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Provides a clear understanding of the application of mixed models, and describes the benefits to be gained from their use as well as the practical implications. Mixed models is becoming a popular method of statistical analysis used for analysing medical data, particularly in the pharmaceutical industry. This method often gives improvements over conventional fixed effect models, especially when data are unbalanced. Presently there is no other book covering the application of mixed models to clinical data, making this book essential reading for those involved in this subject.

Features include:

* Takes a balanced view of mixed models by discussing some of the problems in their use and indicates where more conventional fixed effect models might be preferred.

* Easily accessible to practitioners in any areas where mixed models are used, including medical statisticians and economists

* Illustrated with numerous medical examples which clearly demonstrate the application of the theory

* Extensive coverage of the underlying theory

* Devotes a complete chapter to the use of software procedures and macros to fit mixed models.

This title is aimed at medical, applied and bio-statisticians, along with teachers and students of advanced statistics courses in mixed models. The book is also suitable for medical scientists who need to understand the techniques used and the assumuptions which underpin their use.

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CONTENTS

Preface = xiii

Series Preface = xvii

Mixed Models Notation = xix

1 Introduction = 1

 1.1 The Use of Mixed Models = 1

 1.2 Introductory Example = 3

  1.2.1 Simple model to assess the effects of treatment (Model A) = 3

  1.2.2 A model taking patient effects into account (Model B) = 6

  1.2.3 Random effects model (Model C) = 6

  1.2.4 Estimation (or prediction) of random effects = 11

 1.3 A Multi-Centre Hypertension Trial = 12

  1.3.1 Modelling the data = 14

  1.3.2 Including a baseline covariate (Model B) = 14

  1.3.3 Modelling centre effects (Model C) = 16

  1.3.4 Including centre-by-treatment interaction effects (Model D) = 16

  1.3.5 Modelling centre and centre·treatment effects as random (Model E) = 17

 1.4 Repeated Measures Data = 18

  1.4.1 Covariance pattern models = 19

  1.4.2 Random coefficients models = 20

 1.5 More About Mixed Models = 21

  1.5.1 What is a mixed model? = 22

  1.5.2 Why use mixed models? = 23

  1.5.3 Communicating results = 24

  1.5.4 Mixed models in medicine = 25

  1.5.5 Mixed models in perspective = 25

 1.6 Some Useful Definitions = 26

  1.6.1 Containment = 27

  1.6.2 Balance = 28

  1.6.3 Error strata = 30

2 Normal Mixed Models = 33

 2.1 Model Definition = 33

  2.1.1 The flxed effects model = 33

  2.1.2 The mixed model = 36

  2.1.3 The random effects model covariance structure = 38

  2.1.4 The random coefficients model covariance structure = 40

  2.1.5 The covariance pattern model covariance structure = 42

 2.2 Model Fitting Methods = 44

  2.2.1 The likelihood function and approaches to its maximisation = 45

  2.2.2 Estimation of fixed effects = 48

  2.2.3 Estimation (or prediction) of random effects and coefficients = 49

  2.2.4 Estimating variance parameters = 51

 2.3 The Bayesian Approach = 55

  2.3.1 Introduction = 56

  2.3.2 Determining the posterior density = 57

  2.3.3 Parameter estimation, probability intervals and p-values = 58

  2.3.4 Specifying non-informative prior distributions = 60

  2.3.5 Evaluating the posterior distribution = 64

 2.4 Practical Application and Interpretation = 68

  2.4.1 Negative variance components = 69

  2.4.2 Accuracy of variance parameters = 73

  2.4.3 Bias in fixed and random effects standard errors = 73

  2.4.4 Significance testing = 74

