Books attempting to serve as practical guides on the use of statistical software are always at risk of becoming outdated as the software continues to develop,especially in an area of statistics and data analysis that has received as much research attention as linear mixed models.In fact,much has changed since the first publication of this book in early 2007,and while we tried to keep pace with these changes on the website for this book,the demand for a second edition quickly became clear.There were also a number of topics that were only briefly referenced in the first edition,and we wanted to provide more comprehensive discussions of those topics in a new edition.This second edition of Linear Mired Models:A Practical Guide Using Statistical Software aims to update the casestudies presented in the irst edition using the newest versions of the various software procedures,provide coverage of additional topics in the application of linear mixed models that we believe valuable for data analysts from all fields,and also provide up-to-date information on the options and features of the sofware procedures currently available for fitting linear mixed models in SAS,SPSS, Stata,R/S-plus,and HLM.Based on feedback from readers of the first edition,we have included coverage of the following topics in this second edition:Models with crossed random effects,and software procedures capable of fitting these models(see Chapter 8fora new case study);Power analysis methods for longitudinal and clustered study designs, including software options for power analyses and suggested approaches to writing simulations;Use of t helmer() function in the lme 4 package in R;Fitting linear mixed models to complex sample survey data;Bayesian approaches to making inferences based on linear mixed models;and Updated graphical procedures in the various software packages.We hope that readers will find the updated coverage of these topics helpful for their research activities.We have substantially revised the subject index for the book to enable more e ficient reading and easier location of material on selected topics or software options.We have also added more practical recommendations based on our experiences using the software throughout each of the chapters presenting analysisexamples.New sections discussing overall recommendations can be found at the end of each of these chapters.Finally,we have created an R package named WwG book that contains all of the datasets used in the example

