Models for discrete longitudinal data

The linear mixed model has become the main parametric tool for the analysis of continuous longitudinal data, as the authors discussed in their 2000 book.

Without putting too much emphasis on software, the book shows how the different approaches can be implemented within the SAS software package.

The authors received the American Statistical Association's Excellence in Continuing Education Award based on short courses on longitudinal and incomplete data at the Joint Statistical Meetings of 2002 and 2004.

Author Notes

Geert Molenberghs is Professor of Biostatistics at the Universiteit Hasselt in Belgium and has published methodological work on surrogate markers in clinical trials, categorical data, longitudinal data analysis, and the analysis of non-response in clinical and epidemiological studies
Geert Verbeke is Professor of Biostatistics at the Biostatistical Centre of the Katholieke Universiteit Leuven in Belgium

Preface	p. vii
Acknowledgments	p. ix
I Introductory Material	p. 1
1 Introduction	p. 3
2 Motivating Studies	p. 7
2.1 Introduction	p. 7
2.2 The Analgesic Trial	p. 8
2.3 The Toenail Data	p. 8
2.4 The Fluvoxamine Trial	p. 12
2.5 The Epilepsy Data	p. 14
2.6 The Project on Preterm and Small for Gestational Age Infants (POPS) Study	p. 14
2.7 National Toxicology Program Data	p. 17
2.8 The Sports Injuries Trial	p. 23
2.9 Age Related Macular Degeneration Trial	p. 24
3 Generalized Linear Models	p. 27
3.1 Introduction	p. 27
3.2 The Exponential Family	p. 27
3.3 The Generalized Linear Model (GLM)	p. 28
3.4 Examples	p. 29
3.5 Maximum Likelihood Estimation and Inference	p. 30
3.6 Logistic Regression for the Toenail Data	p. 31
3.7 Poisson Regression for the Epilepsy Data	p. 32
4 Linear Mixed Models for Gaussian Longitudinal Data	p. 35
4.1 Introduction	p. 35
4.2 Marginal Multivariate Model	p. 36
4.3 The Linear Mixed Model	p. 36
4.4 Estimation and Inference for the Marginal Model	p. 39
4.5 Inference for the Random Effects	p. 41
5 Model Families	p. 45
5.1 Introduction	p. 45
5.2 The Gaussian Case	p. 46
5.3 Model Families in General	p. 47
5.4 Inferential Paradigms	p. 52
II Marginal Models	p. 53
6 The Strength of Marginal Models	p. 55
6.1 Introduction	p. 55
6.2 Marginal Models in Contingency Tables	p. 56
6.3 British Occupational Status Study	p. 62
6.4 The Caithness Data	p. 62
6.5 Analysis of the Fluvoxamine Trial	p. 64
6.6 Extensions	p. 68
6.7 Relation to Latent Continuous Densities	p. 79
6.8 Conclusions and Perspective	p. 80
7 Likelihood-based Marginal Models	p. 83
7.1 Notation	p. 84
7.2 The Bahadur Model	p. 86
7.3 A General Framework for Fully Specified Marginal Models	p. 93
7.4 Maximum Likelihood Estimation	p. 99
7.5 An Influenza Study	p. 99
7.6 The Multivariate Probit Model	p. 102
7.7 The Dale Model	p. 113
7.8 Hybrid Marginal-conditional Specification	p. 122
7.9 A Cross-over Trial: An Example in Primary Dysmenorrhoea	p. 127
7.10 Multivariate Analysis of the POPS Data	p. 131
7.11 Longitudinal Analysis of the Fluvoxamine Study	p. 134
7.12 Appendix: Maximum Likelihood Estimation	p. 136
7.13 Appendix: The Multivariate Plackett Distribution	p. 142
7.14 Appendix: Maximum Likelihood Estimation for the Dale Model	p. 147
8 Generalized Estimating Equations	p. 151
8.1 Introduction	p. 151
8.2 Standard GEE Theory	p. 153
8.3 Alternative GEE Methods	p. 161
8.4 Prentice's GEE Method	p. 162
8.5 Second-order Generalized Estimating Equations (GEE2)	p. 164
8.6 GEE with Odds Ratios and Alternating Logistic Regression	p. 165
8.7 GEE2 Based on a Hybrid Marginal-conditional Model	p. 168