  2.4.5 Confidence intervals = 77

  2.4.6 Model checking = 77

  2.4.7 Missing data = 79

 2.5 Example = 79

  2.5.1 Analysis models = 80

  2.5.2 Results = 82

  2.5.3 Discussion of points from Section 2.4 = 83

3 Genera1ised Linear Mixed Models (GLMMs) = 103

 3.1 Generalised Linear Models (GLMs) = 104

  3.1.1 Introduction = 104

  3.1.2 Distributions = 105

  3.1.3 The general form for exponential distributions = 107

  3.1.4 The GLM definition = 108

  3.1.5 Interpreting results from GLMs = 110

  3.1.6 Fitting the GLM = 112

  3.1.7 Expressing individual distributions in the general exponential form = 114

  3.1.8 Conditional logistic regression = 116

 3.2 Generalised Linear Mixed Models (GLMMs) = 116

  3.2.1 The GLMM definition = 116

  3.2.2 The likelihood and quasi-likelihood functions = 118

  3.2.3 Fitting the GLMM = 120

  3.2.4 Some flaws with GLMMs = 124

  3.2.5 Reparameterising random effects models as covariance pattern models = 126

 3.3 Practical Application and Interpretation = 126

  3.3.1 Specifying binary data = 127

  3.3.2 Difficulties with fitting random effects (and random coefficients) models = 127

  3.3.3 Accuracy of variance parameters = 128

  3.3.4 Bias in fixed and random effects standard errors = 129

  3.3.5 Negative variance components = 129

  3.3.6 Uniform fixed effect categories = 130

  3.3.7 Uniform random effect categories = 130

  3.3.8 The dispersion parameter = 130

  3.3.9 Significance testing = 132

  3.3.10 Confidence intervals = 133

  3.3.11 Model checking = 133

 3.4 Example = 134

  3.4.1 Introduction and models fitted = 135

  3.4.2 Results = 136

  3.4.3 Discussion of points from Section 3.3 = 139

4 Mixed Modelㄴ for Categorical Data = 149

 4.1 Ordinal Logistic Regression (Fixed Effects Model) = 149

 4.2 Mixed Ordinal Logistic Regression = 151

  4.2.1 Definition of the mixed ordinal logistic regression model = 151

  4.2.2 Residual variance matrix, R = 154

  4.2.3 Reparameterising random effects models as covariance pattern models = 157

  4.2.4 Likelihood and quasi-likelihood functions = 158

  4.2.5 Model fitting methods = 158

 4.3 Mixed Models for Unordered Categorical Data = 159

  4.3.1 The G matrix = 162

  4.3.2 The R matrix = 162

  4.3.3 Fitting the model = 162

 4.4 Practical Application and Interpretation = 162

  4.4.1 The proportional odds assumption = 163

  4.4.2 Number of covariance parameters = 163

  4.4.3 Choosing a covariance pattern = 163

  4.4.4 Interpreting covariance parameters = 163

  4.4.5 Fixed and random effects estimates = 164

  4.4.6 Checking model assumptions = 164

  4.4.7 The dispersion parameter = 164

  4.4.8 Other points = 164

 4.5 Example = 164

5 Multi-Centre Trials and Meta-Analyses = 171

 5.1 Introduction to Multi-Centre Trials = 171

  5.1.1 What is a multi-centre trial? = 171

  5.1.2 Why use mixed models to analyse multi-centre data? = 171

 5.2 The Implications of Using Different Analysis Models = 172

  5.2.1 Centre and centre·treatment effects fixed = 172

  5.2.2 Centre effects fixed, centre·treatment effects omitted = 173

  5.2.3 Centre and centre·treatment effects random = 174

  5.2.4 Centre effects random, centre·treatment effects omitted = 175

 5.3 Example : A Multi-Centre Trial 176

 5.4 Practical Application and Interpretation = 181

  5.4.1 Plausibility of a centre·treatment interaction = 181

  5.4.2 Generalisation = 182

  5.4.3 Number of centres = 182

  5.4.4 Centre size = 182

  5.4.5 Negative variance components = 183

  5.4.6 Balance = 183

 5.5 Sample Size Estimation = 183

  5.5.1 Normal data = 184

  5.5.2 Non-normal data = 187

 5.6 Meta-Analysis = 189

 5.7 Example : Meta-Analysis = 190

  5.7.1 Analyses = 190

  5.7.2 Results = 191

  5.7.3 Treatment estimates in individual trials = 192

6 Repeated Measures Data = 199

 6.1 Introduction = 199

