Linear and Generalized Linear Mixed Models and Their Applications

by Jiming Jiang

Overview

This book covers two major classes of mixed effects models, linear mixed models and generalized linear mixed models, and it presents an up-to-date account of theory and methods in analysis of these models as well as their applications in various fields. The book offers a systematic approach to inference about non-Gaussian linear mixed models. Furthermore, it has included recently developed methods, such as mixed model diagnostics, mixed model selection, and jackknife method in the context of mixed models.

The book is aimed at students, researchers and other practitioners who are interested in using mixed models for statistical data analysis. The book is suitable for a course in a M.S. program in statistics, provided that the section of further results and technical notes in each of the first four chapters is skipped. If these four sections are included, the book may be used for a course in a Ph.D. program in statistics. A first course in mathematical statistics, the ability to use computers for data analysis, and familiarity with calculus and linear algebra are prerequisites. Additional statistical courses such as regression analysis and a good knowledge about matrices would be helpful.

About the Author:
Jiming Jiang is Professor of Statistics and Director of the Statistical Laboratory at UC-Davis

Product Details

ISBN-13:	9780387479415
Publisher:	Springer New York
Publication date:	03/09/2007
Series:	Springer Series in Statistics
Edition description:	2007
Pages:	257
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

About the Author

Jiming Jiang is Professor of Statistics and a former Director of Statistical Laboratory at the University of California, Davis. He is a prominent researcher in the fields of mixed effects models, small area estimation, model selection, and statistical genetics. He is the author of Large Sample Techniques for Statistics (Springer 2010), Robust Mixed Model Analysis (2019), Asymptotic Analysis of Mixed Effects Models: Theory, Applications, and Open Problems (2017), and The Fence Methods (with T. Nguyen, 2016). He has been editorial board member of The Annals of Statistics and Journal of the American Statistical Association, among others. He is a Fellow of the American Association for the Advancement of Science, the American Statistical Association, and the Institute of Mathematical Statistics; an elected member of the International Statistical Institute; and a Yangtze River Scholar (Chaired Professor, 2017-2020).

Thuan Nguyen is Associate Professor of Biostatistics in the School of Public Health at Oregon Health & Science University, where she teaches and advises graduate students. She is an active researcher in the field of biostatistics, specializing in the analysis of longitudinal data and statistical genetics, as well as small area estimation. She is the coauthor of The Fence Methods (with J. Jiang 2016).