8.8 A Method Based on Linearization	p. 169
8.9 Analysis of the NTP Data	p. 170
8.10 The Heatshock Study	p. 174
8.11 The Sports Injuries Trial	p. 181
9 Pseudo-Likelihood	p. 189
9.1 Introduction	p. 189
9.2 Pseudo-Likelihood: Definition and Asymptotic Properties	p. 190
9.3 Pseudo-Likelihood Inference	p. 192
9.4 Marginal Pseudo-Likelihood	p. 195
9.5 Comparison with Generalized Estimating Equations	p. 199
9.6 Analysis of NTP Data	p. 200
10 Fitting Marginal Models with SAS	p. 203
10.1 Introduction	p. 203
10.2 The Toenail Data	p. 203
10.3 GEE1 with Correlations	p. 204
10.4 Alternating Logistic Regressions	p. 212
10.5 A Method Based on Linearization	p. 215
10.6 Programs for the NTP Data	p. 219
10.7 Alternative Software Tools	p. 221
III Conditional Models	p. 223
11 Conditional Models	p. 225
11.1 Introduction	p. 225
11.2 Conditional Models	p. 226
11.3 Marginal versus Conditional Models	p. 233
11.4 Analysis of the NTP Data	p. 234
11.5 Transition Models	p. 236
12 Pseudo-Likehood	p. 243
12.1 Introduction	p. 243
12.2 Pseudo-Likelihood for a Single Repeated Binary Outcome	p. 244
12.3 Pseudo-Likelihood for a Multivariate Repeated Binary Outcome	p. 245
12.4 Analysis of the NTP Data	p. 246
IV Subject-specific Models	p. 255
13 From Subject-specific to Random-effects Models	p. 257
13.1 Introduction	p. 257
13.2 General Model Formulation	p. 257
13.3 Three Ways to Handle Subject-specific Parameters	p. 258
13.4 Random-effects Models: Special Cases	p. 260
14 The Generalized Linear Mixed Model (GLMM)	p. 265
14.1 Introduction	p. 265
14.2 Model Formulation and Approaches to Estimation	p. 265
14.3 Estimation: Approximation of the Integrand	p. 268
14.4 Estimation: Approximation of the Data	p. 269
14.5 Estimation: Approximation of the Integral	p. 273
14.6 Inference in Generalized Linear Mixed Models	p. 276
14.7 Analyzing the NTP Data	p. 277
14.8 Analyzing the Toenail Data	p. 278
15 Fitting Generalized Linear Mixed Models with SAS	p. 281
15.1 Introduction	p. 281
15.2 The GLIMMIX Procedure for Quasi-Likelihood	p. 282
15.3 The GLIMMIX Macro for Quasi-Likelihood	p. 287
15.4 The NLMIXED Procedure for Numerical Quadrature	p. 290
15.5 Alternative Software Tools	p. 296
16 Marginal versus Random-effects Models	p. 297
16.1 Introduction	p. 297
16.2 Example: The Toenail Data	p. 297
16.3 Parameter Interpretation	p. 298
16.4 Toenail Data: Marginal versus Mixed Models	p. 301
16.5 Analysis of the NTP Data	p. 304
V Case Studies and Extensions	p. 307
17 The Analgesic Trial	p. 309
17.1 Introduction	p. 309
17.2 Marginal Analyses of the Analgesic Trial	p. 310
17.3 Random-effects Analyses of the Analgesic Trial	p. 314
17.4 Comparing Marginal and Random-effects Analyses	p. 317
17.5 Programs for the Analgesic Trial	p. 318
18 Ordinal Data	p. 325
18.1 Regression Models for Ordinal Data	p. 326
18.2 Marginal Models for Repeated Ordinal Data	p. 329
18.3 Random-effects Models for Repeated Ordinal Data	p. 331
18.4 Ordinal Analysis of the Analgesic Trial	p. 332
18.5 Programs for the Analgesic Trial	p. 334
19 The Epilepsy Data	p. 337
19.1 Introduction	p. 337
19.2 A Marginal GEE Analysis	p. 337
19.3 A Generalized Linear Mixed Model	p. 340
19.4 Marginalizing the Mixed Model	p. 342
20 Non-linear Models	p. 347
20.1 Introduction	p. 347
20.2 Univariate Non-linear Models	p. 349
20.3 The Indomethacin Study: Non-hierarchical Analysis	p. 351
20.4 Non-linear Models for Longitudinal Data	p. 355
20.5 Non-linear Mixed Models	p. 357
20.6 The Orange Tree Data	p. 358
20.7 Pharmacokinetic and Pharmacodynamic Models	p. 360
20.8 The Songbird Data	p. 368
20.9 Discrete Outcomes	p. 376
20.10 Hypothesis Testing and Non-linear Models	p. 379
