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Statistical Methods for the Analysis of Repeated Measurements

文件格式:Pdf 可复制性:可复制 TAG标签: analysis Methods Statistical Measurements Repeated 点击次数: 更新时间:2009-10-16 09:19
介绍

1 Introduction 1

1.1 Repeated Measurements . . . . . . . . . . . . . . 1

1.2 Advantages and Disadvantages of Repeated Measurements Designs  2

1.3 Notation for Repeated Measurements . . . . . . . . . . .. 3

1.4 Missing Data . . . . . . . . . . . . . . . . . .. 4

1.5 Sample Size Estimation . . . . . . . . . . . . .  8

1.6 Outline of Topics . . . . . . .. . . . . . . .  9

1.7 Choosing the “Best” Method of Analysis . . . . . . . . . 12

2 Univariate Methods                                           15

2.1 Introduction . . . . . . . . . .. . .15

2.2 One Sample . . . .  . .. . . . . . . .16

2.3 Multiple Samples . . . . .. . . . . 21

2.4 Comments . . . . . . . .. . . . . . . .26

2.5 Problems . . . . . . . . . . . . . . .28

3 Normal-Theory Methods: Unstructured Multivariate Approach     45

3.1 Introduction . . . . . . . . . 45

3.2 Multivariate Normal Distribution Theory . . . . . . . 46

3.2.1 The Multivariate Normal Distribution . . .. . . . .. 46

3.2.2 The Wishart Distribution . . . .. . . . . .. 46

3.2.3 Wishart Matrices . . . . . . . . 47

3.2.4 Hotelling’s T2 Statistic . . . . . .. 47

3.2.5 Hypothesis Tests . . . . . . . .. 48

3.3 One-Sample Repeated Measurements . .  . . . .49

3.3.1 Methodology . . . .. . . . . . . . .. 49

3.3.2 Examples . . .  . . . . . . . . . . 50

3.3.3 Comments . . . . . . . . . . . . .. 54

3.4 Two-Sample Repeated Measurements . . .. . . . .  55

3.4.1 Methodology . . . . . . . . . . . 55

3.4.2 Example . . . . .  . . . . . . . . . 57

3.4.3 Comments . . . . . . . . . . . . .. 60

3.5 Problems . . .  . . . . . . . 61

4 Normal-Theory Methods: Multivariate Analysis of Variance        73

4.1 Introduction . . . .  . . . . 73

4.2 The Multivariate General Linear Model . . . . . 74

4.2.1 Notation and Assumptions  . . . . .74

4.2.2 Parameter Estimation . . . .  . . . . . . . 75

4.2.3 Hypothesis Testing . . . . . . . . 76

4.2.4 Comparisons of Test Statistics . . .. . . . . . . 77

4.3 Pro.le Analysis . . . . . . . . . . . . 78

4.3.1 Methodology . . . . .  . . . . . 78

4.3.2 Example . . . . . . . . . .  . . . . . . . 81

4.4 Growth Curve Analysis . . . . . . . . . . .  83

4.4.1 Introduction . . . . . .  . . . . . .  83

4.4.2 The Growth Curve Model . . . . . . . 83

4.4.3 Examples . . .. . . . . . . . . . . .  87

4.5 Problems . . . . . .. . . . . 94

5 Normal-Theory Methods: Repeated Measures ANOVA 103

5.1 Introduction . . . . . .. . . . . . . 103

5.2 The Fundamental Model . . . . . . . . . . 104

