Sutradhar Dynamic Mixed Models for Familial Longitudinal Data

2011
ISBN: 978-1-4419-8342-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 494 Seiten

Reihe: Springer Series in Statistics

This book provides a theoretical foundation for the analysis of discrete data such as count and binary data in the longitudinal setup. Unlike the existing books, this book uses a class of auto-correlation structures to model the longitudinal correlations for the repeated discrete data that accommodates all possible Gaussian type auto-correlation models as special cases including the equi-correlation models. This new dynamic modelling approach is utilized to develop theoretically sound inference techniques such as the generalized quasi-likelihood (GQL) technique for consistent and efficient estimation of the underlying regression effects involved in the model, whereas the existing 'working' correlations based GEE (generalized
estimating equations) approach has serious theoretical limitations both for consistent and efficient estimation, and the existing random effects based correlations approach is not suitable to model the longitudinal correlations. The book has exploited the random effects carefully only to model the correlations of the familial data. Subsequently, this book has modelled the correlations of the longitudinal data collected from the members of a large number of independent families by using the class of auto-correlation structures conditional on the random effects. The book also provides models and inferences for discrete longitudinal data in the adaptive clinical trial set up.
The book is mathematically rigorous and provides details for the development of estimation approaches under selected familial and longitudinal models. Further, while the book provides special cares for mathematics behind the correlation models, it also presents the
illustrations of the statistical analysis of various real life data.
This book will be of interest to the researchers including graduate students in biostatistics and econometrics, among other applied statistics research areas.
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 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.

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.

Autoren/Hrsg.

Sutradhar, Brajendra C.

Inhaltsverzeichnis

1;Dynamic Mixed Models for Familial Longitudinal Data;3
1.1;Preface;7
1.2;Acknowledgements;11
1.3;Contents;13
1.4;Chapter 1 Introduction;19
1.4.1;1.1 Background of Familial Models;19
1.4.2;1.2 Background of Longitudinal Models;21
1.4.3;References;24
1.5;Chapter 2 Overview of Linear Fixed Models for Longitudinal Data;27
1.5.1;2.1 Estimation of ß;28
1.5.1.1;2.1.1 Method of Moments (MM);28
1.5.1.2;2.1.2 Ordinary Least Squares (OLS) Method;29
1.5.1.2.1;2.1.2.1 Generalized Least Squares (GLS) Method;30
1.5.1.3;2.1.3 OLS Versus GLS Estimation Performance;31
1.5.2;2.2 Estimation of ß Under Stationary General Autocorrelation Structure;32
1.5.2.1;2.2.1 A Class of Autocorrelations;32
