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Dynamic Mixed Models for Familial Longitudinal Data
Brajendra C. Sutradhar
Verlag Springer-Verlag, 2011
ISBN 9781441983428 , 494 Seiten
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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