Contents Preface ix Chapter 1 Introduction 1 1.1 Types of Models That Produce Data 1 1.2 Statistical Models 2 1.3 Fixed and Random Effects 4 1.4 Mixed Models 6 1.5 Typical Studies and the Modeling Issues They Raise 7 1.6 A Typology for Mixed Models 11 1.7 Flowcharts to Select SAS Software to Run Various Mixed Models 13 Chapter 2 Randomized Block Designs 17 2.1 Introduction 18 2.2 Mixed Model for a Randomized Complete Blocks Design 18 2.3 Using PROC MIXED to Analyze RCBD Data 22 2.4 Introduction to Theory of Mixed Models 42 2.5 Example of an Unbalanced Two-Way Mixed Model: Incomplete Block Design 44 2.6 Summary 56 Chapter 3 Random Effects Models 57 3.1 Introduction: Descriptions of Random Effects Models 58 3.2 Example: One-Way Random Effects Treatment Structure 64 3.3 Example: A Simple Conditional Hierarchical Linear Model 75 3.4 Example: Three-Level Nested Design Structure 81 3.5 Example: A Two-Way Random Effects Treatment Structure to Estimate Heritability 88 3.6 Summary 91 Chapter 4 Multi-factor Treatment Designs with Multiple Error Terms 93 4.1 Introduction 94 4.2 Treatment and Experiment Structure and Associated Models 94 4.3 Inference with Mixed Models for Factorial Treatment Designs 102 4.4 Example: A Split-Plot Semiconductor Experiment 113
iv Contents 4.5 Comparison with PROC GLM 130 4.6 Example: Type Dose Response 135 4.7 Example: Variance Component Estimates Equal to Zero 148 4.8 More on PROC GLM Compared to PROC MIXED: Incomplete Blocks, Missing Data, and Estimability 154 4.9 Summary 156 Chapter 5 Analysis of Repeated Measures Data 159 5.1 Introduction 160 5.2 Example: Mixed Model Analysis of Data from Basic Repeated Measures Design 163 5.3 Modeling Covariance Structure 174 5.4 Example: Unequally Spaced Repeated Measures 198 5.5 Summary 202 Chapter 6 Best Linear Unbiased Prediction 205 6.1 Introduction 206 6.2 Examples of BLUP 206 6.3 Basic Concepts of BLUP 210 6.4 Example: Obtaining BLUPs in a Random Effects Model 212 6.5 Example: Two-Factor Mixed Model 219 6.6 A Multilocation Example 226 6.7 Location-Specific Inference in Multicenter Example 234 6.8 Summary 241 Chapter 7 Analysis of Covariance 243 7.1 Introduction 244 7.2 One-Way Fixed Effects Treatment Structure with Simple Linear Regression Models 245 7.3 Example: One-Way Treatment Structure in a Randomized Complete Block Design Structure Equal Slopes Model 251 7.4 Example: One-Way Treatment Structure in an Incomplete Block Design Structure Time to Boil Water 263 7.5 Example: One-Way Treatment Structure in a Balanced Incomplete Block Design Structure 272 7.6 Example: One-Way Treatment Structure in an Unbalanced Incomplete Block Design Structure 281 7.7 Example: Split-Plot Design with the Covariate Measured on the Large-Size Experimental Unit or Whole Plot 286 7.8 Example: Split-Plot Design with the Covariate Measured on the Small-Size Experimental Unit or Subplot 297 7.9 Example: Complex Strip-Plot Design with the Covariate Measured on an Intermediate-Size Experimental Unit 308 7.10 Summary 315