1.5.2.2;2.2.2 Estimation of ß;36
1.5.3;2.3 A Rat Data Example;37
1.5.4;2.4 Alternative Modelling for Time Effects;41
1.5.5;Exercises;42
1.5.6;References;44
1.5.7;Appendix;45
1.6;Chapter 3 Overview of Linear Mixed Models for Longitudinal Data;47
1.6.1;3.1 Linear Longitudinal Mixed Model;48
1.6.1.1;3.1.1 GLS Estimation of ß;49
1.6.1.2;3.1.2 Moment Estimating Equations for s². and .l;50
1.6.1.3;3.1.3 Linear Mixed Models for Rat Data;51
1.6.2;3.2 Linear Dynamic Mixed Models for Balanced Longitudinal Data;54
1.6.2.1;3.2.1 Basic Properties of the Dynamic Dependence Mixed Model (3.21);55
1.6.2.2;3.2.2 Estimation of the Parameters of the Dynamic Mixed Model (3.21);56
1.6.3;3.3 Further Estimation for the Parameters of the Dynamic Mixed Model;60
1.6.3.1;3.3.1 GMM/IMM Estimation Approach;61
1.6.3.2;3.3.2 GQL Estimation Approach;66
1.6.3.3;3.3.3 Asymptotic Efficiency Comparison;70
1.6.4;Exercises;73
1.6.5;References;75
1.7;Chapter 4 Familial Models for Count Data;77
1.7.1;4.1 Poisson Mixed Models and Basic Properties;78
1.7.2;4.2 Estimation for Single Random Effect Based Parametric Mixed Models;81
1.7.2.1;4.2.1 Exact Likelihood Estimation and Drawbacks;81
1.7.2.2;4.2.2 Penalized Quasi-Likelihood Approach;83
1.7.2.3;4.2.3 Small Variance Asymptotic Approach: A Likelihood Approximation (LA);86
1.7.2.3.1;4.2.3.1 A Higher-Order Likelihood Approximation (HOLA);89
1.7.2.4;4.2.4 Hierarchical Likelihood (HL) Approach;93
1.7.2.5;4.2.5 Method of Moments (MM);95
1.7.2.6;4.2.6 Generalized Quasi-Likelihood (GQL) Approach;96
1.7.2.6.1;4.2.6.1 Marginal Generalized Quasi-Likelihood (GQL) Estimation of ß;97
1.7.2.6.2;4.2.6.2 Marginal Generalized Quasi-Likelihood (GQL) Estimation of s².;98
1.7.2.6.3;4.2.6.3 Joint Generalized Quasi-Likelihood (GQL) Estimation for ß and s².;101
1.7.2.7;4.2.7 Efficiency Comparison;103
1.7.2.7.1;4.2.7.1 Efficiency Comparison Between GQL and MM Approaches: A Small Sample Study;103
1.7.2.7.2;4.2.7.2 Efficiency Comparison Between GQL and HL Approaches: A Small Sample Study;106
1.7.2.8;4.2.8 A Health Care Data Utilization Example;109
1.7.3;4.3 Estimation for Multiple Random Effects Based Parametric Mixed Models;112
1.7.3.1;4.3.1 Random Effects in a Two-Way Factorial Design Setup;112
1.7.3.2;4.3.2 One-Way Heteroscedastic Random Effects;112
1.7.3.3;4.3.3 Multiple Independent Random Effects;113
1.7.3.3.1;4.3.3.1 Method of Moments Estimation for ß, s². , and s²t;114
1.7.3.3.2;4.3.3.2 Joint GQL Estimation for ß, s². , and s²t;115
1.7.3.3.3;4.3.3.3 Relative Performances of the GQL Versus MM Approaches: An Asymptotic Efficiency Comparison;117
1.7.3.3.4;4.3.3.4 GQL Versus MM Estimation: A Simulation Study Based on an Asthma Count Data Model with Two Components of Dispersion;120
1.7.3.3.5;4.3.3.5 An Asthma Count Data Model with Four Fixed Covariates and Two Components of Dispersion;120
1.7.4;4.4 Semiparametric Approach;122
1.7.4.1;4.4.1 Computations for µi, .i, Si, and Oi;125
1.7.4.2;4.4.2 Construction of the Estimating Equation for When ß When s². Is Known;128