List of Tables xv

List of Figures xvii

Preface to the Second Edition xix

Preface to the First Edition xxi

The Authors xxiii

Acknowledgments xxv

1Introduction 1

1.1What Are Linear Mixed Models(L MMs) ? 1

      1.1.1Models with Random Effects for Clustered Data 2

      1.1.2Models for Longitudinal or Repeated-Measures Data 2

      1.1.3The Purpose of This Book 3

      1.1.4Outline of Book Contents 4

1.2A Brief History of L MMs 5

      1.2.1Key Theoretical Developments 5

      1.2.2Key Software Developments 7

2.Linear Mixed Models:An Overview 9

2.1Introduction 9

C32.1.1Types and Structures of Data Sets 9

      2.1.1.1Clustered Data vs.Repeated-Measures and Longitudinal

      Data 9

      2.1.1.2Levels of Data 11

      2.1.2Types of Factors and Their Related Effects in an LMM 12

      2.1.2.1Fixed Factors 12

      2.1.2.2Random Factors 12

      2.1.2.3Fixed Factors vs.Random Factors 13

      2.1.2.4Fixed Effects vs.Random Effects 13

      2.1.2.5Nested vs.Crossed Factors and Their Corresponding Ef-fects 14

2.2Specification of LMMs 15

      2.2.1General Specification for an Individual Observation 15

      2.2.2General Matrix Specification 16

      2.2.2.1Covariance Structures for the D Matrix 19

      2.2.2.2Covariance Structures for the R;Matrix 20

      2.2.2.3Group-Specific Covariance Parameter Values for the D and R;Matrices 21

2.2.3Alternative Matrix Specification for All Subjects

2.2.4Hierarchical Linear Model(HLM) Specification of the LMM 22

2.3The Marginal Linear Model 22

2.3.1Specification of the Marginal Model 23

2.3.2The Marginal Model Implied by an LMM 23

2.4Estimation in LMMs 25

      2.4.1Maximum Likelihood(ML) Estimation 25

      2.4.1.1Special Case:Assume θIs Known 26

      2.4.1.2General Case:Assume θIs Unknown 26

      2.4.2REML Estimation 27

      2.4.3REML vs.ML Estimation 28

2.5Computational Issues 29

2.5.1Algorithms for Likelihood Function Optimization 29

2.5.2Computational Problems with Estimation of Covariance Parameters 31

2.6Tools for Model Selection 33

      2.6.1Basic Concepts in Model Selection 34

      2.6.1.1Nested Models 34

      2.6.1.2Hypotheses:Specification and Testing 34

      2.6.2Likelihood Ratio Tests(LRTs) 34

      2.6.2.1Likelihood Ratio Tests for Fixed-Effect Parameters 35

      2.6.2.2Likelihood Ratio Tests for Covariance Parameters 35

      2.6.3Alternative Tests 36

      2.6.3.1Alternative Tests for Fixed-Effect Parameters 36

      2.6.3.2Alternative Tests for Covariance Parameters 38

      2.6.4Information Criteria 38

2.7Model-Building Strategies 39

      2.7.1The Top-Down Strategy 39

      2.7.2The Step-Up Strategy 40

2.8Checking Model Assumptions(Diagnostics) 41

      2.8.1Residual Diagnostics 41

      2.8.1.1Raw Residuals 41

      2.8.1.2Standardized and Studentized Residuals 42

      2.8.2Influence Diagnostics 42

      2.8.3Diagnostics for Random Effects 43

2.9Other Aspects of LMMs 46

