Dynamic Mixed Models for Familial Longitudinal Data (eBook)

Brajendra Sutradhar is a University Research Professor at Memorial University in St. John's, Canada. He is an elected member of the International Statistical Institute and a fellow of the American Statistical Association. He has published about 110 papers in statistics journals in the area of multivariate analysis, time series analysis including forecasting, sampling, survival analysis for correlated failure times, robust inferences in generalized linear mixed models with outliers, and generalized linear longitudinal mixed models with bio-statistical and econometric applications. He has served as an associate editor for six years for Canadian Journal of Statistics and for four years for the Journal of Environmental and Ecological Statistics. He has served for 3 years as a member of the advisory committee on statistical methods in Statistics Canada. Professor Sutradhar was awarded the 2007 distinguished service award of Statistics Society of Canada for his many years of services to the society including his special services for society's annual meetings.

Dynamic Mixed Models for Familial Longitudinal Data 3
Preface 7
Acknowledgements 11
Contents 13
Chapter 1 Introduction 19
1.1 Background of Familial Models 19
1.2 Background of Longitudinal Models 21
References 24
Chapter 2 Overview of Linear Fixed Models for Longitudinal Data 27
2.1 Estimation of ß 28
2.1.1 Method of Moments (MM) 28
2.1.2 Ordinary Least Squares (OLS) Method 29
2.1.2.1 Generalized Least Squares (GLS) Method 30
2.1.3 OLS Versus GLS Estimation Performance 31
2.2 Estimation of ß Under Stationary General Autocorrelation Structure 32
2.2.1 A Class of Autocorrelations 32
2.2.2 Estimation of ß 36
2.3 A Rat Data Example 37
2.4 Alternative Modelling for Time Effects 41
Exercises 42
References 44
Appendix 45
Chapter 3 Overview of Linear Mixed Models for Longitudinal Data 47
3.1 Linear Longitudinal Mixed Model 48
3.1.1 GLS Estimation of ß 49
3.1.2 Moment Estimating Equations for s². and .l 50
3.1.3 Linear Mixed Models for Rat Data 51
3.2 Linear Dynamic Mixed Models for Balanced Longitudinal Data 54
3.2.1 Basic Properties of the Dynamic Dependence Mixed Model (3.21) 55
3.2.2 Estimation of the Parameters of the Dynamic Mixed Model (3.21) 56
3.3 Further Estimation for the Parameters of the Dynamic Mixed Model 60
3.3.1 GMM/IMM Estimation Approach 61
3.3.2 GQL Estimation Approach 66
3.3.3 Asymptotic Efficiency Comparison 70
Exercises 73
References 75
Chapter 4 Familial Models for Count Data 77
4.1 Poisson Mixed Models and Basic Properties 78
4.2 Estimation for Single Random Effect Based Parametric Mixed Models 81
4.2.1 Exact Likelihood Estimation and Drawbacks 81
4.2.2 Penalized Quasi-Likelihood Approach 83
4.2.3 Small Variance Asymptotic Approach: A Likelihood Approximation (LA) 86
4.2.3.1 A Higher-Order Likelihood Approximation (HOLA) 89
4.2.4 Hierarchical Likelihood (HL) Approach 93
4.2.5 Method of Moments (MM) 95
4.2.6 Generalized Quasi-Likelihood (GQL) Approach 96
4.2.6.1 Marginal Generalized Quasi-Likelihood (GQL) Estimation of ß 97
4.2.6.2 Marginal Generalized Quasi-Likelihood (GQL) Estimation of s². 98
4.2.6.3 Joint Generalized Quasi-Likelihood (GQL) Estimation for ß and s². 101
