9780387950273

This book provides a comprehensive treatment of linear mixed models for continuous longitudinal data. Next to model formulation, this edition puts major emphasis on exploratory data analysis for all aspects of the model, such as the marginal model, subject-specific profiles, and residual covariance structure. Further, model diagnostics and missing data receive extensive treatment. Sensitivity analysis for incomplete data is given a prominent place. Several variations to the conventional linear mixed model are discussed (a heterogeity model, condional linear mid models).This book will be of interest to applied statisticians and biomedical researchers in industry, public health organizations, contract research organizations, and academia. The book is explanatory rather than mathematically rigorous. Most analyses were done with the MIXED procedure of the SAS software package, and many of its features are clearly elucidated. How3ever, some other commercially available packages are discussed as well. Great care has been taken in presenting the data analyses in a software-independent fashion.Geert Verbeke is Assistant Professor at the Biostistical Centre of the Katholieke Universiteit Leuven in Belgium. He received the B.S. degree in mathematics (1989) from the Katholieke Universiteit Leuven, the M.S. in biostatistics (1992) from the Limburgs Universitair Centrum, and earned a Ph.D. in biostatistics (1995) from the Katholieke Universiteit Leuven. Dr. Verbeke wrote his dissertation, as well as a number of methodological articles, on various aspects of linear mixed models for longitudinal data analysis. He has held visiting positions at the Gerontology Research Center and the Johns Hopkins University.Geert Molenberghs is Assistant Professor of Biostatistics at the Limburgs Universitair Centrum in Belgium. He received the B.S. degree in mathematics (1988) and a Ph.D. in biostatistics (1993) from the Universiteit Antwerpen. Dr. Molenberghs published methodological work on the analysis of non-response in clinical and epidemiological studies. He serves as an associate editor for Biometrics, Applied Statistics, and Biostatistics, and is an officer of the Belgian Statistical Society. He has held visiting positions at the Harvard School of Public Health.

Preface

vii

Acknowledgments

Introduction

(6)

Examples

(12)

The Rat Data

(2)

The Toenail Data (TDO)

(1)

The Baltimore Longitudinal Study of Aging (BLSA)

(5)

The Prostate Data

(3)

The Hearing Data

(1)

The Vorozole Study

(1)

Heights of Schoolgirls

(1)

Growth Data

(2)

Mastitis in Dairy Cattle

(1)

A Model for Longitudinal Data

(12)

Introduction

(1)

A Two-Stage Analysis

(3)

Stage 1

(1)

Stage 2

(1)

Example: The Rat Data

(1)

Example: The Prostate Data

(1)

Two-Stage Analysis

(1)

The General Linear Mixed-Effects Model

(8)

The Model

(2)

Example: The Rat Data

(1)

Example: The Prostate Data

(1)

A Model for the Residual Covariance Structure

(5)

Exploratory Data Analysis

(10)

Introduction

(1)

Exploring the Marginal Distribution

(4)

The Average Evolution

(2)

The Variance Structure

(1)

The Correlation Structure

(1)

Exploring Subject-Specific Profiles

(6)

Measuring the Overall Goodness-of-Fit

(2)

Testing for the Need of a Model Extension

(1)

Example: The Rat Data

(1)

Example: The Prostate Data

(2)

Estimation of the Marginal Model

(14)

Introduction

(1)

Maximum Likelihood Estimation

(1)

Restricted Maximum Likelihood Estimation

(4)

Variance Estimation of Normal Populations

(1)

Estimation of Residual Variance in Linear Regression

(1)

REML Estimation for the Linear Mixed Model

(2)

Justification of REML Estimation

(1)

Comparison Between ML and REML Estimation

(1)

Model-Fitting Procedures

(1)

Example: The Prostate Data

(2)

Estimation Problems

(5)

Small Variance Components

(2)

Model Misspecifications

(3)

Inference for the Marginal Model

(22)

Introduction

(1)

Inference for the Fixed Effects

(9)

Approximate Wald Tests

(1)

Approximate t-Tests and F-Tests

(1)

Example: The Prostate Data

(4)

Robust Inference

(1)

Likelihood Ratio Tests

(2)

Inference for the Variance Components

(10)

Approximate Wald Tests

(1)

Likelihood Ratio Tests

(1)

Example: The Rat Data

(3)

Marginal Testing for the Need of Random Effects

(3)

Example: The Prostate Data

(2)

Information Criteria

(3)

Inference for the Random Effects

(16)

Introduction

(1)

Empirical Bayes Inference

(1)

