Random Effect and Latent Variable Model Selection (eBook)

Preface Random Effect and Latent Variable Model Selection 5
Contents 8
Part I: Random Effects Models 9
Chapter 1 10
Likelihood Ratio Testing for Zero Variance Components in Linear Mixed Models 10
1 Examples 11
1.1 Loa loa Prevalence in West Africa 11
1.2 Onion Density in Australia 12
1.3 Coronary Sinus Potassium 14
2 Model and Testing Framework 16
3 Standard Asymptotic Results for LMMs 17
4 Finite Sample and Asymptotic Results for General Design LMMs with One Variance Component 17
5 Linear Mixed Models with Multiple Variance Components 18
5.1 Fast Finite Sample Approximation 20
5.2 Mixture Approximation to the Bootstrap 20
6 Revisiting the Applications 21
7 Discussion 22
References 24
Chapter 2 25
Variance Component Testing in Generalized Linear Mixed Models for Longitudinal/Clustered Data and other Related Topics 25
1 Introduction 25
2 Generalized Linear Mixed Models for Longitudinal/Clustered Data 26
3 The Likelihood Ratio Test for Variance Components in GLMMs 27
4 The Score Test for Variance Components in GLMMs 31
5 Simulation Study to Compare the Likelihood Ratio Test and the Score Test for Variance Components 35
6 Polynomial Test in Semiparametric Additive Mixed Models 36
7 Application 39
8 Discussion 40
References 41
Chapter 3 43
Bayesian Model Uncertainty in Mixed EffectsModels 43
1 Introduction 43
1.1 Motivation 43
1.2 Frequentist Literature 44
1.3 Bayesian Approach 45
2 Bayesian Model Uncertainty 46
2.1 Subset Selection in Linear Regression 46
2.2 Bayes Factors and Default Priors 48
3 Bayesian Subset Selection for Mixed Effects Models 49
3.1 Bayes Factor Approximations 49
3.2 Stochastic Search Variable Selection 50
4 Linear Mixed Models 51
4.1 Priors 51
4.2 Posterior Computation 53
5 Binary Logistic Mixed Models 55
5.1 Priors and Posterior Computation 57
5.2 Importance Weights 59
6 Simulation Examples 60
7 Epidemiology Application 63
8 OtherModels 64
8.1 Logistic Models for Ordinal Data 64
8.2 Probit Models 65
9 Discussion 65
References 66
Chapter 4 69
Bayesian Variable Selection in Generalized Linear Mixed Models 69
1 Introduction 69
1.1 Background and Motivation 69
1.2 Time to Pregnancy Application 71
1.3 Background on Model Selection in GLMMs 72
2 Bayesian Subset Selection in GLMMs 73
2.1 Generalized Linear Mixed Models 73
2.2 Description of Approach 75
2.3 Reparameterization and Mixture Prior Specification 76
2.4 An Approximation 78
3 Posterior Computation 80
3.1 General Strategies 80
3.2 Updating Parameters 81
3.3 Calculation of Quantities 83
4 Simulation Examples 84
4.1 Simulation Setup 84
4.2 Results 86
4.3 Assessment of Accuracy of the Approximation 90
5 Time-to-Pregnancy Application 91
5.1 Data and Model Selection Problem 91
5.2 Prior Specification, Implementation and Results 91
6 Discussion 94
References 95
Appendix 97
Part II: Factor Analysis and Structural Equations Models 98
Chapter 5 99
A Unified Approach to Two-Level Structural Equation Models and Linear Mixed Effects Models 99
1 Introduction 99
2 Model Formulation 101
3 The EM Algorithm 104
3.1 Maximum Likelihood Estimation 104
3.2 Asymptotic Properties 110
4 Examples 112
5 Goodness-of-Fit and Related Issues 118
Appendix. EQS Input Program for the Model in Example 2 119
References 122
Chapter 6 124
Bayesian Model Comparison of Structural Equation Models 124
1 Introduction 124
2 Bayes Factor and other Model Comparison Statistics 125
2.1 Bayes Factor 125
2.2 Other Alternatives 127
3 Computation of Bayes Factor through Path Sampling 129
4 Model Comparison of Nonlinear SEMs 130
4.1 Model Description 130
4.2 Model Comparison via Bayes Factor 131
4.3 A Simulation Study 133
5 Model Comparison of an Integrated SEM 136
5.1 The Integrated Model 138
5.2 Model Comparison 139
5.3 An Illustrative Example 141
6 Discussion 146
Appendix: Full Conditional Distributions 148
References 151
Chapter 7 154
Bayesian Model Selection in Factor Analytic Models 154
1 Introduction 154
2 Specification of the Model 156
3 Bayesian Uncertainty in the Number of Factors 158
4 Simulation Study 160
4.1 One-Factor Model 160
4.2 Three-Factor Model 161
5 Application to Rodent Organ Weight Data 161
6 Discussion 163
References 164
Appendix: Full Conditional Distributions for the Gibbs Sampler 165
Index 167