2.9.1Predicting Random Effects:Best Linear Unbiased Predictors 46

2.9.2Intraclass Correlation Coefficients(ICCs) 47

2.9.3Problems with Model Specification(Aliasing) 47

2.9.4Missing Data 49

2.9.5Centering Covariates 50

2.9.6Fitting Linear Mixed Models to Complex Sample Survey Data 50

      2.9.6.1Purely Model-Based Approaches 51

      2.9.6.2Hybrid Design-and Model-Based Approaches 52

      2.9.7Bayesian Analysis of Linear Mixed Models 55

2.10Power Analysis for Linear Mixed Models 56

      2.10.1Direct Power Computations 56

      2.10.2Examining Power via Simulation 57

2.11Chapter Summary 58

3Two-Level Models for Clustered Data:The Rat Pup Example 59

3.1Introduction 59

3.2The Rat Pup Study 60

      3.2.1Study Description 60

      3.2.2Data Summary 62

3.3Overview of the Rat Pup Data Analysis 65

      3.3.1Analysis Steps 66

      3.3.2Model Specification 68

      3.3.2.1General Model Specification 68

      3.3.2.2Hierarchical Model Specification 69

      3.3.3Hypothesis Tests 72

3.4Analysis Steps in the Software Procedures 75

      3.4.1 SAS 75

      3.4.2 SPSS 84

      3.4.3 R 91

      3.4.3.1Analysis Using the lme() Function 91

      3.4.3.2Analysis Using t helmer() Function 96

      3.4.4 Stata 98

      3.4.5 HLM 102

      3.4.5.1DataSet Preparation 102

      3.4.5.2Preparing the Multivariate DataMatrix(MDM) File 103

3.5Results of Hypothesis Tests 107

      3.5.1Likelihood Ratio Tests for Random Effects 107

      3.5.2Likelihood Ratio Tests for Residual Variance 107

      3.5.3F-tests and Likelihood Ratio Tests for Fixed Effects 108

3.6Comparing Results across the Software Procedures 109

      3.6.1Comparing Model 3.1 Results 109

      3.6.2Comparing Model 3.2B Results 114

      3.6.3Comparing Model 3.3 Results 114

3.7Interpreting Parameter Estimates in the Final Model 115

      3.7.1Fixed-Effect Parameter Estimates 115

      3.7.2Covariance Parameter Estimates 118

3.8Estimating the Intraclass Correlation Coefficients(ICCs) 118

3.9Calculating Predicted Values 120

      3.9.1Litter-Specific(Conditional) Predicted Values 120

      3.9.2Population-Averaged(Unconditional) Predicted Values 121

3.10Diagnostics for the Final Model 122

      3.10.1Residual Diagnostics 122

      3.10.1.1 Conditional Residuals 122

      3.10.1.2 Conditional Student ized Residuals 124

      3.10.2Influence Diagnostics 126

      3.10.2.1 Overall Influence Diagnostics 126

      3.10.2.2 Influence on Covariance Parameters 128

      3.10.2.3 Influence on Fixed Effects 129

3.11Software Notes and Recommendations 130

      3.11.1DataStructure 130

      3.11.2Syntax vs.Menus 130

      3.11.3Heterogeneous Residual Variances for Level 2 Groups 130

      3.11.4Display of the Marginal Covariance and Correlation Matrices 130

      3.11.5Differences in Model Fit Criteria 131

      3.11.6Differences in Tests for Fixed Effects 131

      3.11.7Post-Hoc Comparisons of Least Squares(LS) Means(EstimatedMarginal Means 133