Henderson's Mixed-Model Equations

(1)

Best Linear Unbiased Prediction (BLUP)

(1)

Shrinkage

(1)

Example: The Random-Intercepts Model

(1)

Example: The Prostate Data

(1)

The Normality Assumption for Random Effects

(10)

Introduction

(2)

Impact on EB Estimates

(2)

Impact on the Estimation of the Marginal Model

(2)

Checking the Normality Assumption

(4)

Fitting Linear Mixed Models with SAS

(28)

Introduction

(1)

The SAS Program

(10)

The PROC MIXED Statement

(1)

The Class Statement

(1)

The Model Statement

(1)

The ID Statement

(1)

The Random Statement

(1)

The Repeated Statement

(3)

The Contrast Statement

101

(1)

The Estimate Statement

101

(1)

The Make Statement

102

(1)

Some Additional Statements and Options

102

(2)

The SAS Output

104

(10)

Information on the Iteration Procedure

104

(1)

Information on the Model Fit

105

(2)

Information Criteria

107

(1)

Inference for the Variance Components

107

(4)

Inference for the Fixed Effects

111

(2)

Inference for the Random Effects

113

(1)

Note on the Mean Parameterization

114

(3)

The Random and Repeated Statements

117

(2)

Proc Mixed versus Proc GLM

119

(2)

General Guidelines for Model Building

121

(14)

Introduction

121

(2)

Selection of a Preliminary Mean Structure

123

(2)

Selection of a Preliminary Random-Effects Structure

125

(3)

Selection of a Residual Covariance Structure

128

(4)

Model Reduction

132

(3)

Exploring Serial Correlation

135

(16)

Introduction

135

(1)

An Informal Check for Serial Correlation

136

(1)

Flexible Models for Serial Correlation

137

(4)

Introduction

137

(1)

Fractional Polynomials

137

(1)

Example: The Prostate Data

138

(3)

The Semi-Variogram

141

(7)

Introduction

141

(1)

The Semi-Variogram for Random-Intercepts Models

142

(2)

Example: The Vorozole Study

144

(1)

The Semi-Variogram for Random-Effects Models

144

(3)

Example: The Prostate Data

147

(1)

Some Remarks

148

(3)

Local Influence for the Linear Mixed Model

151

(18)

Introduction

151

(2)

Local Influence

153

(5)

The Detection of Influential Subjects

158

(4)

Example: The Prostate Data

162

(5)

Local Influence Under REML Estimation

167

(2)

The Heterogeneity Model

169

(20)

Introduction

169

(2)

The Heterogeneity Model

171

(2)

Estimation of the Heterogeneity Model

173

(4)

Classification of Longitudinal Profiles

177

(1)

Goodness-of-Fit Checks

178

(2)

Example: The Prostate Data

180

(3)

Example: The Heights of Schoolgirls

183

(6)

Conditional Linear Mixed Models

189

(12)

Introduction

189

(1)

A Linear Mixed Model for the Hearing Data

190

(4)

Conditional Linear Mixed Models

194

(3)

Applied to the Hearing Data

197

(1)

Relation with Fixed-Effects Models

198

(3)

Exploring Incomplete Data

201

(8)

Joint Modeling of Measurements and Missingness

209

(12)

Introduction

209

(1)

The Impact of Incompleteness

210

(1)

Simple ad hoc Methods

211

(1)

Modeling Incompleteness

212

(2)

Terminology

214

(1)

Missing Data Patterns

215

(1)

Missing Data Mechanisms

215

(2)

Ignorability

217

(1)

A Special Case: Dropout

218

(3)

Simple Missing Data Methods

221

(10)

Introduction

221

(2)

Complete Case Analysis

223

(1)

Simple Forms of Imputation

223

(4)

Last Observation Carried Forward

224

(1)

Imputing Unconditional Means

225

(1)

Buck's Method: Conditional Mean Imputation

225

(1)

Discussion of Imputation Techniques

226

(1)

Available Case Methods

227

(1)

MCAR Analysis of Toenail Data

227

(4)

Selection Models

231

(44)

Introduction

231

(2)

A Selection Model for the Toenail Data

233

(6)

MAR Analysis

233

(1)

MNAR analysis

234

(5)

Scope of Ignorability

239

(1)

Growth Data

240

(29)

Analysis of Complete Growth Data

240

(16)

Frequentist Analysis of Incomplete Growth Data

256

(1)

Likelihood Analysis of Incomplete Growth Data

257

(10)

Missingness Process for the Growth Data

267

(2)