4.2.7 Efficiency Comparison 103
4.2.7.1 Efficiency Comparison Between GQL and MM Approaches: A Small Sample Study 103
4.2.7.2 Efficiency Comparison Between GQL and HL Approaches: A Small Sample Study 106
4.2.8 A Health Care Data Utilization Example 109
4.3 Estimation for Multiple Random Effects Based Parametric Mixed Models 112
4.3.1 Random Effects in a Two-Way Factorial Design Setup 112
4.3.2 One-Way Heteroscedastic Random Effects 112
4.3.3 Multiple Independent Random Effects 113
4.3.3.1 Method of Moments Estimation for ß, s². , and s²t 114
4.3.3.2 Joint GQL Estimation for ß, s². , and s²t 115
4.3.3.3 Relative Performances of the GQL Versus MM Approaches: An Asymptotic Efficiency Comparison 117
4.3.3.4 GQL Versus MM Estimation: A Simulation Study Based on an Asthma Count Data Model with Two Components of Dispersion 120
4.3.3.5 An Asthma Count Data Model with Four Fixed Covariates and Two Components of Dispersion 120
4.4 Semiparametric Approach 122
4.4.1 Computations for µi, .i, Si, and Oi 125
4.4.2 Construction of the Estimating Equation for When ß When s². Is Known 128
4.5 Monte Carlo Based Likelihood Estimation 129
4.5.1 MCEM Approach 131
4.5.2 MCNR Approach 131
Exercises 132
References 135
Chapter 5 Familial Models for Binary Data 137
5.1 Binary Mixed Models and Basic Properties 138
5.1.1 Computational Formulas for Binary Moments 141
5.2 Estimation for Single Random Effect Based Parametric Mixed Models 142
5.2.1 Method of Moments (MM) 142
5.2.2 An Improved Method of Moments (IMM) 144
5.2.2.1 Can There Be an Optimal B Free from Third and Fourth-Order Moments Under Simple Binary Logistic Mixed Models? 145
5.2.2.2 Effect of Mis-specification For Optimal Choice 148
5.2.3 Generalized Quasi-Likelihood (GQL) Approach 149
5.2.3.1 Marginal Generalized Quasi-Likelihood Estimation of ß 149
5.2.3.2 Marginal Generalized Quasi-Likelihood Estimation of s. 150
5.2.3.3 Joint Generalized Quasi-Likelihood (GQL) Estimation for ß and s. 152
5.2.4 Maximum Likelihood (ML) Estimation 153
5.2.5 Asymptotic Efficiency Comparison 156
5.2.5.1 Asymptotic variance of the IMM Estimator 156
5.2.5.2 Asymptotic Variance of the GQL Estimator 157
5.2.5.3 Asymptotic Variance of the ML Estimator 158
5.2.5.4 Numerical Comparison 160
5.2.6 COPD Data Analysis: A Numerical Illustration 161
5.3 Binary Mixed Models with Multidimensional Random Effects 164
5.3.1 Models in Two-Way Factorial Design Setup and Basic Properties 164
5.3.1.1 Unconditional Mean 165
5.3.1.2 Unconditional Covariances and Correlations in a Two-Way Design Setup 166
5.3.2 Estimation of Parameters 167
5.3.2.1 Estimation of Regression Effects ß 167
5.3.2.2 Estimation of the Variance Component s². Due to Factor A 169
5.3.2.3 Estimation of the Variance Component s² a Due to Factor B 173
5.3.3 Salamander Mating Data Analysis 178
5.3.3.1 Data Description 178
5.3.3.2 Binary Mixed Model for Salamander Data 179
5.3.3.3 Model Parameters Estimation and Interpretation 180
5.4 Semiparametric Approach 182
5.4.1 GQL Estimation 182
5.4.2 A Marginal Quasi-Likelihood (MQL) Approach 184
5.4.3 Asymptotic Efficiency Comparison: An Empirical Study 185
5.5 Monte Carlo Based Likelihood Estimation 187
Exercises 187
References 190
Appendix 192
Chapter 6 Longitudinal Models for Count Data 198