1.7.5;4.5 Monte Carlo Based Likelihood Estimation;129
1.7.5.1;4.5.1 MCEM Approach;131
1.7.5.2;4.5.2 MCNR Approach;131
1.7.6;Exercises;132
1.7.7;References;135
1.8;Chapter 5 Familial Models for Binary Data;137
1.8.1;5.1 Binary Mixed Models and Basic Properties;138
1.8.1.1;5.1.1 Computational Formulas for Binary Moments;141
1.8.2;5.2 Estimation for Single Random Effect Based Parametric Mixed Models;142
1.8.2.1;5.2.1 Method of Moments (MM);142
1.8.2.2;5.2.2 An Improved Method of Moments (IMM);144
1.8.2.2.1;5.2.2.1 Can There Be an Optimal B Free from Third and Fourth-Order Moments Under Simple Binary Logistic Mixed Models?;145
1.8.2.2.2;5.2.2.2 Effect of Mis-specification For Optimal Choice;148
1.8.2.3;5.2.3 Generalized Quasi-Likelihood (GQL) Approach;149
1.8.2.3.1;5.2.3.1 Marginal Generalized Quasi-Likelihood Estimation of ß;149
1.8.2.3.2;5.2.3.2 Marginal Generalized Quasi-Likelihood Estimation of s.;150
1.8.2.3.3;5.2.3.3 Joint Generalized Quasi-Likelihood (GQL) Estimation for ß and s.;152
1.8.2.4;5.2.4 Maximum Likelihood (ML) Estimation;153
1.8.2.5;5.2.5 Asymptotic Efficiency Comparison;156
1.8.2.5.1;5.2.5.1 Asymptotic variance of the IMM Estimator;156
1.8.2.5.2;5.2.5.2 Asymptotic Variance of the GQL Estimator;157
1.8.2.5.3;5.2.5.3 Asymptotic Variance of the ML Estimator;158
1.8.2.5.4;5.2.5.4 Numerical Comparison;160
1.8.2.6;5.2.6 COPD Data Analysis: A Numerical Illustration;161
1.8.3;5.3 Binary Mixed Models with Multidimensional Random Effects;164
1.8.3.1;5.3.1 Models in Two-Way Factorial Design Setup and Basic Properties;164
1.8.3.1.1;5.3.1.1 Unconditional Mean;165
1.8.3.1.2;5.3.1.2 Unconditional Covariances and Correlations in a Two-Way Design Setup;166
1.8.3.2;5.3.2 Estimation of Parameters;167
1.8.3.2.1;5.3.2.1 Estimation of Regression Effects ß;167
1.8.3.2.2;5.3.2.2 Estimation of the Variance Component s². Due to Factor A;169
1.8.3.2.3;5.3.2.3 Estimation of the Variance Component s² a Due to Factor B;173
1.8.3.3;5.3.3 Salamander Mating Data Analysis;178
1.8.3.3.1;5.3.3.1 Data Description;178
1.8.3.3.2;5.3.3.2 Binary Mixed Model for Salamander Data;179
1.8.3.3.3;5.3.3.3 Model Parameters Estimation and Interpretation;180
1.8.4;5.4 Semiparametric Approach;182
1.8.4.1;5.4.1 GQL Estimation;182
1.8.4.2;5.4.2 A Marginal Quasi-Likelihood (MQL) Approach;184
1.8.4.3;5.4.3 Asymptotic Efficiency Comparison: An Empirical Study;185
1.8.5;5.5 Monte Carlo Based Likelihood Estimation;187
1.8.6;Exercises;187
1.8.7;References;190
1.8.8;Appendix;192
1.9;Chapter 6 Longitudinal Models for Count Data;198
1.9.1;6.1 Marginal Model;199
1.9.2;6.2 Marginal Model Based Estimation of Regression Effects;200
1.9.3;6.3 Correlation Models for Stationary Count Data;202
1.9.3.1;6.3.1 Poisson AR(1) Model;203
1.9.3.2;6.3.2 Poisson MA(1) Model;204
1.9.3.3;6.3.3 Poisson Equicorrelation Model;204
1.9.4;6.4 Inferences for Stationary Correlation Models;205
1.9.4.1;6.4.1 Likelihood Approach and Complexity;205
1.9.4.2;6.4.2 GQL Approach;206