      3.11.8Calculation of Student ized Residuals and Influence Statistics 133

      3.11.9Calculation of EBLUPs 133

      3.11.10Tests for Covariance Parameters 133

      3.11.11Reference Categories for Fixed Factors 134

4Three-Level Models for Clustered Data: The Classroom Example 135

4.1Introduction 135

4.2The Classroom Study 137

      4.2.1Study Description 137

      4.2.2Data Summary139

      4.2.2.1DataSet Preparation 139

      4.2.2.2Preparing the Multivariate DataMatrix(MDM) File 139

4.3Overview of the Classroom Data Analysis 142

      4.3.1Analysis Steps 142

      4.3.2Model Specification 146

      4.3.2.1General Model Specification146

      4.3.2.2Hierarchical Model Specification 146

      4.3.3Hypothesis Tests 148

4.4Analysis Steps in the Software Procedures 151

      4.4.1 SAS 151

      4.4.2 SPSS 157

      4.4.3 R 162

      4.4.3.1Analysis Using the lme() Function 162

      4.4.3.2Analysis Using t helmer() Function 165

      4.4.4 Stata 168

      4.4.5 HLM 171

4.5Results of Hypothesis Tests 177

      4.5.1Likelihood Ratio Tests for Random Effects 177

      4.5.2Likelihood Ratio Tests and t-Tests for Fixed Effects 177

4.6Compaa ring Results across the Software Procedures 179

      4.6.1Comparing Model 4.1 Results 179

      4.6.2Comparing Model 4.2 Results 181

      4.6.3Comparing Model 4.3 Results 181

      4.6.4Comparing Model 4.4 Results 181

4.7Interpreting Parameter Estimates in the Final Model 185

      4.7.1Fixed-Effect Parameter Estimates 185

      4.7.2Covariance Parameter Estimates 186

4.8Estimating the Intraclass Correlation Coefficients(ICCs) 187

4.9Calculating Predicted Values 189

      4.9.1Conditional and Marginal Predicted Values 189

      4.9.2Plotting Predicted Values Using HLM 190

4.10Diagnostics for the Final Model 191

      4.10.1Plots of the EBLUPs 191

      4.10.2Residual Diagnostics 191

4.11Software Notes 195

      4.11.1REML vs.ML Estimation 195

      4.11.2Setting up Three-Level Models in HLM 195

      4.11.3Calculation of Degrees of Freedom fort-Tests in HLM 196

      4.11.4Analyzing Cases with Complete Data 196

      4.11.5Miscellaneous Differences 197

4.12Recommendations 198

5Models for Repeated-Measures Data:The Rat Brain Example 199

5.1Introduction 199

5.2The Rat Brain Study 199

      5.2.1Study Description 199

      5.2.2Data Summary 202

5.3Overview of the Rat Brain Data Analysis 204

      5.3.1Analysis Steps 204

      5.3.2Model Specification 206

      5.3.2.1General Model Specification 206

      5.3.2.2Hierarchical Model Specification 207

      5.3.3Hypothesis Tests 210

5.4Analysis Steps in the Software Procedures 212

      5.4.1 SAS 213

      5.4.2 SPSS 215

      5.4.3 R 218

      5.4.3.1Analysis Using the lme() Function 219

      5.4.3.2Analysis Using t helmer() Function 221

      5.4.4 Stata 223

      5.4.5 HLM 226

      5.4.5.1DataSet Preparation 226

      5.4.5.2Preparing the MDM File 227

5.5Results of Hypothesis Tests 231

      5.5.1Likelihood Ratio Tests for Random Effects 231

      5.5.2Likelihood Ratio Tests for Residual Variance 231

      5.5.3F-Tests for Fixed Effects 232

5.6Comparing Results across the Software Procedures 232

      5.6.1Comparing Model 5.1 Results 233

      5.6.2Comparing Model 5.2 Results 233

5.7Interpreting Parameter Estimates in the Final Model 233

      5.7.1Fixed-Effect Parameter Estimates 233

      5.7.2Covariance Parameter Estimates 239

5.8The Implied Marginal Variance-Covariance Matrix for the Final Model 240

5.9Diagnostics for the Final Model 241

5.10Software Notes 241

      5.10.1Heterogeneous Residual Variances for Level 1 Groups 243

      5.10.2EBL UPs for Multiple Random Effects 244

5.11Other Analytic Approaches 244

      5.11.1Kronecker Product for More Flexible Residual Covariance Structures 244

      5.11.2Fitting the Marginal Model 246

      5.11.3Repeated-Measures ANOVA 247

5.12Recommendations 247

6Random Coefficient Models for Longitudinal Data：The Autism Example

6.1Introduction249

6.2The Autism Study249

      6.2.1Study Description 249

      6.2.2Data Summary 251

6.3Overview of the Autism Data Analysis 255

      6.3.1Analysis Steps 255

      6.3.2Model Specification 265

      6.3.2.1General Model Specification 257

      6.3.2.2 Hierarchical Model Specification 260

      6.3.3Hypothesis Tests 261

6.4Analysis Steps in the Software Procedures 263

      6.4.1 SAS 263

      6.4.2 SPSS 267

      6.4.3 R 270

      6.4.3.1Analysis Using the lme() Function 271

      6.4.3.2Analysis Using t helmer() Function 271

      6.4.4 Stata 276

      6.4.5 HLM 278

      6.4.5.1DataSet Preparation278

      6.4.5.2Preparing the MDM File 279

6.5Results of Hypothesis Tests 284

      Likelihood Ratio Test for Random Effects 284

      Likelihood Ratio Tests for Fixed Effects 285

6.6Comparing Results across the Software Procedures 285

      6.6.1Comparing Model 6.1 Results 285

      6.6.2Comparing Model 6.2 Results 289

      6.6.3Comparing Model 6.3 Results 289

6.7Interpreting Parameter Estimates in the Final Model 289

      6.7.1Fixed-Effect Parameter Estimates 289

      6.7.2Covariance Parameter Estimates 291

6.8Calculating Predicted Values 293

      6.8.1Marginal Predicted Values 293

      6.8.2Conditional Predicted Values 295

6.9Diagnostics for the Final Model 297

      6.9.1Residual Diagnostics 297

      6.9.2Diagnostics for the Random Effects 298

      6.9.3Observed and Predicted Values 300

6.10Software Note:Computational Problems with the D Matrix 301

6.10.1Recommendations 302

6.11An Alternative Approach:Fittingthe Marginal Model with an Unstructured Covariance Matrix 302