6.1 Marginal Model 199
6.2 Marginal Model Based Estimation of Regression Effects 200
6.3 Correlation Models for Stationary Count Data 202
6.3.1 Poisson AR(1) Model 203
6.3.2 Poisson MA(1) Model 204
6.3.3 Poisson Equicorrelation Model 204
6.4 Inferences for Stationary Correlation Models 205
6.4.1 Likelihood Approach and Complexity 205
6.4.2 GQL Approach 206
6.4.2.1 Asymptotic Distribution of the GQL Estimator 207
6.4.2.2 ‘Working’ Independence Assumption Based GQL Estimation 208
6.4.2.3 Efficiency of the Independence Assumption Based Estimator 208
6.4.2.4 Performance of the GQL Estimation: A Simulation Example 210
6.4.3 GEE Approach and Limitations 213
6.4.3.1 Efficiency of the GEE Based Estimator Under Correlation Structure Mis-specification 213
6.5 Nonstationary Correlation Models 218
6.5.1 Nonstationary Correlation Models with the Same Specified Marginal Mean and Variance Functions 219
6.5.1.1 Nonstationary AR(1) Models 219
6.5.1.2 Nonstationary MA(1) Models 220
6.5.1.3 Nonstationary EQC Models 220
6.5.2 Estimation of Parameters 222
6.5.2.1 Estimation of r Parameter Under AR(1) Model 222
6.5.2.2 Estimation of r Parameter Under MA(1) Correlation Model 223
6.5.2.3 Estimation of . Parameter Under Exchangeable (EQC) Correlation Model 223
6.5.3 Model Selection 224
6.6 More Nonstationary Correlation Models 226
6.6.1 Models with Variable Marginal Means and Variances 226
6.6.1.1 Nonstationary MA(1) Models 226
6.6.2 Estimation of Parameters 228
6.6.2.1 GQL Estimation for Regression Effects ß 228
6.6.2.2 Moment Estimation for the Correlation Parameter . 229
6.6.3 Model Selection 230
6.6.4 Estimation and Model Selection: A Simulation Example 232
6.6.4.1 Simulated Estimates Under the True and Misspecified Models 232
6.6.4.2 Model Selection 233
6.7 A Data Example: Analyzing Health Care Utilization Count Data 234
6.8 Models for Count Data from Longitudinal Adaptive Clinical Trials 236
6.8.1 Adaptive Longitudinal Designs 237
6.8.1.1 Simple Longitudinal Play-the-Winner (SLPW) Rule to Formulate wi 239
6.8.1.2 Bivariate Random Walk (BRW) Design 240
6.8.2 Performance of the SLPW and BRW Designs For Treatment Selection: A Simulation Study 241
6.8.3 Weighted GQL Estimation for Treatment Effects and Other Regression Parameters 244
6.8.3.1 Formulas for µi(wi0), and Si* (wi0,.) : 244
6.8.3.2 Weighted GQL Estimation of ß 246
Exercises 248
References 251
Appendix 253
Chapter 7 Longitudinal Models for Binary Data 258
7.1 Marginal Model 260
7.1.1 Marginal Model Based Estimation for Regression Effects 261
7.2 Some Selected Correlation Models for Longitudinal Binary Data 262
7.2.1 Bahadur Multivariate Binary Density (MBD) Based Model 263
7.2.1.1 Stationary Case 263
7.2.1.2 Nonstationary Case 265
7.2.2 Kanter Observation-Driven Dynamic (ODD) Model 266
7.2.2.1 Stationary Case 266
7.2.2.2 Non-stationary Case 268
7.2.3 A Linear Dynamic Conditional Probability (LDCP) Model 269
7.2.3.1 Stationary Case 269
7.2.3.2 Nonstationary Case 271
7.2.4 A Numerical Comparison of Range Restrictions for Correlation Index Parameter Under Stationary Binary Models 271
7.3 Low-Order Autocorrelation Models for Stationary Binary Data 273
7.3.1 Binary AR(1) Model 273
7.3.2 Binary MA(1) Model 273
7.3.3 Binary Equicorrelation (EQC) Model 276