1.9.4.2.1;6.4.2.1 Asymptotic Distribution of the GQL Estimator;207
1.9.4.2.2;6.4.2.2 ‘Working’ Independence Assumption Based GQL Estimation;208
1.9.4.2.3;6.4.2.3 Efficiency of the Independence Assumption Based Estimator;208
1.9.4.2.4;6.4.2.4 Performance of the GQL Estimation: A Simulation Example;210
1.9.4.3;6.4.3 GEE Approach and Limitations;213
1.9.4.3.1;6.4.3.1 Efficiency of the GEE Based Estimator Under Correlation Structure Mis-specification;213
1.9.5;6.5 Nonstationary Correlation Models;218
1.9.5.1;6.5.1 Nonstationary Correlation Models with the Same Specified Marginal Mean and Variance Functions;219
1.9.5.1.1;6.5.1.1 Nonstationary AR(1) Models;219
1.9.5.1.2;6.5.1.2 Nonstationary MA(1) Models;220
1.9.5.1.3;6.5.1.3 Nonstationary EQC Models;220
1.9.5.2;6.5.2 Estimation of Parameters;222
1.9.5.2.1;6.5.2.1 Estimation of r Parameter Under AR(1) Model;222
1.9.5.2.2;6.5.2.2 Estimation of r Parameter Under MA(1) Correlation Model;223
1.9.5.2.3;6.5.2.3 Estimation of . Parameter Under Exchangeable (EQC) Correlation Model;223
1.9.5.3;6.5.3 Model Selection;224
1.9.6;6.6 More Nonstationary Correlation Models;226
1.9.6.1;6.6.1 Models with Variable Marginal Means and Variances;226
1.9.6.1.1;6.6.1.1 Nonstationary MA(1) Models;226
1.9.6.2;6.6.2 Estimation of Parameters;228
1.9.6.2.1;6.6.2.1 GQL Estimation for Regression Effects ß;228
1.9.6.2.2;6.6.2.2 Moment Estimation for the Correlation Parameter .;229
1.9.6.3;6.6.3 Model Selection;230
1.9.6.4;6.6.4 Estimation and Model Selection: A Simulation Example;232
1.9.6.4.1;6.6.4.1 Simulated Estimates Under the True and Misspecified Models;232
1.9.6.4.2;6.6.4.2 Model Selection;233
1.9.7;6.7 A Data Example: Analyzing Health Care Utilization Count Data;234
1.9.8;6.8 Models for Count Data from Longitudinal Adaptive Clinical Trials;236
1.9.8.1;6.8.1 Adaptive Longitudinal Designs;237
1.9.8.1.1;6.8.1.1 Simple Longitudinal Play-the-Winner (SLPW) Rule to Formulate wi;239
1.9.8.1.2;6.8.1.2 Bivariate Random Walk (BRW) Design;240
1.9.8.2;6.8.2 Performance of the SLPW and BRW Designs For Treatment Selection: A Simulation Study;241
1.9.8.3;6.8.3 Weighted GQL Estimation for Treatment Effects and Other Regression Parameters;244
1.9.8.3.1;6.8.3.1 Formulas for µi(wi0), and Si* (wi0,.) :;244
1.9.8.3.2;6.8.3.2 Weighted GQL Estimation of ß;246
1.9.9;Exercises;248
1.9.10;References;251
1.9.11;Appendix;253
1.10;Chapter 7 Longitudinal Models for Binary Data;258
1.10.1;7.1 Marginal Model;260
1.10.1.1;7.1.1 Marginal Model Based Estimation for Regression Effects;261
1.10.2;7.2 Some Selected Correlation Models for Longitudinal Binary Data;262
1.10.2.1;7.2.1 Bahadur Multivariate Binary Density (MBD) Based Model;263
1.10.2.1.1;7.2.1.1 Stationary Case;263
1.10.2.1.2;7.2.1.2 Nonstationary Case;265
1.10.2.2;7.2.2 Kanter Observation-Driven Dynamic (ODD) Model;266
1.10.2.2.1;7.2.2.1 Stationary Case;266
1.10.2.2.2;7.2.2.2 Non-stationary Case;268
1.10.2.3;7.2.3 A Linear Dynamic Conditional Probability (LDCP) Model;269
1.10.2.3.1;7.2.3.1 Stationary Case;269
1.10.2.3.2;7.2.3.2 Nonstationary Case;271
