Dynamic Mixed Models for Familial Longitudinal Data

Dynamic Mixed Models for Familial Longitudinal Data

Preface

Acknowledgements

Contents

Chapter 1 Introduction

1.1 Background of Familial Models

1.2 Background of Longitudinal Models

References

Chapter 2 Overview of Linear Fixed Models for Longitudinal Data

2.1 Estimation of ß

2.1.1 Method of Moments (MM)

2.1.2 Ordinary Least Squares (OLS) Method

2.1.2.1 Generalized Least Squares (GLS) Method

2.1.3 OLS Versus GLS Estimation Performance

2.2 Estimation of ß Under Stationary General Autocorrelation Structure

2.2.1 A Class of Autocorrelations

2.2.2 Estimation of ß

2.3 A Rat Data Example

2.4 Alternative Modelling for Time Effects

Exercises

References

Appendix

Chapter 3 Overview of Linear Mixed Models for Longitudinal Data

3.1 Linear Longitudinal Mixed Model

3.1.1 GLS Estimation of ß

3.1.2 Moment Estimating Equations for s². and .l

3.1.3 Linear Mixed Models for Rat Data

3.2 Linear Dynamic Mixed Models for Balanced Longitudinal Data

3.2.1 Basic Properties of the Dynamic Dependence Mixed Model (3.21)

3.2.2 Estimation of the Parameters of the Dynamic Mixed Model (3.21)

3.3 Further Estimation for the Parameters of the Dynamic Mixed Model

3.3.1 GMM/IMM Estimation Approach

3.3.2 GQL Estimation Approach

3.3.3 Asymptotic Efficiency Comparison

Exercises

References

Chapter 4 Familial Models for Count Data

4.1 Poisson Mixed Models and Basic Properties

4.2 Estimation for Single Random Effect Based Parametric Mixed Models

4.2.1 Exact Likelihood Estimation and Drawbacks

4.2.2 Penalized Quasi-Likelihood Approach

4.2.3 Small Variance Asymptotic Approach: A Likelihood Approximation (LA)

4.2.3.1 A Higher-Order Likelihood Approximation (HOLA)

4.2.4 Hierarchical Likelihood (HL) Approach

4.2.5 Method of Moments (MM)

4.2.6 Generalized Quasi-Likelihood (GQL) Approach

4.2.6.1 Marginal Generalized Quasi-Likelihood (GQL) Estimation of ß

4.2.6.2 Marginal Generalized Quasi-Likelihood (GQL) Estimation of s².

4.2.6.3 Joint Generalized Quasi-Likelihood (GQL) Estimation for ß and s².

4.2.7 Efficiency Comparison

4.2.7.1 Efficiency Comparison Between GQL and MM Approaches: A Small Sample Study

4.2.7.2 Efficiency Comparison Between GQL and HL Approaches: A Small Sample Study

4.2.8 A Health Care Data Utilization Example

4.3 Estimation for Multiple Random Effects Based Parametric Mixed Models

4.3.1 Random Effects in a Two-Way Factorial Design Setup

4.3.2 One-Way Heteroscedastic Random Effects

4.3.3 Multiple Independent Random Effects

4.3.3.1 Method of Moments Estimation for ß, s². , and s²t

4.3.3.2 Joint GQL Estimation for ß, s². , and s²t

4.3.3.3 Relative Performances of the GQL Versus MM Approaches: An Asymptotic Efficiency Comparison

4.3.3.4 GQL Versus MM Estimation: A Simulation Study Based on an Asthma Count Data Model with Two Components of Dispersion