7.3.4 Complexity in Likelihood Inferences Under Stationary Binary Correlation Models 277
7.3.5 GQL Estimation Approach 278
7.3.5.1 Efficiency of the Independence Assumption Based Estimation 279
7.3.6 GEE Approach and Its Limitations for Binary Data 281
7.4 Inferences in Nonstationary Correlation Models for Repeated Binary Data 283
7.4.1 Nonstationary AR(1) Correlation Model 283
7.4.2 Nonstationary MA(1) Correlation Model 285
7.4.3 Nonstationary EQC Model 286
7.4.4 Nonstationary Correlations Based GQL Estimation 287
7.4.4.1 Estimation of . Parameter Under Binary AR(1) Model 289
7.4.4.2 Estimation of . Parameter Under Binary MA(1) Correlation Model 289
7.4.4.3 Estimation of . Parameter Under Exchangeable (EQC) Correlation Model 290
7.4.5 Model Selection 290
7.5 SLID Data Example 291
7.5.1 Introduction to the SLID Data 291
7.5.2 Analysis of the SLID Data 293
7.6 Application to an Adaptive Clinical Trial Setup 295
7.6.1 Binary Response Based Adaptive Longitudinal Design 295
7.6.1.1 Simple Longitudinal Play-the-Winner (SLPW) Rule to Formulate wi 297
7.6.1.2 Performance of the Adaptive Design 299
7.6.2 Construction of the Adaptive Design Weights Based Weighted GQL Estimation 302
7.6.2.1 Computation of Unconditional Expectation of di : wi0 302
7.6.2.2 WGQL Estimating Equations for Regression Parameters Including the Treatment Effects 303
7.6.2.2.1 Moment Estimates for Longitudinal Correlations 306
7.6.2.2.2 Asymptotic Variances of the WGQL Regression Estimates 307
7.7 More Nonstationary Binary Correlation Models 307
7.7.1 Linear Binary Dynamic Regression (LBDR) Model 307
7.7.1.1 Autocorrelation Structure 308
7.7.1.2 GQL and Conditional GQL (CGQL) Approaches for Parameter Estimation 309
7.7.2 A Binary Dynamic Logit (BDL) Model 312
7.7.2.1 Basic Properties of the Lag 1 Dependence Model (7.142) 312
7.7.2.2 Estimation of the Parameters of the BDL Model 314
7.7.2.2.1 GQL Estimation 315
7.7.2.2.2 OGQL Estimation 316
7.7.2.2.3 Likelihood Estimation 321
7.7.2.3 Fitting Asthma Data to the BDL Model: An Illustration 322
7.7.3 Application of the Binary Dynamic Logit (BDL) Model in an Adaptive Clinical Trial Setup 324
7.7.3.1 Random Treatments Based BDL Model 324
7.7.3.1.1 Unconditional Moments Up to Order Four 325
7.7.3.1.2 Extended WGQL (EWGQL) or Weighted OGQL (WOGQL) Estimating Equation 328
Exercises 331
References 333
Appendix 335
Chapter 8 Longitudinal Mixed Models for Count Data 338
8.1 A Conditional Serially Correlated Model 338
8.1.1 Unconditional Mean, Variance, and Correlations Under Serially Correlated Model 340
8.2 Parameter Estimation 340
8.2.1 Estimation of the Regression Effects ß 341
8.2.1.1 GMM/IMM Approach 341
8.2.1.2 GQL Approach 342
8.2.1.3 Conditional Maximum Likelihood (CML) Approach 343
8.2.1.4 Instrumental Variables Based GMM (IVBGMM) Estimation Approach 344
8.2.1.5 A Simulation Study 346
8.2.2 Estimation of the Random Effects Variance s². : 349
8.2.2.1 GMM Estimation for s². 349
8.2.2.2 GQL Estimation for s². : 351
8.2.2.3 Asymptotic Efficiency Comparison : GMM versus GQL 352
8.2.2.3.1 Asymptotic Variances of the GMM Estimators 352
8.2.2.3.2 Asymptotic Variances of the GQL Estimators 352
8.2.2.3.3 Asymptotic Efficiency Computation 353
8.2.3 Estimation of the Longitudinal Correlation Parameter . 354