1.10.2.4;7.2.4 A Numerical Comparison of Range Restrictions for Correlation Index Parameter Under Stationary Binary Models;271
1.10.3;7.3 Low-Order Autocorrelation Models for Stationary Binary Data;273
1.10.3.1;7.3.1 Binary AR(1) Model;273
1.10.3.2;7.3.2 Binary MA(1) Model;273
1.10.3.3;7.3.3 Binary Equicorrelation (EQC) Model;276
1.10.3.4;7.3.4 Complexity in Likelihood Inferences Under Stationary Binary Correlation Models;277
1.10.3.5;7.3.5 GQL Estimation Approach;278
1.10.3.5.1;7.3.5.1 Efficiency of the Independence Assumption Based Estimation;279
1.10.3.6;7.3.6 GEE Approach and Its Limitations for Binary Data;281
1.10.4;7.4 Inferences in Nonstationary Correlation Models for Repeated Binary Data;283
1.10.4.1;7.4.1 Nonstationary AR(1) Correlation Model;283
1.10.4.2;7.4.2 Nonstationary MA(1) Correlation Model;285
1.10.4.3;7.4.3 Nonstationary EQC Model;286
1.10.4.4;7.4.4 Nonstationary Correlations Based GQL Estimation;287
1.10.4.4.1;7.4.4.1 Estimation of . Parameter Under Binary AR(1) Model;289
1.10.4.4.2;7.4.4.2 Estimation of . Parameter Under Binary MA(1) Correlation Model;289
1.10.4.4.3;7.4.4.3 Estimation of . Parameter Under Exchangeable (EQC) Correlation Model;290
1.10.4.5;7.4.5 Model Selection;290
1.10.5;7.5 SLID Data Example;291
1.10.5.1;7.5.1 Introduction to the SLID Data;291
1.10.5.2;7.5.2 Analysis of the SLID Data;293
1.10.6;7.6 Application to an Adaptive Clinical Trial Setup;295
1.10.6.1;7.6.1 Binary Response Based Adaptive Longitudinal Design;295
1.10.6.1.1;7.6.1.1 Simple Longitudinal Play-the-Winner (SLPW) Rule to Formulate wi;297
1.10.6.1.2;7.6.1.2 Performance of the Adaptive Design;299
1.10.6.2;7.6.2 Construction of the Adaptive Design Weights Based Weighted GQL Estimation;302
1.10.6.2.1;7.6.2.1 Computation of Unconditional Expectation of di : wi0;302
1.10.6.2.2;7.6.2.2 WGQL Estimating Equations for Regression Parameters Including the Treatment Effects;303
1.10.6.2.2.1;7.6.2.2.1 Moment Estimates for Longitudinal Correlations;306
1.10.6.2.2.2;7.6.2.2.2 Asymptotic Variances of the WGQL Regression Estimates;307
1.10.7;7.7 More Nonstationary Binary Correlation Models;307
1.10.7.1;7.7.1 Linear Binary Dynamic Regression (LBDR) Model;307
1.10.7.1.1;7.7.1.1 Autocorrelation Structure;308
1.10.7.1.2;7.7.1.2 GQL and Conditional GQL (CGQL) Approaches for Parameter Estimation;309
1.10.7.2;7.7.2 A Binary Dynamic Logit (BDL) Model;312
1.10.7.2.1;7.7.2.1 Basic Properties of the Lag 1 Dependence Model (7.142);312
1.10.7.2.2;7.7.2.2 Estimation of the Parameters of the BDL Model;314
1.10.7.2.2.1;7.7.2.2.1 GQL Estimation;315
1.10.7.2.2.2;7.7.2.2.2 OGQL Estimation;316
1.10.7.2.2.3;7.7.2.2.3 Likelihood Estimation;321
1.10.7.2.3;7.7.2.3 Fitting Asthma Data to the BDL Model: An Illustration;322
1.10.7.3;7.7.3 Application of the Binary Dynamic Logit (BDL) Model in an Adaptive Clinical Trial Setup;324
1.10.7.3.1;7.7.3.1 Random Treatments Based BDL Model;324
1.10.7.3.1.1;7.7.3.1.1 Unconditional Moments Up to Order Four;325