6.11.1Recommendations 305

7Models for Clustered Longitudinal Data:The Dental Veneer Example 307

7.1Introduction 307

7.2The Dental Veneer Study 309

      7.2.1Study Description 309

      7.2.2Data Summary 310

7.3Overview of the Dental Veneer Data Analysis 312

      7.3.1Analysis Steps 312

      7.3.2Model Specification 314

      7.3.2.1General Model Specification 314

      7.3.2.2Hierarchical Model Specification 317

7.3.3Hypothesis Tests 320

7.4Analysis Steps in the Software Procedures 322

      7.4.1 SAS 322

      7.4.2 SPSS 327

      7.4.3 R 330

      7.4.3.1Analysis Using the lme() Function 331

      7.4.3.2Analysis Using the lmer() Function 334

      7.4.4 Stata 337

      7.4.5 HLM 341

      7.4.5.1DataSet Preparation 341

      7.4.5.2Preparing the Multivariate DataMatrix(MDM) File 342

7.5Resultss of Hypothesis Tests 346

      7.5.1Likelihood Ratio Tests for Random Effects 346

      7.5.2Likelihood Ratio Tests for Residual Variance 347

      7.5.3Likelihood Ratio Tests for Fixed Effects 347

7.6Comparing Results across the Software Procedures 348

      7.6.1Comparing Model 7.1 Results 348

      7.6.2Comparing Results for Models 7.2A,7.2B,and 7.2C 348

      7.6.3Comparing Model 7.3 Results 349

7.7Interpreting Parameter Estimates in the Final Model 355

      7.7.1Fixed-Effect Parameter Estimates 355

      7.7.2Covariance Parameter Estimates 356

7.8The Implied Marginal Variance-Covariance Matrix for the Final Model 357

7.9Diagnostics for the Final Model 359

      7.9.1Residual Diagnostics 359

      7.9.2Diagnostics for the Random Effects 360

7.10SoftwareNotes and Recommendations 363

      7.10.1ML vs.REML Estimation 363

      7.10.2The Ability to Remove Random Effects from a Model 364

      7.10.3Considering Alternative Residual Covariance Structures 364

      7.10.4Aliasing of Covariance Parameters 365

      7.10.5Displaying the Marginal Covariance and Correlation Matrices 365

      7.10.6Miscellaneous Software Notes 366

7.11OtherAnalytic Approaches 366

      7.11.1Modeling the Covariance Structure 366

      7.11.2The Step-Up vs.Step-Down Approach to Model Building 367

      7.11.3Alternative Uses of Baseline Values for the Dependent Variable 367

8Models for Data with Crossed Random Factors:The SAT Score Example 369

8.1Introduction 369

8.2The SAT Score Study 369

      8.2.1Study Description 369

      8.2.2Data Summary 371

8.3Overview of the SAT Score Data Analysis 373

      8.3.1Model Specification 374

      8.3.1.1General Model Specification 374

      8.3.1.2Hierarchical Model Specification 374

      8.3.2Hypothesis Tests 375

8.4Analysis Steps in the Software Procedures 376

      8.4.1 SAS 376

      8.4.2 SPSS 378

      8.4.3 R 380

      8.4.4 Sta

      ta 382

      8.4.5 HLM 384

      8.4.5.1DataSet Preparation 384

      8.4.5.2Preparing the MDM File 385

      8.4.5.3Model Fitting 385

8.5Results of Hypothesis Tests 387

      8.5.1Likelihood Ratio Tests for Random Effects 387

      8.5.2 Testing the Fixed Year Effect 387

8.6Comparing Results across the Software Procedures 387

8.7Interpreting Parameter Estimates in the Final Model 380

      8.7.1Fixed-Effect Parameter Estimates 380

      8.7.2Covariance Parameter Estimates 390

8.8The Implied Marginal Variance-Covariance Matrix for the Final Model 391

8.9Recommended Diagnostics for the Final Model 392

8.10Software Notes and Additional Recommendations 393

A Statistical Software Resources 395

A.1Descriptions/Availability of Software Packages 395

      A.1.1SAS 395

      A.1.2IBM SPSS Statistics 395

      A.1.3R 395

      A.1.4Stata 396

      A.1.5HLM 396

A.2Useful Internet Links 396

B Calculation of the Marginal Variance-Covariance Matrix 397

C Acronyms/Abbreviations 399

Bibliography 401

Index 407

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