution Theory for Matrices 509 References 513 Index 519