  6.1.1 Reasons for repeated measurements = 199

  6.1.2 Analysis objectives = 200

  6.1.3 Fixed effects approaches = 200

  6.1.4 Mixed model approaches = 202

 6.2 Covariance Pattern Models = 202

  6.2.1 Covariance patterns = 203

  6.2.2 Choice of covariance pattern = 207

  6.2.3 Choice of fixed effects = 209

  6.2.4 General points = 210

 6.3 Example : Covariance Pattern Models for Normal Data = 211

  6.3.1 Analysis models = 212

  6.3.2 Selection of covariance pattern = 212

  6.3.3 Assessing fixed effects = 214

  6.3.4 Model checking = 216

 6.4 Example : Covariance Pattern Models for Count Data = 226

  6.4.1 GLMM analysis models = 227

  6.4.2 Analysis using a categorical mixed model = 230

 6.5 Random Coefficients Models = 235

  6.5.1 Introduction = 235

  6.5.2 General points = 237

  6.5.3 Comparisons with fixed effects approaches = 238

 6.6 Examples of Random Coefficients Models = 239

  6.6.1 A linear random coefficients model = 239

  6.6.2 A polynomial random coefficients model = 241

 6.7 Sample Size Estimation = 256

  6.7.1 Normal data = 256

  6.7.2 Non-normal data = 258

  6.7.3 Categorical data = 259

7 Cross-Over Trials = 261

 7.1 Introduction = 261

 7.2 Advantages of Mixed Models in Cross-Over Trials = 261

 7.3 The AB／BA Cross-Over Trial = 262

  7.3.1 Example : AB／BA cross-over design = 264

 7.4 Higher Order Complete Block Designs = 269

  7.4.1 Inclusion of carry-over effects = 269

  7.4.2 Example : Four-period, four-treatment cross-over trial = 269

 7.5 Incomplete Block Designs = 272

  7.5.1 The three-treatment, two-period design (Koch's design) = 273

  7.5.2 Example : Two-period cross-over trial = 274

 7.6 Optimal Designs = 276

  7.6.1 Example : Balaam's design = 276

 7.7 Covariance Pattern Models = 278

  7.7.1 Structured by period = 278

  7.7.2 Structured by treatment = 279

  7.7.3 Example : Four-way cross-over trial = 279

 7.8 Analysis of Binary Data = 286

 7.9 Analysis of Categorical Data = 290

 7.10 Use of Results from Random Effects Models in Trial Design = 291

  7.10.1 Example = 292

 7.11 General Points = 293

8 Other Applications of Mixed Models = 295

 8.1 Trials with Repeated Measurements Within Visits = 295

  8.1.1 Covariance pattern models = 296

  8.1.2 Example = 300

  8.1.3 Random coefficients models = 306

  8.1.4 Example : Random coefficients models = 308

 8.2 Multi-Centre Trials with Repeated Measures = 312

  8.2.1 Example : Multi-centre hypertension trial = 313

  8.2.2 Covariance pattern models = 314

 8.3 Multi-Centre Cross-Over Trials = 318

 8.4 Hierarchical Multi-Centre Trials and Meta-Analysis = 319

 8.5 Matched Case-Control Studies = 320

  8.5.1 Example = 321

  8.5.2 Analysis of a quantitative variable = 321

  8.5.3 Check of model assumptions = 323

  8.5.4 Analysis of binary variables = 325

 8.6 Different Variances for Treatment Groups in a Simple Between-Patient Trial = 333

  8.6.1 Example = 334

 8.7 Estimating Variance Components in an Animal Physiology Trial = 336

  8.7.1 Sample size estimation for a future experiment = 337

 8.8 Inter- and Intra-Observer Variation in Foetal Scan Measurements = 342

 8.9 Components of Variation and Mean Estimates in a Cardiology Experiment = 344

 8.10 Cluster Sample Surveys = 346

  8.10.1 Example : Cluster sample survey = 346

 8.11 Small Area Mortality Estimates = 348

 8.12 Estimating Surgeon Performance = 353

 8.13 Event History Analysis = 355

  8.13.1 Example = 355

9 Software for Fitting Mixed Models = 359

 9.1 Packages for Fitting Mixed Models = 359

 9.2 Basic Use of PROC MIXED = 360

  9.2.1 Syntax = 360

  9.2.2 PROC MIXED statement options = 362

 9.3 Basic Use of PROC GENMOD and the GLIMMIX Macro = 379

  9.3.1 PROC GENMOD = 379

  9.3.2 The GLIMMIX macro = 381

Glossary = 383

References = 387

Contacts = 391

Index = 393