1.10.7.3.1.2;7.7.3.1.2 Extended WGQL (EWGQL) or Weighted OGQL (WOGQL) Estimating Equation;328
1.10.8;Exercises;331
1.10.9;References;333
1.10.10;Appendix;335
1.11;Chapter 8 Longitudinal Mixed Models for Count Data;338
1.11.1;8.1 A Conditional Serially Correlated Model;338
1.11.1.1;8.1.1 Unconditional Mean, Variance, and Correlations Under Serially Correlated Model;340
1.11.2;8.2 Parameter Estimation;340
1.11.2.1;8.2.1 Estimation of the Regression Effects ß;341
1.11.2.1.1;8.2.1.1 GMM/IMM Approach;341
1.11.2.1.2;8.2.1.2 GQL Approach;342
1.11.2.1.3;8.2.1.3 Conditional Maximum Likelihood (CML) Approach;343
1.11.2.1.4;8.2.1.4 Instrumental Variables Based GMM (IVBGMM) Estimation Approach;344
1.11.2.1.5;8.2.1.5 A Simulation Study;346
1.11.2.2;8.2.2 Estimation of the Random Effects Variance s². :;349
1.11.2.2.1;8.2.2.1 GMM Estimation for s².;349
1.11.2.2.2;8.2.2.2 GQL Estimation for s². :;351
1.11.2.2.3;8.2.2.3 Asymptotic Efficiency Comparison : GMM versus GQL;352
1.11.2.2.3.1;8.2.2.3.1 Asymptotic Variances of the GMM Estimators;352
1.11.2.2.3.2;8.2.2.3.2 Asymptotic Variances of the GQL Estimators;352
1.11.2.2.3.3;8.2.2.3.3 Asymptotic Efficiency Computation;353
1.11.2.3;8.2.3 Estimation of the Longitudinal Correlation Parameter .;354
1.11.2.3.1;8.2.3.1 GMM Estimation for .;354
1.11.2.3.2;8.2.3.2 . Estimation Under the GQL Approach;355
1.11.2.4;8.2.4 A Simulation Study;356
1.11.2.4.1;8.2.4.1 Estimation Under the ‘Working’ Conditional Independence (. = 0) Model;360
1.11.2.4.2;8.2.4.2 Estimation Under the ‘Working’ Longitudinal Fixed (s². = 0) Model;362
1.11.2.5;8.2.5 An Illustration: Analyzing Health Care Utilization Count Data by Using Longitudinal Fixed and Mixed Models;363
1.11.3;8.3 A Mean Deflated Conditional Serially Correlated Model;365
1.11.3.1;8.3.1 First and Second-Order Raw Response Based GQL Estimation;366
1.11.3.1.1;8.3.1.1 GQL(I) Approach for s². Estimation;366
1.11.3.1.2;8.3.1.2 GQL(N) Approach for s². Estimation;366
1.11.3.2;8.3.2 Corrected Response (CR) Based GQL Estimation;368
1.11.3.2.1;8.3.2.1 GQL(CR-I) Estimation for s².;368
1.11.3.2.2;8.3.2.2 GQL(CR-N) Estimation s².;370
1.11.3.3;8.3.3 Relative Performances of GQL(I) and GQL(N) Estimation Approaches: A Simulation Study;371
1.11.3.3.1;8.3.3.1 Performance for Overdispersion Estimation;371
1.11.3.3.2;8.3.3.2 Performance for Regression Effects Estimation;372
1.11.3.3.3;8.3.3.3 Performance for Correlation Index Estimation;374
1.11.3.4;8.3.4 A Further Application: Analyzing Patent Count Data;374
1.11.4;8.4 Longitudinal Negative Binomial Fixed Model and Estimationof Parameters;379
1.11.4.1;8.4.1 Inferences in Stationary Negative Binomial CorrelationModels;380
1.11.4.1.1;8.4.1.1 Estimation of Parameters;381
1.11.4.1.1.1;8.4.1.1.1 GQL Estimation for ß;381
1.11.4.1.1.2;8.4.1.1.2 Estimation of c*;382
1.11.4.1.1.3;8.4.1.1.3 Moment Estimation of .;384
1.11.4.2;8.4.2 A Data Example: Analyzing Epileptic Count Data by Using Poisson and Negative Binomial Longitudinal Models;384