20.11 Flexible Functions	p. 379
20.12 Using SAS for Non-linear Mixed-effects Models	p. 384
21 Pseudo-Likelihood for a Hierarchical Model	p. 393
21.1 Introduction	p. 393
21.2 Pseudo-Likelihood Estimation	p. 394
21.3 Two Binary Endpoints	p. 397
21.4 A Meta-analysis of Trials in Schizophrenic Subjects	p. 401
21.5 Concluding Remarks	p. 403
22 Random-effects Models with Serial Correlation	p. 405
22.1 Introduction	p. 405
22.2 A Multilevel Probit Model with Autocorrelation	p. 406
22.3 Parameter Estimation for the Multilevel Probit Model	p. 408
22.4 A Generalized Linear Mixed Model with Autocorrelation	p. 410
22.5 A Meta-analysis of Trials in Schizophrenic Subjects	p. 412
22.6 SAS Code for Random-effects Models with Autocorrelation	p. 415
22.7 Concluding Remarks	p. 417
23 Non-Gaussian Random Effects	p. 419
23.1 Introduction	p. 419
23.2 The Heterogeneity Model	p. 421
23.3 Estimation and Inference	p. 423
23.4 Empirical Bayes Estimation and Classification	p. 427
23.5 The Verbal Aggression Data	p. 428
23.6 Concluding Remarks	p. 435
24 Joint Continuous and Discrete Responses	p. 437
24.1 Introduction	p. 437
24.2 A Continuous and a Binary Endpoint	p. 439
24.3 Hierarchical Joint Models	p. 445
24.4 Age Related Macular Degeneration Trial	p. 448
24.5 Joint Models in SAS	p. 455
24.6 Concluding Remarks	p. 464
25 High-dimensional Joint Models	p. 467
25.1 Introduction	p. 467
25.2 Joint Mixed Model	p. 469
25.3 Model Fitting and Inference	p. 471
25.4 A Study in Psycho-Cognitive Functioning	p. 473
VI Missing Data	p. 479
26 Missing Data Concepts	p. 481
26.1 Introduction	p. 481
26.2 A Formal Taxonomy	p. 482
27 Simple Methods, Direct Likelihood, and WGEE	p. 489
27.1 Introduction	p. 489
27.2 Longitudinal Analysis or Not?	p. 490
27.3 Simple Methods	p. 491
27.4 Bias in LOCF, CC, and Ignorable Likelihood	p. 495
27.5 Weighted Generalized Estimating Equations	p. 498
27.6 The Depression Trial	p. 499
27.7 Age Related Macular Degeneration Trial	p. 503
27.8 The Analgesic Trial	p. 507
28 Multiple Imputation and the EM Algorithm	p. 511
28.1 Introduction	p. 511
28.2 Multiple Imputation	p. 511
28.3 The Expectation-Maximization Algorithm	p. 516
28.4 Which Method to Use?	p. 526
28.5 Age Related Macular Degeneration Study	p. 527
28.6 Concluding Remarks	p. 529
29 Selection Models	p. 531
29.1 Introduction	p. 531
29.2 An MNAR Dale Model	p. 532
29.3 A Model for Non-monotone Missingness	p. 543
29.4 Concluding Remarks	p. 552
30 Pattern-mixture Models	p. 555
30.1 Introduction	p. 555
30.2 Pattern-mixture Modeling Approach	p. 556
30.3 Identifying Restriction Strategies	p. 557
30.4 A Unifying Framework for Selection and Pattern-mixture Models	p. 561
30.5 Selection Models versus Pattern-mixture Models	p. 563
30.6 Analysis of the Fluvoxamine Data	p. 567
30.7 Concluding Remarks	p. 572
31 Sensitivity Analysis	p. 575
31.1 Introduction	p. 575
31.2 Sensitivity Analysis for Selection Models	p. 576
31.3 A Local Influence Approach for Ordinal Data with Dropout	p. 578
31.4 A Local Influence Approach for Incomplete Binary Data	p. 585
31.5 Interval of Ignorance	p. 590
31.6 Sensitivity Analysis and Pattern-mixture Models	p. 604
31.7 Concluding Remarks	p. 605
32 Incomplete Data and SAS	p. 607
32.1 Introduction	p. 607
32.2 Complete Case Analysis	p. 607
32.3 Last Observation Carried Forward	p. 609
32.4 Direct Likelihood	p. 611
32.5 Weighted Estimating Equations (WGEE)	p. 613
32.6 Multiple Imputation	p. 618
32.7 The EM Algorithm	p. 633
32.8 MNAR Models and Sensitivity Analysis Tools	p. 635
References	p. 637
Index	p. 671

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