4.3.3.5 An Asthma Count Data Model with Four Fixed Covariates and Two Components of Dispersion

4.4 Semiparametric Approach

4.4.1 Computations for µi, .i, Si, and Oi

4.4.2 Construction of the Estimating Equation for When ß When s². Is Known

4.5 Monte Carlo Based Likelihood Estimation

4.5.1 MCEM Approach

4.5.2 MCNR Approach

Exercises

References

Chapter 5 Familial Models for Binary Data

5.1 Binary Mixed Models and Basic Properties

5.1.1 Computational Formulas for Binary Moments

5.2 Estimation for Single Random Effect Based Parametric Mixed Models

5.2.1 Method of Moments (MM)

5.2.2 An Improved Method of Moments (IMM)

5.2.2.1 Can There Be an Optimal B Free from Third and Fourth-Order Moments Under Simple Binary Logistic Mixed Models?

5.2.2.2 Effect of Mis-specification For Optimal Choice

5.2.3 Generalized Quasi-Likelihood (GQL) Approach

5.2.3.1 Marginal Generalized Quasi-Likelihood Estimation of ß

5.2.3.2 Marginal Generalized Quasi-Likelihood Estimation of s.

5.2.3.3 Joint Generalized Quasi-Likelihood (GQL) Estimation for ß and s.

5.2.4 Maximum Likelihood (ML) Estimation

5.2.5 Asymptotic Efficiency Comparison

5.2.5.1 Asymptotic variance of the IMM Estimator

5.2.5.2 Asymptotic Variance of the GQL Estimator

5.2.5.3 Asymptotic Variance of the ML Estimator

5.2.5.4 Numerical Comparison

5.2.6 COPD Data Analysis: A Numerical Illustration

5.3 Binary Mixed Models with Multidimensional Random Effects

5.3.1 Models in Two-Way Factorial Design Setup and Basic Properties

5.3.1.1 Unconditional Mean

5.3.1.2 Unconditional Covariances and Correlations in a Two-Way Design Setup

5.3.2 Estimation of Parameters

5.3.2.1 Estimation of Regression Effects ß

5.3.2.2 Estimation of the Variance Component s². Due to Factor A

5.3.2.3 Estimation of the Variance Component s² a Due to Factor B

5.3.3 Salamander Mating Data Analysis

5.3.3.1 Data Description

5.3.3.2 Binary Mixed Model for Salamander Data

5.3.3.3 Model Parameters Estimation and Interpretation

5.4 Semiparametric Approach

5.4.1 GQL Estimation

5.4.2 A Marginal Quasi-Likelihood (MQL) Approach

5.4.3 Asymptotic Efficiency Comparison: An Empirical Study

5.5 Monte Carlo Based Likelihood Estimation

Exercises

References

Appendix

Chapter 6 Longitudinal Models for Count Data

6.1 Marginal Model

6.2 Marginal Model Based Estimation of Regression Effects

6.3 Correlation Models for Stationary Count Data

6.3.1 Poisson AR(1) Model

6.3.2 Poisson MA(1) Model

6.3.3 Poisson Equicorrelation Model

6.4 Inferences for Stationary Correlation Models

6.4.1 Likelihood Approach and Complexity

6.4.2 GQL Approach

6.4.2.1 Asymptotic Distribution of the GQL Estimator

6.4.2.2 ‘Working’ Independence Assumption Based GQL Estimation

6.4.2.3 Efficiency of the Independence Assumption Based Estimator

6.4.2.4 Performance of the GQL Estimation: A Simulation Example

6.4.3 GEE Approach and Limitations

6.4.3.1 Efficiency of the GEE Based Estimator Under Correlation Structure Mis-specification

6.5 Nonstationary Correlation Models

6.5.1 Nonstationary Correlation Models with the Same Specified Marginal Mean and Variance Functions

6.5.1.1 Nonstationary AR(1) Models

6.5.1.2 Nonstationary MA(1) Models

6.5.1.3 Nonstationary EQC Models

6.5.2 Estimation of Parameters

6.5.2.1 Estimation of r Parameter Under AR(1) Model

6.5.2.2 Estimation of r Parameter Under MA(1) Correlation Model

6.5.2.3 Estimation of . Parameter Under Exchangeable (EQC) Correlation Model

6.5.3 Model Selection

6.6 More Nonstationary Correlation Models

6.6.1 Models with Variable Marginal Means and Variances

6.6.1.1 Nonstationary MA(1) Models