A Selection Model for Nonrandom Dropout

269

(1)

A Selection Model for the Vorozole Study

270

(5)

Pattern-Mixture Models

275

(20)

Introduction

275

(5)

A Simple Illustration

275

(3)

A Paradox

278

(2)

Pattern-Mixture Models

280

(1)

Pattern-Mixture Model for the Toenail Data

281

(6)

A Pattern-Mixture Model for the Vorozole Study

287

(4)

Some Reflections

291

(4)

Sensitivity Analysis for Selection Models

295

(36)

Introduction

295

(2)

A Modified Selection Model for Nonrandom Dropout

297

(1)

Local Influence

298

(9)

Review of the Theory

299

(1)

Applied to the Model of Diggle and Kenward

300

(2)

Special Case: Compound Symmetry

302

(4)

Serial Correlation

306

(1)

Analysis of the Rat Data

307

(5)

Mastitis in Dairy Cattle

312

(14)

Informal Sensitivity Analysis

312

(7)

Local Influence Approach

319

(7)

Alternative Local Influence Approaches

326

(2)

Random-Coefficient-based Models

328

(2)

Concluding Remarks

330

(1)

Sensitivity Analysis for Pattern-Mixture Models

331

(44)

Introduction

331

(1)

Pattern-Mixture Models and MAR

332

(4)

MAR and ACMV

333

(2)

Nonmonotone Patterns: A Counterexample

335

(1)

Multiple Imputation

336

(3)

Parameter and Precision Estimation

338

(1)

Hypothesis Testing

338

(1)

Pattern-Mixture Models and Sensitivity Analysis

339

(4)

Identifying Restrictions Strategies

343

(9)

Strategy Outline

343

(1)

Identifying Restrictions

344

(3)

ACMV Restrictions

347

(3)

Drawing from the Conditional Densities

350

(2)

Analysis of the Vorozole Study

352

(21)

Fitting a Model

352

(14)

Hypothesis Testing

366

(5)

Model Reduction

371

(2)

Thoughts

373

(2)

How Ignorable Is Missing At Random?

375

(12)

Introduction

375

(2)

Information and Sampling Distributions

377

(2)

Illustration

379

(4)

Example

383

(2)

Implications for PROC Mixed

385

(2)

The Expectation-Maximization Algorithm

387

(4)

Design Considerations

391

(14)

Introduction

391

(1)

Power Calculations Under Linear Mixed Models

392

(1)

Example: The Rat Data

393

(1)

Power Calculations When Dropout Is to Be Expected

394

(3)

Example: The Rat Data

397

(8)

Constant pj,k\≥k, Varying nj

399

(2)

Constant pj,k\≥k, Constant nj

401

(1)

Increasing pj,k\≥k over Time, Constant nj

402

(3)

Case Studies

405

(118)

Blood Pressures

405

(6)

The Heat Shock Study

411

(9)

Introduction

411

(4)

Analysis of Heat Shock Data

415

(5)

The Validation of Surrogate Endpoints from Multiple Trials

420

(26)

Introduction

420

(1)

Validation Criteria

421

(3)

Notation and Motivating Examples

424

(5)

A Meta-Analytic Approach

429

(5)

Data Analysis

434

(5)

Computational Issues

439

(3)

Extensions

442

(1)

Reflections on Surrogacy

443

(1)

Precdiction Intervals

444

(1)

SAS Code for Random-Effects Model

445

(1)

The Milk Protein Content Trial

446

(24)

Introduction

446

(2)

Informal Sensitivity Analaysis

448

(9)

Formal Sensitivity Analysis

457

(13)

Hepatitis B Vaccination

470

(15)

Time Evolution of Antibodies

472

(9)

Prediction at Year 12

481

(1)

SAS Code for Vaccination Models

482

(3)

Appendix

A Software

485

(30)

A.1 The SAS System

485

(1)

A.1.1 Standard Applications

485

(1)

A.1.2 New Features in SAS Version 7.0

485

(4)

A.2 Fitting Mixed Models Using MLwiN

489

(4)

A.3 Fitting Mixed Models Using SPlus

493

(1)

A.3.1 Standard SPlus Functions

494

(3)

A.3.2 OSWALD for Nonrandom Nonresponse

497

(18)

B Technical Details for Sensitivity Analysis

515

(8)

B.1 Local Influence: Derivation of Components of Δi

515

(3)

B.2 Proof of Theorem 20.1

518

(5)

References

523

(31)

Index

554

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Linear Mixed Models for Longitudinal Data

0387950273

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