Abstract.In the preface (p. xix) the authors state “the main aims of this book are to provide an introduction to the principles of modeling as applied to longitudinal data from panel and related studies with the necessary statistical theory, and to describe the application of these principles to the analysis of a wide range of examples using the Sabre software from within R.” As the title indicates, this book focuses on mixed models (also called multilevel or hierarchical models), which have become a popular choice for modeling clustered and longitudinal data, and in particular describes use of SabreR, a R version of the Sabre software package. This program can be used to analyze outcomes that are continuous, counts, dichotomous, or ordinal, and the book nicely includes applications for all of these outcome types. Overall, the authors nicely succeed in achieving their stated aims.
In the preface (p. xix) the authors state “the main aims of this book are to provide an introduction to the principles of modeling as applied to longitudinal data from panel and related studies with the necessary statistical theory, and to describe the application of these principles to the analysis of a wide range of examples using the Sabre software from within R.” As the title indicates, this book focuses on mixed models (also called multilevel or hierarchical models), which have become a popular choice for modeling clustered and longitudinal data, and in particular describes use of SabreR, a R version of the Sabre software package. This program can be used to analyze outcomes that are continuous, counts, dichotomous, or ordinal, and the book nicely includes applications for all of these outcome types. Overall, the authors nicely succeed in achieving their stated aims.
Many different datasets are used to illustrate and describe the methods presented. Indeed, the use of so many datasets, which are obtained with the program, is a major asset of the book. The reader clearly gets a sense of the broad applicability of mixed models for data analysis. Worked examples are described in every chapter, as well as statistical theory about the models that are described. For the worked examples, the statistical model is first written out and defined, followed by SabreR code that is used to estimate the model, and finally portions of the computer output. For most of the examples, the authors do a good job of describing the meaning of the estimates that the program produces.
In terms of statistical theory, a useful feature is that likelihood functions are presented for all of the models described in the book. This mixture of application and theory is nicely balanced, so that those that are primarily interested in the applications, say, can easily skip the more technical material. Alternatively, the more technical readers and students, in particular, will benefit from the sections on the likelihood functions and estimation details. Throughout, the book emphasizes full-likelihood estimation of the model parameters, using Gaussian quadrature, either standard or adaptive, for integration over the random effect distribution(s). At the end of all chapters, several data analysis exercises are provided, making it a useful text for a course or workshop on this topic. For such a course or workshop, the book would only work in conjunction with use of SabreR. Given that R and SabreR are freely available, this does make it a viable choice for students.
The book is well organized and nicely builds in complexity. The introduction includes descriptions of the sixteen datasets that are used in the book. These datasets come from many different research areas including psychology, education, economics, psychiatry and other medical fields. Thus, the book is well-suited to quantitative social scientists (particularly econometricians), biostatisticians, and applied statisticians in general.
The first four chapters describe generalized linear models without random effects, including multiple linear regression, logistic and probit regression, Poisson regression, and ordered logit regression (i.e., the proportional odds model) for analysis of continuous, dichotomous, counts, and ordered responses, respectively. Two-level mixed models for these outcome types are then detailed in Chapters 5 through 9. In the multilevel jargon, two-level refers to situations of repeated observations (level-1) nested within subjects (level-2) or subjects (level-1) nested within clusters (level-2). For all outcome types, the authors nicely describe the decomposition of the model in terms of the level-1 and level-2 submodels. This decomposition is common in the education and social science literatures, and aids in the understanding of the model parameters and interpretation of their estimates. In terms of the random effect structure, all of the examples are of random-intercept models of one kind or another. Thus, random intercept and trend models, say, or more general random coefficient models are not considered. However, SabreR can estimate up to three-level models, and these are briefly described in Chapter 10. A three-level example treating a continuous response variable is presented and described, while the chapter exercises include problems with binary and count outcomes.
