Linear and Generalized Linear Mixed Models and Their Applications

✍ Scribed by Jiming Jiang, Thuan Nguyen

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 352
• Series: Springer Series in Statistics
• Edition: 2
• Category: Library

✦ Table of Contents

Preface
Contents
List of Notations
1 Linear Mixed Models: Part I
1.1 Introduction
1.1.1 Effect of Air Pollution Episodes on Children
1.1.2 Genome-Wide Association Study
1.1.3 Small Area Estimation of Income
1.2 Types of Linear Mixed Models
1.2.1 Gaussian Mixed Models
1.2.1.1 Mixed ANOVA Model
1.2.1.2 Longitudinal Model
1.2.1.3 Marginal Model
1.2.1.4 Hierarchical Models
1.2.2 Non-Gaussian Linear Mixed Models
1.2.2.1 Mixed ANOVA Model
1.2.2.2 Longitudinal Model
1.2.2.3 Marginal Model
1.3 Estimation in Gaussian Mixed Models
1.3.1 Maximum Likelihood
1.3.1.1 Point Estimation
1.3.1.2 Asymptotic Covariance Matrix
1.3.2 Restricted Maximum Likelihood (REML)
1.3.2.1 Point Estimation
1.3.2.2 Historical Note
1.3.2.3 Asymptotic Covariance Matrix
1.4 Estimation in Non-Gaussian Linear Mixed Models
1.4.1 Quasi-Likelihood Method
1.4.2 Partially Observed Information
1.4.3 Iterative Weighted Least Squares
1.4.3.1 Balanced Case
1.4.3.2 Unbalanced Case
1.4.4 Jackknife Method
1.4.5 High-Dimensional Misspecified Mixed Model Analysis
1.5 Other Methods of Estimation
1.5.1 Analysis of Variance Estimation
1.5.1.1 Balanced Data
1.5.1.2 Unbalanced Data
1.5.2 Minimum Norm Quadratic Unbiased Estimation
1.6 Notes on Computation and Software
1.6.1 Notes on Computation
1.6.1.1 Computation of the ML and REML Estimators
1.6.1.2 The EM Algorithm
1.6.2 Notes on Software
1.7 Real-Life Data Examples
1.7.1 Analysis of Birth Weights of Lambs
1.7.2 Analysis of Hip Replacements Data
1.7.3 Analyses of High-Dimensional GWAS Data
1.8 Further Results and Technical Notes
1.8.1 A Note on Finding the MLE
1.8.2 Note on Matrix X Not Being Full Rank
1.8.3 Asymptotic Behavior of ML and REML Estimators in Non-Gaussian Mixed ANOVA Models
1.8.4 Truncated Estimator
1.8.5 POQUIM in General
1.9 Exercises
2 Linear Mixed Models: Part II
2.1 Tests in Linear Mixed Models
2.1.1 Tests in Gaussian Mixed Models
2.1.1.1 Exact Tests
2.1.1.2 Optimal Tests
2.1.1.3 Likelihood-Ratio Tests
2.1.2 Tests in Non-Gaussian Linear Mixed Models
2.1.2.1 Empirical Method of Moments
2.1.2.2 Partially Observed Information
2.1.2.3 Jackknife Method
2.1.2.4 Robust Versions of Classical Tests
2.2 Confidence Intervals in Linear Mixed Models
2.2.1 Confidence Intervals in Gaussian Mixed Models
2.2.1.1 Exact Confidence Intervals for Variance Components
2.2.1.2 Approximate Confidence Intervals for Variance Components
2.2.1.3 Simultaneous Confidence Intervals
2.2.1.4 Confidence Intervals for Fixed Effects
2.2.2 Confidence Intervals in Non-Gaussian Linear MixedModels
2.2.2.1 ANOVA Models
2.2.2.2 Longitudinal Models
2.3 Prediction
2.3.1 Best Prediction
2.3.2 Best Linear Unbiased Prediction
2.3.2.1 Empirical BLUP
2.3.3 Observed Best Prediction
2.3.4 Prediction of Future Observation
2.3.4.1 Distribution-Free Prediction Intervals
2.3.4.2 Standard Linear Mixed Models
2.3.4.3 Nonstandard Linear Mixed Models
2.3.4.4 A Simulated Example
2.3.5 Classified Mixed Model Prediction
2.3.5.1 CMMP of Mixed Effects
2.3.5.2 CMMP of Future Observation
2.3.5.3 CMMP When the Actual Match Does Not Exist
2.3.5.4 Empirical Demonstration
2.3.5.5 Incorporating Covariate Information in Matching
2.3.5.6 More Empirical Demonstration
2.3.5.7 Prediction Interval
2.4 Model Checking and Selection
2.4.1 Model Diagnostics
2.4.1.1 Diagnostic Plots
2.4.1.2 Goodness-of-Fit Tests
2.4.2 Information Criteria
2.4.2.1 Selection with Fixed Random Factors
2.4.2.2 Selection with Random Factors
2.4.3 The Fence Methods
2.4.3.1 The Effective Sample Size
2.4.3.2 The Dimension of a Model
2.4.3.3 Unknown Distribution
2.4.3.4 Finite-Sample Performance and the Effect of a Constant
2.4.3.5 Criterion of Optimality
2.4.4 Shrinkage Mixed Model Selection
2.5 Bayesian Inference
2.5.1 Inference About Variance Components
2.5.2 Inference About Fixed and Random Effects
2.6 Real-Life Data Examples
2.6.1 Reliability of Environmental Sampling
2.6.2 Hospital Data