5.3 One Sample . . . .. . . . . . . . . .106

5.3.1 Repeated Measures ANOVA Model . . . . . . . . . 106

5.3.2 Sphericity Condition . . . . . . . . . . 109

5.3.3 Example . . . . . . . . . . . . . 111

5.4 Multiple Samples . . . . . . . . . . . . 112

5.4.1 Repeated Measures ANOVA Model . . . . . . . . . . 112

5.4.2 Example . . . .  . . . . . . . . . . . .115

5.5 Problems . . . . . .. . . . . . . . . . . .116

6 Normal-Theory Methods: Linear Mixed Models 125

6.1 Introduction . . . . . . . . . . . . .125

6.2 The Linear Mixed Model . . . . . . . . . . 126

6.2.1 The Usual Linear Model . . . . .  . . . . .126

6.2.2 The Mixed Model . . . . . . . .  .. . . . . 126

6.2.3 Parameter Estimation . . . . .. . . .  . .127

6.2.4 Background on REML Estimation . . . . .. . .  .128

6.3 Application to Repeated Measurements . . . . . 130

6.4 Examples . .. . .  . . . .134

6.4.1 Two Groups, Four Time Points, No Missing Data . . . . 134

6.4.2 Three Groups, 24 Time Points, No Missing Data . . . .  139

6.4.3 Four Groups, Unequally Spaced Repeated Measurements,

Time-Dependent Covariate . .. 145

6.5 Comments . . . . . . . . .. 149

6.5.1 Use of the Random Intercept and Slope Model . . . . . .. 149

6.5.2 E.ects of Choice of Covariance Structure on Estimates and Tests151

6.5.3 Performance of Linear Mixed Model Test Statisticsand Estimators155

6.6 Problems . . . .  . . . .156

7 Weighted Least Squares Analysis of Repeated Categorical Outcomes169

7.1 Introduction . . .  . . . . . 169

7.2 Background . . . . . . . . . 170

7.2.1 The Multinomial Distribution . . . .. . . . 170

7.2.2 Linear Models Using Weighted Least Squares . . . . .   .171

7.2.3 Analysis of Categorical Data Using Weighted Least Squares . . . . .175

7.2.4 Taylor Series Variance Approximations for Nonlinear Response Functions . 178

7.3 Application to Repeated Measurements . . . . . . 184

7.3.1 Overview . . . . . . . . . 184

7.3.2 One Population, Dichotomous Response, Repeated Measurements Factor Is Unordered 184

7.3.3 One Population, Dichotomous Response, Repeated Measurements Factor Is Ordered. 187

7.3.4 One Population, Polytomous Response . . . . . . . . 191

7.3.5 Multiple Populations, Dichotomous Response . . . . 196

7.4 Accommodation of Missing Data . . . . . . . . . . . . . . . 204

7.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 204

7.4.2 Ratio Estimation for Proportions . . . . . . . . . . . 204

7.4.3 One Population, Dichotomous Response . . . . . . . 205

7.4.4 Multiple Populations, Dichotomous Response . . . . 209

7.4.5 Assessing the Missing-Data Mechanism . . . . . . . 214

7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8 Randomization Model Methods for One-Sample Repeated Measurements 239