8.2.3.1 GMM Estimation for . 354
8.2.3.2 . Estimation Under the GQL Approach 355
8.2.4 A Simulation Study 356
8.2.4.1 Estimation Under the ‘Working’ Conditional Independence (. = 0) Model 360
8.2.4.2 Estimation Under the ‘Working’ Longitudinal Fixed (s². = 0) Model 362
8.2.5 An Illustration: Analyzing Health Care Utilization Count Data by Using Longitudinal Fixed and Mixed Models 363
8.3 A Mean Deflated Conditional Serially Correlated Model 365
8.3.1 First and Second-Order Raw Response Based GQL Estimation 366
8.3.1.1 GQL(I) Approach for s². Estimation 366
8.3.1.2 GQL(N) Approach for s². Estimation 366
8.3.2 Corrected Response (CR) Based GQL Estimation 368
8.3.2.1 GQL(CR-I) Estimation for s². 368
8.3.2.2 GQL(CR-N) Estimation s². 370
8.3.3 Relative Performances of GQL(I) and GQL(N) Estimation Approaches: A Simulation Study 371
8.3.3.1 Performance for Overdispersion Estimation 371
8.3.3.2 Performance for Regression Effects Estimation 372
8.3.3.3 Performance for Correlation Index Estimation 374
8.3.4 A Further Application: Analyzing Patent Count Data 374
8.4 Longitudinal Negative Binomial Fixed Model and Estimationof Parameters 379
8.4.1 Inferences in Stationary Negative Binomial CorrelationModels 380
8.4.1.1 Estimation of Parameters 381
8.4.1.1.1 GQL Estimation for ß 381
8.4.1.1.2 Estimation of c* 382
8.4.1.1.3 Moment Estimation of . 384
8.4.2 A Data Example: Analyzing Epileptic Count Data by Using Poisson and Negative Binomial Longitudinal Models 384
8.4.3 Nonstationary Negative Binomial Correlation Models and Estimation of Parameters 386
8.4.3.1 First Two Moments Based Negative Binomial Autoregression Model 386
8.4.3.1.1 Nonstationary Mean Variance Structure 387
8.4.3.1.2 Non-stationary Correlation Structure 388
8.4.3.2 A Proposed Conditional GQL (CGQL) Estimation Approach 388
8.4.3.2.1 CGQL Estimation for ß 389
8.4.3.2.2 CGQL Estimation for c* 390
8.4.3.2.3 MMs Equation for . 392
Exercises 392
References 394
Appendix 396
Chapter 9 Longitudinal Mixed Models for Binary Data 405
9.1 A Conditional Serially Correlated Model 406
9.1.1 Basic Properties of the Model 406
9.1.2 Parameter Estimation 408
9.1.2.1 GQL Estimation of the Regression Effects ß 408
9.1.2.2 GQL Estimation of the Random Effects Variance s². 409
9.1.2.2.1 GQL(I) Estimation of s². 410
9.1.2.2.2 GQL(N) Estimation of s². 410
9.1.2.3 Estimation of . Under the GQL Approach 411
9.2 Binary Dynamic Mixed Logit (BDML) Model 412
9.2.1 GMM/IMM Estimation 414
9.2.1.1 Construction of the Unbiased Moment Functions 414
9.2.1.1.1 Formula for pit 415
9.2.1.1.2 Formula for .iut 415
9.2.1.2 GMM Estimating Equation for a = (ß', ., s². )' 416
9.2.1.2.1 Computation of the C Matrix 417
9.2.1.2.2 Computation of ..'/.a 419
9.2.2 GQL Estimation 419
9.2.2.1 Computation of Oi 420
9.2.3 Efficiency Comparison: GMM Versus GQL 421
9.2.3.1 Asymptotic Distribution of the GMM Estimator 421
9.2.3.2 Asymptotic Distribution of the GQL Estimator 422
9.2.3.3 Asymptotic Efficiency Comparison 422
9.2.3.4 Small Sample Efficiency Comparison: A Simulation Study 424
9.2.4 Fitting the Binary Dynamic Mixed Logit Model to the SLID data 425
9.2.5 GQL Versus Maximum Likelihood (ML) Estimation for BDML Model 427
9.2.5.1 ML Estimation 428