Contents v Chapter 8 Random Coefficient Models 317 8.1 Introduction 317 8.2 Example: One-Way Random Effects Treatment Structure in a Completely Randomized Design Structure 320 8.3 Example: Random Student Effects 326 8.4 Example: Repeated Measures Growth Study 330 8.5 Summary 341 Chapter 9 Heterogeneous Variance Models 343 9.1 Introduction 344 9.2 Example: Two-Way Analysis of Variance with Unequal Variances 345 9.3 Example: Simple Linear Regression Model with Unequal Variances 354 9.4 Example: Nested Model with Unequal Variances for a Random Effect 366 9.5 Example: Within-Subject Variability 374 9.6 Example: Combining Between- and Within-Subject Heterogeneity 393 9.7 Example: Log-Linear Variance Models 402 9.8 Summary 411 Chapter 10 Mixed Model Diagnostics 413 10.1 Introduction 413 10.2 From Linear to Linear Mixed Models 415 10.3 The Influence Diagnostics 424 10.4 Example: Unequally Spaced Repeated Measures 426 10.5 Summary 435 Chapter 11 Spatial Variability 437 11.1 Introduction 438 11.2 Description 438 11.3 Spatial Correlation Models 440 11.4 Spatial Variability and Mixed Models 442 11.5 Example: Estimating Spatial Covariance 447 11.6 Using Spatial Covariance for Adjustment: Part 1, Regression 457 11.7 Using Spatial Covariance for Adjustment: Part 2, Analysis of Variance 460 11.8 Example: Spatial Prediction Kriging 471 11.9 Summary 478 Chapter 12 Power Calculations for Mixed Models 479 12.1 Introduction 479 12.2 Power Analysis of a Pilot Study 480 12.3 Constructing Power Curves 483 12.4 Comparing Spatial Designs 486
vi Contents 12.5 Power via Simulation 489 12.6 Summary 495 Chapter 13 Some Bayesian Approaches to Mixed Models 497 13.1 Introduction and Background 497 13.2 P-Values and Some Alternatives 499 13.3 Bayes Factors and Posterior Probabilities of Null Hypotheses 502 13.4 Example: Teaching Methods 507 13.5 Generating a Sample from the Posterior Distribution with the PRIOR Statement 509 13.6 Example: Beetle Fecundity 511 13.7 Summary 524 Chapter 14 Generalized Linear Mixed Models 525 14.1 Introduction 526 14.2 Two Examples to Illustrate When Generalized Linear Mixed Models Are Needed 527 14.3 Generalized Linear Model Background 529 14.4 From GLMs to GLMMs 538 14.5 Example: Binomial Data in a Multi-center Clinical Trial 542 14.6 Example: Count Data in a Split-Plot Design 557 14.7 Summary 566 Chapter 15 Nonlinear Mixed Models 567 15.1 Introduction 568 15.2 Background on PROC NLMIXED 569 15.3 Example: Logistic Growth Curve Model 571 15.4 Example: Nested Nonlinear Random Effects Models 587 15.5 Example: Zero-Inflated Poisson and Hurdle Poisson Models 589 15.6 Example: Joint Survival and Longitudinal Model 595 15.7 Example: One-Compartment Pharmacokinetic Model 607 15.8 Comparison of PROC NLMIXED and the %NLINMIX Macro 623 15.9 Three General Fitting Methods Available in the %NLINMIX Macro 625 15.10 Troubleshooting Nonlinear Mixed Model Fitting 629 15.11 Summary 634 Chapter 16 Case Studies 637 16.1 Introduction 638 16.2 Response Surface Experiment in a Split-Plot Design 639 16.3 Response Surface Experiment with Repeated Measures 643
Contents vii 16.4 A Split-Plot Experiment with Correlated Whole Plots 650 16.5 A Complex Split Plot: Whole Plot Conducted as an Incomplete Latin Square 659 16.6 A Complex Strip-Split-Split-Plot Example 667 16.7 Unreplicated Split-Plot Design 674 16.8 2 3 Treatment Structure in a Split-Plot Design with the Three-Way Interaction as the Whole-Plot Comparison 684 16.9 2 3 Treatment Structure in an Incomplete Block Design Structure with Balanced Confounding 694 16.10 Product Acceptability Study with Crossover and Repeated Measures 699 16.11 Random Coefficients Modeling of an AIDS Trial 716 16.12 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