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 23 Treatment Structure in a Split-Plot Design with the
Three-Way Interaction as the Whole-Plot
Comparison 684
16.9 23 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
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