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

8.2 The Hypergeometric Distribution and Large-Sample Tests of Randomness for 2 × 2 Tables  240

8.2.1 The Hypergeometric Distribution . . . . . . . . . . . 240

8.2.2 Test of Randomness for a 2 × 2 Contingency Table . 241

8.2.3 Test of Randomness for s 2 × 2 Contingency Tables 242

8.3 Application to Repeated Measurements: Binary Response,

Two Time Points .. . . 244

8.4 The Multiple Hypergeometric Distribution and

Large-Sample Tests of Randomness for r × c Tables . . . . . . . . 246

8.4.1 The Multiple Hypergeometric Distribution . . . . . 247

8.4.2 Test of Randomness for an r × c Contingency Table 248

8.4.3 Test of Randomness for s r × c Tables . . . . . . . . 249

8.4.4 Cochran–Mantel–Haenszel Mean Score Statistic . . . 251

8.4.5 Cochran–Mantel–Haenszel Correlation Statistic . . . 253

8.5 Application to Repeated Measurements: Polytomous Response, Multiple Time Points . 253

8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 253

8.5.2 The General Association Statistic QG . . . . . . . . 255

8.5.3 The Mean Score Statistic QM and the Correlation Statistic QC . 255

8.6 Accommodation of Missing Data . . . . . . . . . . . . . . . 258

8.6.1 General Association Statistic QG . . . . . . . . . . . 258

8.6.2 Mean Score Statistic QM . . . . . . . . . . . . . . . 260

8.6.3 Correlation Statistic
    QC . . . . . . . . . . . . . . . . 262

8.7 Use of Mean Score and Correlation Statistics for Continuous Data263

8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

9 Methods Based on Extensions of Generalized Linear Models 273

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

9.2 Univariate Generalized Linear Models . . . . . . . . . . . . 274

9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 274

9.2.2 Random Component . . . . . . . . . . . . . . . . . . 275

9.2.3 Systematic Component . . . . . . . . . . . . . . . . 279

9.2.4 Link Function . . . . . . . . . . . . . . . . . . . . . . 279

9.2.5 Canonical Links . . . . . . . . . . . . . . . . . . . . 279

9.2.6 Parameter Estimation . . . . . . . . . . . . . . . . . 281

9.3 Quasilikelihood . . . . . . . . . . . . . . . . . . . . . . . . . 286

9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 286

9.3.2 Construction of a Quasilikelihood Function . . . . . 287

9.3.3 Quasilikelihood Estimating Equations . . . . . . . . 289

9.3.4 Comparison Between Quasilikelihood and Generalized Linear Models 291

9.4 Overview of Methods for the Analysis of Repeated Measurements .. . . 291

9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 291

9.4.2 Marginal Models . . . . . . . . . . . . . . . . . . . . 292

9.4.3 Random-E.ects Models . . . . . . . . . . . . . . . . 293

9.4.4 Transition Models . . . . . . . . . . . . . . . . . . . 293

9.4.5 Comparisons of the Three Approaches . . . . . . . . 294

9.5 The GEE Method . . . . . . . . . . . . . . . . . . . . . . . 295

9.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 295

9.5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . 296

9.5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . 301

9.5.4 Hypothesis Tests Using Wald Statistics . . . . . . . 308

9.5.5 Assessing Model Adequacy . . . . . . . . . . . . . . 309

9.5.6 Sample Size Estimation . . . . . . . . . . . . . . . . 310

9.5.7 Studies of the Properties of GEE . . . . . . . . . . . 311

9.5.8 Computer Software . . . . . . . . . . . . . . . . . . . 312

9.5.9 Cautions Concerning the Use of GEE . . . . . . . . 313

9.6 Subsequent Developments . . . . . . . . . . . . . . . . . . . 314

9.6.1 Alternative Procedures for Estimation of GEE Association Parameters 314

9.6.2 Other Developments and Extensions . . . . . . . . . 316

9.6.3 GEE1 and GEE2 . . . . . . . . . . . . . . . . . . . . 316

9.6.4 Extended Generalized Estimating Equations (EGEE) 317

9.6.5 Likelihood-Based Approaches . . . . . . . . . . . . . 318

9.7 Random-E.ects Models . . . . . . . . . . . . . . . . . . . . 318

9.8 Methods for the Analysis of Ordered Categorical Repeated Measurements 320

9.8.1 Introduction . . . . . . .  . . .  320

9.8.2 Univariate Cumulative Logit Models for Ordered Categorical Outcomes 321

9.8.3 The Univariate Proportional-Odds Model . . . . . . . . . . . . . 322

9.8.4 The Stram–Wei–Ware Methodology for the Analysis of

Ordered Categorical Repeated Measurements . . . . . . . . . . . . . 324

9.8.5 Extension of GEE to Ordered Categorical Outcomes  . . . . 331

9.9 Problems . . . . . . . . . . . . .. . . . . 332

10 Nonparametric Methods 347

10.1 Introduction . . . . . . . . . . . . 347

10.2 Overview . . . . . . . . . . . . . .. 348

10.3 Multivariate One-Sample and Multisample Tests for Complete Data .. 350

10.3.1 One Sample . . . .  . . . . . . .. 350

10.3.2 Multiple Samples . . . . . . . . . . 350

10.4 Two-Sample Tests for Incomplete Data . . . . . 355

10.4.1 Introduction . . . . .  . . .355

10.4.2 The Wei–Lachin Method . . . . . .355

10.4.3 The Wei–Johnson Method . . . . 356

10.4.4 Examples . . . . . . . .. . . . . . . . . ..362

10.5 Problems . . . . . . . . . .  . . . . . . .  364

Bibliography 373

Author Index 405

Subject Index 412

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