Chapter 4 Multi-factor Treatment Designs with Multiple Error Terms 93

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1 Contents Preface ix Chapter 1 Introduction Types of Models That Produce Data Statistical Models Fixed and Random Effects Mixed Models Typical Studies and the Modeling Issues They Raise A Typology for Mixed Models Flowcharts to Select SAS Software to Run Various Mixed Models 13 Chapter 2 Randomized Block Designs Introduction Mixed Model for a Randomized Complete Blocks Design Using PROC MIXED to Analyze RCBD Data Introduction to Theory of Mixed Models Example of an Unbalanced Two-Way Mixed Model: Incomplete Block Design Summary 56 Chapter 3 Random Effects Models Introduction: Descriptions of Random Effects Models Example: One-Way Random Effects Treatment Structure Example: A Simple Conditional Hierarchical Linear Model Example: Three-Level Nested Design Structure Example: A Two-Way Random Effects Treatment Structure to Estimate Heritability Summary 91 Chapter 4 Multi-factor Treatment Designs with Multiple Error Terms Introduction Treatment and Experiment Structure and Associated Models Inference with Mixed Models for Factorial Treatment Designs Example: A Split-Plot Semiconductor Experiment 113

2 iv Contents 4.5 Comparison with PROC GLM Example: Type Dose Response Example: Variance Component Estimates Equal to Zero More on PROC GLM Compared to PROC MIXED: Incomplete Blocks, Missing Data, and Estimability Summary 156 Chapter 5 Analysis of Repeated Measures Data Introduction Example: Mixed Model Analysis of Data from Basic Repeated Measures Design Modeling Covariance Structure Example: Unequally Spaced Repeated Measures Summary 202 Chapter 6 Best Linear Unbiased Prediction Introduction Examples of BLUP Basic Concepts of BLUP Example: Obtaining BLUPs in a Random Effects Model Example: Two-Factor Mixed Model A Multilocation Example Location-Specific Inference in Multicenter Example Summary 241 Chapter 7 Analysis of Covariance Introduction One-Way Fixed Effects Treatment Structure with Simple Linear Regression Models Example: One-Way Treatment Structure in a Randomized Complete Block Design Structure Equal Slopes Model Example: One-Way Treatment Structure in an Incomplete Block Design Structure Time to Boil Water Example: One-Way Treatment Structure in a Balanced Incomplete Block Design Structure Example: One-Way Treatment Structure in an Unbalanced Incomplete Block Design Structure Example: Split-Plot Design with the Covariate Measured on the Large-Size Experimental Unit or Whole Plot Example: Split-Plot Design with the Covariate Measured on the Small-Size Experimental Unit or Subplot Example: Complex Strip-Plot Design with the Covariate Measured on an Intermediate-Size Experimental Unit Summary 315

3 Contents v Chapter 8 Random Coefficient Models Introduction Example: One-Way Random Effects Treatment Structure in a Completely Randomized Design Structure Example: Random Student Effects Example: Repeated Measures Growth Study Summary 341 Chapter 9 Heterogeneous Variance Models Introduction Example: Two-Way Analysis of Variance with Unequal Variances Example: Simple Linear Regression Model with Unequal Variances Example: Nested Model with Unequal Variances for a Random Effect Example: Within-Subject Variability Example: Combining Between- and Within-Subject Heterogeneity Example: Log-Linear Variance Models Summary 411 Chapter 10 Mixed Model Diagnostics Introduction From Linear to Linear Mixed Models The Influence Diagnostics Example: Unequally Spaced Repeated Measures Summary 435 Chapter 11 Spatial Variability Introduction Description Spatial Correlation Models Spatial Variability and Mixed Models Example: Estimating Spatial Covariance Using Spatial Covariance for Adjustment: Part 1, Regression Using Spatial Covariance for Adjustment: Part 2, Analysis of Variance Example: Spatial Prediction Kriging Summary 478 Chapter 12 Power Calculations for Mixed Models Introduction Power Analysis of a Pilot Study Constructing Power Curves Comparing Spatial Designs 486

4 vi Contents 12.5 Power via Simulation Summary 495 Chapter 13 Some Bayesian Approaches to Mixed Models Introduction and Background P-Values and Some Alternatives Bayes Factors and Posterior Probabilities of Null Hypotheses Example: Teaching Methods Generating a Sample from the Posterior Distribution with the PRIOR Statement Example: Beetle Fecundity Summary 524 Chapter 14 Generalized Linear Mixed Models Introduction Two Examples to Illustrate When Generalized Linear Mixed Models Are Needed Generalized Linear Model Background From GLMs to GLMMs Example: Binomial Data in a Multi-center Clinical Trial Example: Count Data in a Split-Plot Design Summary 566 Chapter 15 Nonlinear Mixed Models Introduction Background on PROC NLMIXED Example: Logistic Growth Curve Model Example: Nested Nonlinear Random Effects Models Example: Zero-Inflated Poisson and Hurdle Poisson Models Example: Joint Survival and Longitudinal Model Example: One-Compartment Pharmacokinetic Model Comparison of PROC NLMIXED and the %NLINMIX Macro Three General Fitting Methods Available in the %NLINMIX Macro Troubleshooting Nonlinear Mixed Model Fitting Summary 634 Chapter 16 Case Studies Introduction Response Surface Experiment in a Split-Plot Design Response Surface Experiment with Repeated Measures 643

5 Contents vii 16.4 A Split-Plot Experiment with Correlated Whole Plots A Complex Split Plot: Whole Plot Conducted as an Incomplete Latin Square A Complex Strip-Split-Split-Plot Example Unreplicated Split-Plot Design Treatment Structure in a Split-Plot Design with the Three-Way Interaction as the Whole-Plot Comparison Treatment Structure in an Incomplete Block Design Structure with Balanced Confounding Product Acceptability Study with Crossover and Repeated Measures Random Coefficients Modeling of an AIDS Trial Microarray Example 727 Appendix 1 Linear Mixed Model Theory 733 A1.1 Introduction 734 A1.2 Matrix Notation 734 A1.3 Formulation of the Mixed Model 735 A1.4 Estimating Parameters, Predicting Random Effects 742 A1.5 Statistical Properties 751 A1.6 Model Selection 752 A1.7 Inference and Test Statistics 754 Appendix 2 Data Sets 757 A2.2 Randomized Block Designs 759 A2.3 Random Effects Models 759 A2.4 Analyzing Multi-level and Split-Plot Designs 761 A2.5 Analysis of Repeated Measures Data 762 A2.6 Best Linear Unbiased Prediction 764 A2.7 Analysis of Covariance 765 A2.8 Random Coefficient Models 768 A2.9 Heterogeneous Variance Models 769 A2.10 Mixed Model Diagnostics 771 A2.11 Spatial Variability 772 A2.13 Some Bayesian Approaches to Mixed Models 773 A2.14 Generalized Linear Mixed Models 774 A2.15 Nonlinear Mixed Models 775 A2.16 Case Studies 776 References 781 Index 795

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