9.2.5.2 Relative Performances of the GQL and ML Approaches for BDML model: A Simulation Study 429
9.3 A Binary Dynamic Mixed Probit (BDMP) Model 431
9.3.1 GQL Estimation for BDMP Model 432
9.3.2 GQL Estimation Performance for BDMP Model: A Simulation Study 433
9.3.2.1 Random Effects Mis-specification: True t Versus Working Normal Distributions For Random Effects 434
Exercises 436
References 437
Chapter 10 Familial Longitudinal Models for Count Data 439
10.1 An Autocorrelation Class of Familial Longitudinal Models 439
10.1.1 Marginal Mean and Variance 440
10.1.1.1 Conditional Marginal Mean and Variance 440
10.1.1.1 Unconditional Marginal Mean and Variance 440
10.1.2 Nonstationary Autocorrelation Models 441
10.1.2.1 Conditional AR(1) Model 441
10.1.2.1.1. Unconditional Mean, Variance, and Correlation Structure 442
10.1.2.2 Conditional MA(1) Model 442
10.1.2.2.1. Unconditional Mean, Variance, and Correlation Structure 443
10.1.2.3 An Alternative Conditional MA(1) Model 443
10.1.2.3.1 Unconditional First and Second-Order Moments 444
10.1.2.4 Conditional EQC Model 444
10.1.2.4.1. Unconditional Mean, Variance, and Correlation Structure 445
10.2 Parameter Estimation 445
10.2.1 Estimation of Parameters Under Conditional AR(1) Model 446
10.2.1.1 GQL Estimation of Regression Parameter ß 446
10.2.1.2 GQL Estimation of Familial Correlation Index Parameter s². 447
10.2.1.2.1 GQL(I) Estimation of s². 449
10.2.1.2.2 GQL(N) Estimation of s². 451
10.2.1.3 Estimation of Longitudinal Correlation Index Parameter . 454
10.2.2 Performance of the GQL Approach: A Simulation Study 455
10.2.2.1 Simulation Study with p = 1 Covariate 455
10.2.2.2 Simulation Study with p = 2 Covariates 457
10.2.2.3 Effects of Partial Model Fitting: A Further Simulation Study with p = 2 Covariates 459
10.3 Analyzing Health Care Utilization Data by Using GLLMM 462
10.4 Some Remarks on Model Identification 465
10.4.1 An Exploratory Identification 466
10.4.2 A Further Improved Identification 467
Exercises 467
References 469
Chapter 11 Familial Longitudinal Models for Binary Data 471
11.1 LDCCP Models 472
11.1.1 Conditional-Conditional (CC) AR(1) Model 472
11.1.1.1 Conditional Mean, Variance, and Correlation Structure 472
11.1.1.2 Unconditional Mean, Variance, and Correlation Structure 473
11.1.2 CC MA(1) Model 474
11.1.3 CC EQC Model 475
11.1.4 Estimation of the AR(1) Model Parameters 476
11.1.4.1 GQL Estimation of Regression Parameter ß 476
11.1.4.2 GQL Estimation of Familial Correlation Index Parameter s². 478
11.1.4.3 Moment Estimation of Longitudinal Correlation Index Parameter . 483
11.2 Application toWaterloo Smoking Prevention Data 484
11.3 Family Based BDML Models for Binary Data 487
11.3.1 FBDML Model and Basic Properties 488
11.3.1.1 Conditional Mean, Variance, and Correlation Structures 488
11.3.1.2 Unconditional Mean, Variance, and Correlation Structures 489
11.3.2 Quasi-Likelihood Estimation in the Familial Longitudinal Setup 490
11.3.2.1 Joint GQL Estimation of Parameters 490
11.3.2.2 Asymptotic Covariance Matrix of the Joint GQL Estimator 494
11.3.3 Likelihood Based Estimation 495
11.3.3.1 Likelihood Function for the FBDML Model 495
11.3.3.2 Likelihood Estimating Equations 495
11.3.3.3 Asymptotic Covariance of the Joint ML Estimator 497
Exercises 499
References 503
Index 505