2.6.3 Baseball Example
2.6.4 Iowa Crops Data
2.6.5 Analysis of High-Speed Network Data
2.7 Further Results and Technical Notes
2.7.1 Robust Versions of Classical Tests
2.7.2 Existence of Moments of ML/REML Estimators
2.7.3 Existence of Moments of EBLUE and EBLUP
2.7.4 The Definition of Σn(θ) in Sect.2.4.1.2
2.8 Exercises
3 Generalized Linear Mixed Models: Part I
3.1 Introduction
3.2 Generalized Linear Mixed Models
3.3 Real-Life Data Examples
3.3.1 Salamander Mating Experiments
3.3.2 A Log-Linear Mixed Model for Seizure Counts
3.3.3 Small Area Estimation of Mammography Rates
3.4 Likelihood Function Under GLMM
3.5 Approximate Inference
3.5.1 Laplace Approximation
3.5.2 Penalized Quasi-likelihood Estimation
3.5.2.1 Derivation of PQL
3.5.2.2 Computational Procedures
3.5.2.3 Variance Components
3.5.2.4 Inconsistency of PQL Estimators
3.5.3 Tests of Zero Variance Components
3.5.4 Maximum Hierarchical Likelihood
3.5.5 Note on Existing Software
3.6 GLMM Prediction
3.6.1 Joint Estimation of Fixed and Random Effects
3.6.1.1 Maximum a Posterior
3.6.1.2 Computation of MPE
3.6.1.3 Penalized Generalized WLS
3.6.1.4 Maximum Conditional Likelihood
3.6.1.5 Quadratic Inference Function
3.6.2 Empirical Best Prediction
3.6.2.1 Empirical Best Prediction Under GLMM
3.6.2.2 Model-Assisted EBP
3.6.3 A Simulated Example
3.6.4 Classified Mixed Logistic Model Prediction
3.6.5 Best Look-Alike Prediction
3.6.5.1 BLAP of a Discrete/Categorical Random Variable
3.6.5.2 BLAP of a Zero-Inflated Random Variable
3.7 Real-Life Data Example Follow-Ups and More
3.7.1 Salamander Mating Data
3.7.2 Seizure Count Data
3.7.3 Mammography Rates
3.7.4 Analysis of ECMO Data
3.7.4.1 Prediction of Mixed Effects of Interest
3.8 Further Results and Technical Notes
3.8.1 More on NLGSA
3.8.2 Asymptotic Properties of PQWLS Estimators
3.8.3 MSPE of EBP
3.8.4 MSPE of the Model-Assisted EBP
3.9 Exercises
4 Generalized Linear Mixed Models: Part II
4.1 Likelihood-Based Inference
4.1.1 A Monte Carlo EM Algorithm for Binary Data
4.1.1.1 The EM Algorithm
4.1.1.2 Monte Carlo EM via Gibbs Sampler
4.1.2 Extensions
4.1.2.1 MCEM with Metropolis–Hastings Algorithm
4.1.2.2 Monte Carlo Newton–Raphson Procedure
4.1.2.3 Simulated ML
4.1.3 MCEM with i.i.d. Sampling
4.1.3.1 Importance Sampling
4.1.3.2 Rejection Sampling
4.1.4 Automation
4.1.5 Data Cloning
4.1.6 Maximization by Parts
4.1.7 Bayesian Inference
4.2 Estimating Equations
4.2.1 Generalized Estimating Equations (GEE)
4.2.2 Iterative Estimating Equations
4.2.3 Method of Simulated Moments
4.2.4 Robust Estimation in GLMM
4.3 GLMM Diagnostics and Selection
4.3.1 A Goodness-of-Fit Test for GLMM Diagnostics
4.3.1.1 Tailoring
4.3.1.2 χ2-Test
4.3.1.3 Application to GLMM
4.3.2 Fence Methods for GLMM Selection
4.3.2.1 Maximum Likelihood (ML) Model Selection
4.3.2.2 Mean and Variance/Covariance (MVC) Model Selection
4.3.2.3 Extended GLMM Selection
4.3.3 Two Examples with Simulation
4.3.3.1 A Simulated Example of GLMM Diagnostics
4.3.3.2 A Simulated Example of GLMM Selection
4.4 Real-Life Data Examples
4.4.1 Fetal Mortality in Mouse Litters
4.4.2 Analysis of Gc Genotype Data
4.4.3 Salamander Mating Experiments Revisited
4.4.4 The National Health Interview Survey
4.5 Further Results and Technical Notes
4.5.1 Proof of Theorem 4.3
4.5.2 Linear Convergence and Asymptotic Properties of IEE
4.5.2.1 Linear Convergence
4.5.2.2 Asymptotic Behavior of IEEE
4.5.3 Incorporating Informative Missing Data in IEE
4.5.4 Consistency of MSM Estimator
4.5.5 Asymptotic Properties of First- and Second-StepEstimators
4.5.6 Further Details Regarding the Fence Methods
4.5.6.1 Estimation of σM,M* in Case of Clustered Observations
4.5.6.2 Consistency of the Fence
4.5.7 Consistency of MLE in GLMM with Crossed Random Effects
4.6 Exercises
A Matrix Algebra
A.1 Kronecker Products
A.2 Matrix Differentiation
A.3 Projection and Related Results
A.4 Inverse and Generalized Inverse
A.5 Decompositions of Matrices
A.6 The Eigenvalue Perturbation Theory
B Some Results in Statistics
B.1 Multivariate Normal Distribution
B.2 Quadratic Forms
B.3 OP and oP
B.4 Convolution
B.5 Exponential Family and Generalized Linear Models
References
Index