6.6.2 Estimation of Parameters

6.6.2.1 GQL Estimation for Regression Effects ß

6.6.2.2 Moment Estimation for the Correlation Parameter .

6.6.3 Model Selection

6.6.4 Estimation and Model Selection: A Simulation Example

6.6.4.1 Simulated Estimates Under the True and Misspecified Models

6.6.4.2 Model Selection

6.7 A Data Example: Analyzing Health Care Utilization Count Data

6.8 Models for Count Data from Longitudinal Adaptive Clinical Trials

6.8.1 Adaptive Longitudinal Designs

6.8.1.1 Simple Longitudinal Play-the-Winner (SLPW) Rule to Formulate wi

6.8.1.2 Bivariate Random Walk (BRW) Design

6.8.2 Performance of the SLPW and BRW Designs For Treatment Selection: A Simulation Study

6.8.3 Weighted GQL Estimation for Treatment Effects and Other Regression Parameters

6.8.3.1 Formulas for µi(wi0), and Si* (wi0,.) :

6.8.3.2 Weighted GQL Estimation of ß

Exercises

References

Appendix

Chapter 7 Longitudinal Models for Binary Data

7.1 Marginal Model

7.1.1 Marginal Model Based Estimation for Regression Effects

7.2 Some Selected Correlation Models for Longitudinal Binary Data

7.2.1 Bahadur Multivariate Binary Density (MBD) Based Model

7.2.1.1 Stationary Case

7.2.1.2 Nonstationary Case

7.2.2 Kanter Observation-Driven Dynamic (ODD) Model

7.2.2.1 Stationary Case

7.2.2.2 Non-stationary Case

7.2.3 A Linear Dynamic Conditional Probability (LDCP) Model

7.2.3.1 Stationary Case

7.2.3.2 Nonstationary Case

7.2.4 A Numerical Comparison of Range Restrictions for Correlation Index Parameter Under Stationary Binary Models

7.3 Low-Order Autocorrelation Models for Stationary Binary Data

7.3.1 Binary AR(1) Model

7.3.2 Binary MA(1) Model

7.3.3 Binary Equicorrelation (EQC) Model

7.3.4 Complexity in Likelihood Inferences Under Stationary Binary Correlation Models

7.3.5 GQL Estimation Approach

7.3.5.1 Efficiency of the Independence Assumption Based Estimation

7.3.6 GEE Approach and Its Limitations for Binary Data

7.4 Inferences in Nonstationary Correlation Models for Repeated Binary Data

7.4.1 Nonstationary AR(1) Correlation Model

7.4.2 Nonstationary MA(1) Correlation Model

7.4.3 Nonstationary EQC Model

7.4.4 Nonstationary Correlations Based GQL Estimation

7.4.4.1 Estimation of . Parameter Under Binary AR(1) Model

7.4.4.2 Estimation of . Parameter Under Binary MA(1) Correlation Model

7.4.4.3 Estimation of . Parameter Under Exchangeable (EQC) Correlation Model

7.4.5 Model Selection

7.5 SLID Data Example

7.5.1 Introduction to the SLID Data

7.5.2 Analysis of the SLID Data

7.6 Application to an Adaptive Clinical Trial Setup

7.6.1 Binary Response Based Adaptive Longitudinal Design

7.6.1.1 Simple Longitudinal Play-the-Winner (SLPW) Rule to Formulate wi

7.6.1.2 Performance of the Adaptive Design

7.6.2 Construction of the Adaptive Design Weights Based Weighted GQL Estimation

7.6.2.1 Computation of Unconditional Expectation of di : wi0

7.6.2.2 WGQL Estimating Equations for Regression Parameters Including the Treatment Effects

7.6.2.2.1 Moment Estimates for Longitudinal Correlations

7.6.2.2.2 Asymptotic Variances of the WGQL Regression Estimates

7.7 More Nonstationary Binary Correlation Models

7.7.1 Linear Binary Dynamic Regression (LBDR) Model

7.7.1.1 Autocorrelation Structure

7.7.1.2 GQL and Conditional GQL (CGQL) Approaches for Parameter Estimation

7.7.2 A Binary Dynamic Logit (BDL) Model

7.7.2.1 Basic Properties of the Lag 1 Dependence Model (7.142)

7.7.2.2 Estimation of the Parameters of the BDL Model

7.7.2.2.1 GQL Estimation

7.7.2.2.2 OGQL Estimation

7.7.2.2.3 Likelihood Estimation

7.7.2.3 Fitting Asthma Data to the BDL Model: An Illustration