Table of Contents

Preface     VII
Linear Mixed Models: Part I     1
Introduction     1
Effect of Air Pollution Episodes on Children     2
Prediction of Maize Single-Cross Performance     3
Small Area Estimation of Income     3
Types of Linear Mixed Models     4
Gaussian Mixed Models     4
Non-Gaussian Linear Mixed Models     8
Estimation in Gaussian Models     9
Maximum Likelihood     9
Restricted Maximum Likelihood     12
Estimation in Non-Gaussian Models     15
Quasi-Likelihood Method     16
Partially Observed Information     18
Iterative Weighted Least Squares     20
Jackknife Method     24
Other Methods of Estimation     25
Analysis of Variance Estimation     25
Minimum Norm Quadratic Unbiased Estimation     28
Notes on Computation and Software     29
Notes on Computation     29
Notes on Software     33
Real-Life Data Examples     34
Analysis of Birth Weights of Lambs     35
Analysis of Hip Replacements Data     37
Further Results and Technical Notes     39
Exercises     48
Linear Mixed Models: Part II     51
Tests in Linear Mixed Models     51
Tests in Gaussian Mixed Models     51
Tests in Non-Gaussian Linear Mixed Models     56
Confidence Intervals in Linear Mixed Models     66
Confidence Intervals in Gaussian Mixed Models     66
Confidence Intervals in Non-Gaussian Linear Mixed Models     72
Prediction     74
Prediction of Mixed Effect     74
Prediction of Future Observation     80
Model Checking and Selection     88
Model Diagnostics     88
Model Selection     93
Bayesian Inference     99
Inference about Variance Components     100
Inference about Fixed and Random Effects     101
Real-Life Data Examples     102
Analysis of the Birth Weights of Lambs (Continued)     102
The Baseball Example     103
Further Results and Technical Notes     105
Exercises     113
Generalized Linear Mixed Models: Part I     119
Introduction     119
Generalized Linear Mixed Models     120
Real-Life Data Examples     122
The Salamander Mating Experiments     122
A Log-Linear Mixed Model for Seizure Counts     124
Small Area Estimation of Mammography Rates     124
Likelihood Function under GLMM     125
Approximate Inference     127
Laplace Approximation     127
Penalized Quasi-Likelihood Estimation     128
Tests of Zero Variance Components     132
Maximum Hierarchical Likelihood     134
Prediction of Random Effects     136
Joint Estimation of Fixed and Random Effects     136
Empirical Best Prediction     142
A Simulated Example     149
Further Results and Technical Notes     151
More on NLGSA     151
Asymptotic Properties of PQWLS Estimators     152
MSE of EBP     155
MSPE of the Model-Assisted EBP     158
Exercises     161
Generalized Linear Mixed Models: Part II     163
Likelihood-Based Inference     163
A Monte Carlo EM Algorithm for Binary Data     164
Extensions     167
MCEM with I.I.D. Sampling     170
Automation     171
Maximization by Parts     174
Bayesian Inference     178
Estimating Equations     183
Generalized Estimating Equations (GEE)     184
Iterative Estimating Equations     186
Method of Simulated Moments     190
Robust Estimation in GLMM     196
GLMM Selection     199
A General Principle for Model Selection     200
A Simulated Example     203
Real-Life Data Examples     205
Fetal Mortality in Mouse Litters     205
Analysis of Ge Genotype Data: An Application of the Fence Method     207
The Salamander-Mating Experiments: Various Applications of GLMM     209
Further Results and Technical Notes     214
Proof of Theorem 4.3     214
Linear Convergence and Asymptotic Properties of IEE     214
Incorporating Informative Missing Data in IEE     217
Consistency of MSM Estimator     218
Asymptotic Properties of First and Second-Step Estimators     221
Further Results of the Fence Method     225
Exercises     229
List of Notations     231
Matrix Algebra     233
Kronecker Products     233
Matrix Differentiation     233
Projection     234
Generalized Inverse      235
Decompositions of Matrices     235
The Eigenvalue Perturbation Theory     236
Some Results in Statistics     237
Multivariate Normal Distribution     237
Quadratic Forms     237
Op and op     238
Convolution     238
Exponential Family and Generalized Linear Models     239
References     241
Index     255

Editorial Reviews

From the reviews:

"This book is an up to date description of linear mixed models, LMM, and generalized linear mixed models, GLMM. The material is complete enough to cover a course in a Ph.D. program in statistics. The contribution of this book is that of pointing and developing the inference and estimation issues for non-Gaussion LMMs." (Nicoleta Breaz, Zentrablatt MATH, 2009, 1152)

"The book deals with Gaussian and non-Gaussian linear mixed models. … This book is suitable for a course in statistics at the MSc level … . This book contains many examples, exercises and some useful appendices, making it suitable for use in statistics courses. … The book has a nice lay-out and the index make it easy to jump to a topic of interest. … A nice feature of the book are the many real-life data examples." (M. Moerbeek, Kwantitatieve Methoden, August, 2007)

"This book, which has grown out of the author's research on this area, deserves close attention. It provides a good reference source for an advanced graduate course and would prove useful for research workers who wish to learn about theoretical developments in this area...[T]his book will be a useful source for obtaining the theoreteical skills required for further developments in this area." (Youngjo Lee, Biometrics, December 2007)

"As noted by the author, there have been many new developments in mixed effects models in the past decade. This volume is intended to provide an up-to-date treatment of both theory and methods. … it is full of important results and examples, including significant contributions by the author to the treatment of mixed effects models. As a textbook, it is aimed at MS students in statistics, but includes supplementary material more suitable for PhD candidates. … be useful as such for many GLMM users." (Donald E. Myers, Technometrics, Vol. 50 (1), 2008)

"The book under review covers both LMMs and GLMMs and offersan up-to-date account of theory and methods in the analysis of the models as well as their applications in biological and the medical research, animal and human genetics, and small area estimation. The examples of applications appear near the end of each chapter. … The book is aimed at students, researchers and other practitioners who are interested in using mixed models for statistical data analysis." (Alexander G. Kukush, Mathematical Reviews, Issue 2007 m)

"Jiming Jiang’s book on mixed models covers a lot of material in surprisingly few pages. … On the whole, it is a rather technical book, both in terms of the mathematical level and in terms of notation … . Undoubtedly there is a lot one may learn from the book … . useful for the professional who wishes to know more about the technical results of the topic … ." (Søren Feodor Nielsen, Journal of Applied Statistics, Vol. 35 (4), 2008)

From the Publisher