1.11.4.3;8.4.3 Nonstationary Negative Binomial Correlation Models and Estimation of Parameters;386
1.11.4.3.1;8.4.3.1 First Two Moments Based Negative Binomial Autoregression Model;386
1.11.4.3.1.1;8.4.3.1.1 Nonstationary Mean Variance Structure;387
1.11.4.3.1.2;8.4.3.1.2 Non-stationary Correlation Structure;388
1.11.4.3.2;8.4.3.2 A Proposed Conditional GQL (CGQL) Estimation Approach;388
1.11.4.3.2.1;8.4.3.2.1 CGQL Estimation for ß;389
1.11.4.3.2.2;8.4.3.2.2 CGQL Estimation for c*;390
1.11.4.3.2.3;8.4.3.2.3 MMs Equation for .;392
1.11.5;Exercises;392
1.11.6;References;394
1.11.7;Appendix;396
1.12;Chapter 9 Longitudinal Mixed Models for Binary Data;405
1.12.1;9.1 A Conditional Serially Correlated Model;406
1.12.1.1;9.1.1 Basic Properties of the Model;406
1.12.1.2;9.1.2 Parameter Estimation;408
1.12.1.2.1;9.1.2.1 GQL Estimation of the Regression Effects ß;408
1.12.1.2.2;9.1.2.2 GQL Estimation of the Random Effects Variance s².;409
1.12.1.2.2.1;9.1.2.2.1 GQL(I) Estimation of s².;410
1.12.1.2.2.2;9.1.2.2.2 GQL(N) Estimation of s².;410
1.12.1.2.3;9.1.2.3 Estimation of . Under the GQL Approach;411
1.12.2;9.2 Binary Dynamic Mixed Logit (BDML) Model;412
1.12.2.1;9.2.1 GMM/IMM Estimation;414
1.12.2.1.1;9.2.1.1 Construction of the Unbiased Moment Functions;414
1.12.2.1.1.1;9.2.1.1.1 Formula for pit;415
1.12.2.1.1.2;9.2.1.1.2 Formula for .iut;415
1.12.2.1.2;9.2.1.2 GMM Estimating Equation for a = (ß', ., s². )';416
1.12.2.1.2.1;9.2.1.2.1 Computation of the C Matrix;417
1.12.2.1.2.2;9.2.1.2.2 Computation of ..'/.a;419
1.12.2.2;9.2.2 GQL Estimation;419
1.12.2.2.1;9.2.2.1 Computation of Oi;420
1.12.2.3;9.2.3 Efficiency Comparison: GMM Versus GQL;421
1.12.2.3.1;9.2.3.1 Asymptotic Distribution of the GMM Estimator;421
1.12.2.3.2;9.2.3.2 Asymptotic Distribution of the GQL Estimator;422
1.12.2.3.3;9.2.3.3 Asymptotic Efficiency Comparison;422
1.12.2.3.4;9.2.3.4 Small Sample Efficiency Comparison: A Simulation Study;424
1.12.2.4;9.2.4 Fitting the Binary Dynamic Mixed Logit Model to the SLID data;425
1.12.2.5;9.2.5 GQL Versus Maximum Likelihood (ML) Estimation for BDML Model;427
1.12.2.5.1;9.2.5.1 ML Estimation;428
1.12.2.5.2;9.2.5.2 Relative Performances of the GQL and ML Approaches for BDML model: A Simulation Study;429
1.12.3;9.3 A Binary Dynamic Mixed Probit (BDMP) Model;431
1.12.3.1;9.3.1 GQL Estimation for BDMP Model;432
1.12.3.2;9.3.2 GQL Estimation Performance for BDMP Model: A Simulation Study;433
1.12.3.2.1;9.3.2.1 Random Effects Mis-specification: True t Versus Working Normal Distributions For Random Effects;434
1.12.4;Exercises;436
1.12.5;References;437
1.13;Chapter 10 Familial Longitudinal Models for Count Data;439
1.13.1;10.1 An Autocorrelation Class of Familial Longitudinal Models;439
1.13.1.1;10.1.1 Marginal Mean and Variance;440
1.13.1.1.1;10.1.1.1 Conditional Marginal Mean and Variance;440
1.13.1.1.2;10.1.1.1 Unconditional Marginal Mean and Variance;440
1.13.1.2;10.1.2 Nonstationary Autocorrelation Models;441
1.13.1.2.1;10.1.2.1 Conditional AR(1) Model;441