7.7.3 Application of the Binary Dynamic Logit (BDL) Model in an Adaptive Clinical Trial Setup

7.7.3.1 Random Treatments Based BDL Model

7.7.3.1.1 Unconditional Moments Up to Order Four

7.7.3.1.2 Extended WGQL (EWGQL) or Weighted OGQL (WOGQL) Estimating Equation

Exercises

References

Appendix

Chapter 8 Longitudinal Mixed Models for Count Data

8.1 A Conditional Serially Correlated Model

8.1.1 Unconditional Mean, Variance, and Correlations Under Serially Correlated Model

8.2 Parameter Estimation

8.2.1 Estimation of the Regression Effects ß

8.2.1.1 GMM/IMM Approach

8.2.1.2 GQL Approach

8.2.1.3 Conditional Maximum Likelihood (CML) Approach

8.2.1.4 Instrumental Variables Based GMM (IVBGMM) Estimation Approach

8.2.1.5 A Simulation Study

8.2.2 Estimation of the Random Effects Variance s². :

8.2.2.1 GMM Estimation for s².

8.2.2.2 GQL Estimation for s². :

8.2.2.3 Asymptotic Efficiency Comparison : GMM versus GQL

8.2.2.3.1 Asymptotic Variances of the GMM Estimators

8.2.2.3.2 Asymptotic Variances of the GQL Estimators

8.2.2.3.3 Asymptotic Efficiency Computation

8.2.3 Estimation of the Longitudinal Correlation Parameter .

8.2.3.1 GMM Estimation for .

8.2.3.2 . Estimation Under the GQL Approach

8.2.4 A Simulation Study

8.2.4.1 Estimation Under the ‘Working’ Conditional Independence (. = 0) Model

8.2.4.2 Estimation Under the ‘Working’ Longitudinal Fixed (s². = 0) Model

8.2.5 An Illustration: Analyzing Health Care Utilization Count Data by Using Longitudinal Fixed and Mixed Models

8.3 A Mean Deflated Conditional Serially Correlated Model

8.3.1 First and Second-Order Raw Response Based GQL Estimation

8.3.1.1 GQL(I) Approach for s². Estimation

8.3.1.2 GQL(N) Approach for s². Estimation

8.3.2 Corrected Response (CR) Based GQL Estimation

8.3.2.1 GQL(CR-I) Estimation for s².

8.3.2.2 GQL(CR-N) Estimation s².

8.3.3 Relative Performances of GQL(I) and GQL(N) Estimation Approaches: A Simulation Study

8.3.3.1 Performance for Overdispersion Estimation

8.3.3.2 Performance for Regression Effects Estimation

8.3.3.3 Performance for Correlation Index Estimation

8.3.4 A Further Application: Analyzing Patent Count Data

8.4 Longitudinal Negative Binomial Fixed Model and Estimationof Parameters

8.4.1 Inferences in Stationary Negative Binomial CorrelationModels

8.4.1.1 Estimation of Parameters

8.4.1.1.1 GQL Estimation for ß

8.4.1.1.2 Estimation of c*

8.4.1.1.3 Moment Estimation of .

8.4.2 A Data Example: Analyzing Epileptic Count Data by Using Poisson and Negative Binomial Longitudinal Models

8.4.3 Nonstationary Negative Binomial Correlation Models and Estimation of Parameters

8.4.3.1 First Two Moments Based Negative Binomial Autoregression Model

8.4.3.1.1 Nonstationary Mean Variance Structure

8.4.3.1.2 Non-stationary Correlation Structure

8.4.3.2 A Proposed Conditional GQL (CGQL) Estimation Approach

8.4.3.2.1 CGQL Estimation for ß

8.4.3.2.2 CGQL Estimation for c*

8.4.3.2.3 MMs Equation for .

Exercises

References

Appendix

Chapter 9 Longitudinal Mixed Models for Binary Data

9.1 A Conditional Serially Correlated Model

9.1.1 Basic Properties of the Model

9.1.2 Parameter Estimation

9.1.2.1 GQL Estimation of the Regression Effects ß

9.1.2.2 GQL Estimation of the Random Effects Variance s².

9.1.2.2.1 GQL(I) Estimation of s².

9.1.2.2.2 GQL(N) Estimation of s².

9.1.2.3 Estimation of . Under the GQL Approach

9.2 Binary Dynamic Mixed Logit (BDML) Model

9.2.1 GMM/IMM Estimation

9.2.1.1 Construction of the Unbiased Moment Functions

9.2.1.1.1 Formula for pit

9.2.1.1.2 Formula for .iut

9.2.1.2 GMM Estimating Equation for a = (ß', ., s². )'