A distinguishing feature of this book is the treatment of the more advanced topics covered in the later chapters. Chapter 11 presents multivariate models, specifically presenting applications for bivariate count and ordinal outcomes. The ordinal example is for two repeated ordinal outcomes, and the authors describes how the dataset needs to be created for the bivariate longitudinal analysis using SabreR. The authors also describe an application in which the link function is different for the two outcome variables, specifically describing an analysis of a longitudinal continuous and a longitudinal binary outcome. Although only bivariate examples are presented in the book, SabreR can handle up to three dimensions, and so presumably a trivariate longitudinal model could be estimated. Chapter 12 details models for duration and event history data, which are also called survival analysis models. When random effects are included in these models, they are sometimes called frailty models. The treatment is in terms of discrete-time models in which time is represented by a finite number of time intervals. Besides the basic mixed effects duration model, the chapter also includes discussion and examples of renewal and competing risk models. Chapter 13 describes situations in which the random effect distribution has a mass point, with some probability, at the lower or upper extreme (or both). This type of distribution is used in mover-stayer models of migration patterns in which some subjects never move (i.e. are stayers) and emit the same 0 response at every timepoint. Essentially, this augments the standard normal distribution of the random effects with a spike at negative infinity. Likewise, a spike at positive infinity is also possible for subjects that always emit the highest valued response at all timepoints. Such models are especially useful for binary data, in which some subjects are always 0 and others are always 1.
The two final chapters present material that is common to econometricians, but statisticians in other fields might also find these issues relevant in their areas. Chapter 14 details approaches for distinguishing state dependence, initial conditions, and subject heterogeneity for longitudinal binary outcomes. The idea behind state dependence is akin to a first-order Markov process, in which the state a person is in at a given timepoint is related to their previous state. Initial conditions is more about baseline measurements (only) influencing subsequent measures. Heterogeneity refers to individual differences which can explain a person’s longitudinal data. These three sources can be distinguished to some degree with longitudinal data, and the authors provide SabreR examples for how to do this. Also, this chapter highlights some of the more novel ways in which the SabreR program can be used to estimate joint models. The final Chapter 15 provides a nice description comparing random to fixed effects models for longitudinal data. This debate is one that is frequently encountered in econometrics, but has wider applicability. In the fixed effects approach, dummy variables are used for each subject as fixed regressors (either explicitly or implicitly), rather than including the subject effects as random. A benefit of the fixed effects approach is that these subject dummy variables can represent the effects of all subject variables, either observed or not. However, if there are many subjects, and not many timepoints, then estimation of these dummies can be computationally demanding. For such situations, SabreR includes an implicit fixed effects estimator, which is much less computationally demanding than the ”brute force” method of explicitly including dummies for each subject in a longitudinal dataset. The authors (perhaps wisely) do not really get into the philosophical differences between the two approaches, but rather illustrate how both can be performed with SabreR for the interested reader.
Two appendices are included in the book. Appendix A describes SabreR installation for obtaining this free software program via the Sabre site
http://sabre.lancs.ac.uk/ and SabreR commands. The authors do not go into details about the various options associated with these commands in this book, however one can obtain more information about this at the Sabre website. The Gaussian quadrature approach that is used throughout to achieve the maximum likelihood solution for the various models is also described in Appendix A. In particular, the authors do a nice job of detailing the differences between standard and adaptive Gaussian quadrature approaches. Finally, Appendix B includes an introduction to R and usage of SabreR. This is very useful material, and in some ways, depending on one’s experience with R, might be the first thing that readers look over before using SabreR.
Given the fairly large number of different models presented, the presentation in this book is somewhat on the terse side, without a great deal of explanation. None of the chapters are lengthy, and several include a single illustrated analysis. On the plus side, this means that it doesn’t take a great deal of time to read the text and get a reasonable introductory understanding for the many models presented. However, if one is looking for rather detailed presentation and interpretation of particular models, then this book is perhaps not ideal. These comments are more directed towards the first half of the book which covers the more elementary topics. The chapters on the more advanced topics, beginning with Chapter 11 and including the appendices, are in greater detail. This is reasonable given that many readers will most likely be more familiar with the basic 2- and 3-level generalized linear mixed model that comprise the (roughly) first half of the book, and not so familiar with the advanced topics considered in the second half. Overall, this is a useful book for statisticians and data analysts, especially those in the social sciences, that use or want to use R for analysis of correlated data.
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