1.13.1.2.1.1;10.1.2.1.1. Unconditional Mean, Variance, and Correlation Structure;442
1.13.1.2.2;10.1.2.2 Conditional MA(1) Model;442
1.13.1.2.2.1;10.1.2.2.1. Unconditional Mean, Variance, and Correlation Structure;443
1.13.1.2.3;10.1.2.3 An Alternative Conditional MA(1) Model;443
1.13.1.2.3.1;10.1.2.3.1 Unconditional First and Second-Order Moments;444
1.13.1.2.4;10.1.2.4 Conditional EQC Model;444
1.13.1.2.4.1;10.1.2.4.1. Unconditional Mean, Variance, and Correlation Structure;445
1.13.2;10.2 Parameter Estimation;445
1.13.2.1;10.2.1 Estimation of Parameters Under Conditional AR(1) Model;446
1.13.2.1.1;10.2.1.1 GQL Estimation of Regression Parameter ß;446
1.13.2.1.2;10.2.1.2 GQL Estimation of Familial Correlation Index Parameter s².;447
1.13.2.1.2.1;10.2.1.2.1 GQL(I) Estimation of s².;449
1.13.2.1.2.2;10.2.1.2.2 GQL(N) Estimation of s².;451
1.13.2.1.3;10.2.1.3 Estimation of Longitudinal Correlation Index Parameter .;454
1.13.2.2;10.2.2 Performance of the GQL Approach: A Simulation Study;455
1.13.2.2.1;10.2.2.1 Simulation Study with p = 1 Covariate;455
1.13.2.2.2;10.2.2.2 Simulation Study with p = 2 Covariates;457
1.13.2.2.3;10.2.2.3 Effects of Partial Model Fitting: A Further Simulation Study with p = 2 Covariates;459
1.13.3;10.3 Analyzing Health Care Utilization Data by Using GLLMM;462
1.13.4;10.4 Some Remarks on Model Identification;465
1.13.4.1;10.4.1 An Exploratory Identification;466
1.13.4.2;10.4.2 A Further Improved Identification;467
1.13.5;Exercises;467
1.13.6;References;469
1.14;Chapter 11 Familial Longitudinal Models for Binary Data;471
1.14.1;11.1 LDCCP Models;472
1.14.1.1;11.1.1 Conditional-Conditional (CC) AR(1) Model;472
1.14.1.1.1;11.1.1.1 Conditional Mean, Variance, and Correlation Structure;472
1.14.1.1.2;11.1.1.2 Unconditional Mean, Variance, and Correlation Structure;473
1.14.1.2;11.1.2 CC MA(1) Model;474
1.14.1.3;11.1.3 CC EQC Model;475
1.14.1.4;11.1.4 Estimation of the AR(1) Model Parameters;476
1.14.1.4.1;11.1.4.1 GQL Estimation of Regression Parameter ß;476
1.14.1.4.2;11.1.4.2 GQL Estimation of Familial Correlation Index Parameter s².;478
1.14.1.4.3;11.1.4.3 Moment Estimation of Longitudinal Correlation Index Parameter .;483
1.14.2;11.2 Application toWaterloo Smoking Prevention Data;484
1.14.3;11.3 Family Based BDML Models for Binary Data;487
1.14.3.1;11.3.1 FBDML Model and Basic Properties;488
1.14.3.1.1;11.3.1.1 Conditional Mean, Variance, and Correlation Structures;488
1.14.3.1.2;11.3.1.2 Unconditional Mean, Variance, and Correlation Structures;489
1.14.3.2;11.3.2 Quasi-Likelihood Estimation in the Familial Longitudinal Setup;490
1.14.3.2.1;11.3.2.1 Joint GQL Estimation of Parameters;490
1.14.3.2.2;11.3.2.2 Asymptotic Covariance Matrix of the Joint GQL Estimator;494
1.14.3.3;11.3.3 Likelihood Based Estimation;495
1.14.3.3.1;11.3.3.1 Likelihood Function for the FBDML Model;495
1.14.3.3.2;11.3.3.2 Likelihood Estimating Equations;495
1.14.3.3.3;11.3.3.3 Asymptotic Covariance of the Joint ML Estimator;497
1.14.4;Exercises;499
1.14.5;References;503
1.15;Index;505