9.2.1.2.1 Computation of the C Matrix

9.2.1.2.2 Computation of ..'/.a

9.2.2 GQL Estimation

9.2.2.1 Computation of Oi

9.2.3 Efficiency Comparison: GMM Versus GQL

9.2.3.1 Asymptotic Distribution of the GMM Estimator

9.2.3.2 Asymptotic Distribution of the GQL Estimator

9.2.3.3 Asymptotic Efficiency Comparison

9.2.3.4 Small Sample Efficiency Comparison: A Simulation Study

9.2.4 Fitting the Binary Dynamic Mixed Logit Model to the SLID data

9.2.5 GQL Versus Maximum Likelihood (ML) Estimation for BDML Model

9.2.5.1 ML Estimation

9.2.5.2 Relative Performances of the GQL and ML Approaches for BDML model: A Simulation Study

9.3 A Binary Dynamic Mixed Probit (BDMP) Model

9.3.1 GQL Estimation for BDMP Model

9.3.2 GQL Estimation Performance for BDMP Model: A Simulation Study

9.3.2.1 Random Effects Mis-specification: True t Versus Working Normal Distributions For Random Effects

Exercises

References

Chapter 10 Familial Longitudinal Models for Count Data

10.1 An Autocorrelation Class of Familial Longitudinal Models

10.1.1 Marginal Mean and Variance

10.1.1.1 Conditional Marginal Mean and Variance

10.1.1.1 Unconditional Marginal Mean and Variance

10.1.2 Nonstationary Autocorrelation Models

10.1.2.1 Conditional AR(1) Model

10.1.2.1.1. Unconditional Mean, Variance, and Correlation Structure

10.1.2.2 Conditional MA(1) Model

10.1.2.2.1. Unconditional Mean, Variance, and Correlation Structure

10.1.2.3 An Alternative Conditional MA(1) Model

10.1.2.3.1 Unconditional First and Second-Order Moments

10.1.2.4 Conditional EQC Model

10.1.2.4.1. Unconditional Mean, Variance, and Correlation Structure

10.2 Parameter Estimation

10.2.1 Estimation of Parameters Under Conditional AR(1) Model

10.2.1.1 GQL Estimation of Regression Parameter ß

10.2.1.2 GQL Estimation of Familial Correlation Index Parameter s².

10.2.1.2.1 GQL(I) Estimation of s².

10.2.1.2.2 GQL(N) Estimation of s².

10.2.1.3 Estimation of Longitudinal Correlation Index Parameter .

10.2.2 Performance of the GQL Approach: A Simulation Study

10.2.2.1 Simulation Study with p = 1 Covariate

10.2.2.2 Simulation Study with p = 2 Covariates

10.2.2.3 Effects of Partial Model Fitting: A Further Simulation Study with p = 2 Covariates

10.3 Analyzing Health Care Utilization Data by Using GLLMM

10.4 Some Remarks on Model Identification

10.4.1 An Exploratory Identification

10.4.2 A Further Improved Identification

Exercises

References

Chapter 11 Familial Longitudinal Models for Binary Data

11.1 LDCCP Models

11.1.1 Conditional-Conditional (CC) AR(1) Model

11.1.1.1 Conditional Mean, Variance, and Correlation Structure

11.1.1.2 Unconditional Mean, Variance, and Correlation Structure

11.1.2 CC MA(1) Model

11.1.3 CC EQC Model

11.1.4 Estimation of the AR(1) Model Parameters

11.1.4.1 GQL Estimation of Regression Parameter ß

11.1.4.2 GQL Estimation of Familial Correlation Index Parameter s².

11.1.4.3 Moment Estimation of Longitudinal Correlation Index Parameter .

11.2 Application toWaterloo Smoking Prevention Data

11.3 Family Based BDML Models for Binary Data

11.3.1 FBDML Model and Basic Properties

11.3.1.1 Conditional Mean, Variance, and Correlation Structures

11.3.1.2 Unconditional Mean, Variance, and Correlation Structures

11.3.2 Quasi-Likelihood Estimation in the Familial Longitudinal Setup

11.3.2.1 Joint GQL Estimation of Parameters

11.3.2.2 Asymptotic Covariance Matrix of the Joint GQL Estimator

11.3.3 Likelihood Based Estimation

11.3.3.1 Likelihood Function for the FBDML Model

11.3.3.2 Likelihood Estimating Equations

11.3.3.3 Asymptotic Covariance of the Joint ML Estimator

Exercises

References

Index