Generalized, Linear, and Mixed Models

This book is another  ne contribution to the statistics literature from these respected authors. The book ties together standard mixed model results with advances that have been made in the last decade. It shows how recent advances  t into the original context of earlier results. It also collects these advances into a single coherent package. This is very useful to those of us who have followed the advances in the journals, but have not really put all of the pieces together, or who were not privy to some of the more private motivations for the work. There is a good mix of mathematical detail and verbal explanation in all of the chapters. There are very few theorems, because the authors easily switch between textual and mathematical explanation. Derivations are natural and insightful. Experienced readers will  nd this text fairly fun to read, but new students will  nd it challenging. The problems at the end of the chapters range between simple and dif cult. The book serves well as a text or reference. The book contains 11 chapters and 2 appendices. Much of the information in Chapters 1–4 is found easily elsewhere, but the way in which it is presented establishes the reasoning behind results presented in later chapters. Chapter 1 has a good set of guidelines about how to determine whether an effect is  xed or random. Chapter 2 is on one-way classi cations. There are two major sections: normal models and binary models. Both sections show how to incorporate and interpret random effects in these models. Careful attention is paid to the problems resulting when estimates are outside of the parameter space. Chapter 3 is on single-predictor regression. Most of the chapter is spent on estimation and interpretation of the random-intercepts model in normal data. The second part of the chapter covers adding normally distributed random effects to logistic regression. There are several well-written interpretations of the parameters in logistic regression and how they relate to odds ratios. Chapter 4 is a thumbnail discussion of linear models. It includes a good discussion of the uses of generalized inverses and estimable functions. Chapter 5 covers generalized linear models. It gives a short history of the early ideas that lead to the development of generalized linear models, then develops the general form from the example. The similarities of generalized linear models to likelihood estimation are discussed, as is the simpli cation in model speci cation from using quasi-likelihood. All of this is based on several nicely laid out details in the derivation of maximum likelihood in iid samples from members of the exponential family. Chapter 6 covers linear mixed models. These are fairly standard results, especially for users of Proc Mixed in SAS. The maximum likelihood (ML) and restricted maximum likelihood (REML) estimators are developed. Hypothesis testing under known and estimated covariance matrices is discussed. Prediction of random effects is also discussed in both cases. Analysis of variance (ANOVA) estimators are discussed brie y. Some mention of Hocking’s use of balanced-data ANOVA estimators via the EM algorithm for unbalanced data could salvage the discussion on the use of ANOVA estimators in unbalanced data. The authors also compare the merits of ML and REML estimation, falling on the side of REML. Chapter 7 covers longitudinal models. This chapter mainly considers cases where R D E6var6y—u77 is not based on the identity matrix, and a compound symmetric covariance structure is proposed for var(E6y—u7). Clearly, there are several combinations of choices for which parameters in these matrices are 0. This chapter carefully outlines these cases. The analytical details for each case are presented for the prediction of random effects and estimators where possible. Chapter 8 covers generalized linear mixed models. This is a valuable chapter because it lucidly describes how to incorporate random effects into generalized linear models. It also explains the rationale for using marginal or conditional models, and shows how the use of random effects induces correlations among observations with common random effects. This is a natural concept in linear mixed models, and it is nice to have it demonstrated for generalized mixed models. The chapter also discusses methods of estimation and hypothesis testing, both good and bad. Chapters 9, 10, and 11 are more conceptual than detailed. Chapter 9 considers prediction through best prediction, best linear prediction, and best linear unbiased prediction. Apart from best linear unbiased prediction in normal models, these subjects are best described as “thorny. ” The ideas that differentiate each of these prediction methods are discussed. Some estimation methods are discussed for the linear predictors. As for Chapter 10, a whole book could be written on computing estimators for all of the different methods presented in the book. The authors correctly chose to summarize the methods so that the breadth and relationship of the topics can be viewed concisely. This is a strength of the book, bringing the dispersed research on these topics into a common forum. Section 10.3 is most interesting, because it contains descriptions of many of the newer estimation methods in generalized linear mixed models. Chapter 11 covers nonlinear mixed models. It is brief. All in all, Generalized, Linear, and Mixed Models is a good book. My only criticism is that it is a little uneven in its detail levels. That being said, the book helped me organize many of the methods that I have randomly explored over the years. I found it educational.