Linear Mixed Models for Longitudinal Data

Mixed-effects models (or mixed models) provide a  exible and powerful tool for the analysis of data with a complex variance structure, such as correlated data. Linear mixed models originated speci cally in the area of application. The motivation of this book is to satisfy the great demand by users from various applied backgrounds for clearer guidance on using the available methodology (e.g., theoretical concepts and their software implementation) more effectively. The authors refer to this book as a second version of their  rst book Linear Mixed Models in Practice (Verbeke and Molenberghs 1997). This book, however, is not presented as a second edition since a large range of new topics has been added and material kept from the  rst version has been reworked. The authors adopt the view that each type of outcome should be analyzed using instruments that exploit the nature of the data (i.e., each model family requires its own speci c software tools). This book provides a comprehensive treatment of linear mixed models for continuous longitudinal data. Over 125 illustrations are included in the book. This book gives emphasis to practice rather than mathematical rigor. Although enough theory is covered in the text to understand the strengths and weaknesses of mixed models, the authors emphasize the applied aspects of these. Hence, this book is explanatory rather than research oriented. It is mentioned that the authors attempt to target applied statisticians and biomedical researchers in industry, public health organizations, contract research organizations, and academia. I do believe that the book may serve as a useful reference to a broader audience (e.g., researchers in reliability engineering). Since practical examples are provided as well as discussion of the leading software utilization, it may also be appropriate as a textbook in an advanced undergraduate-level or a graduate-level course in an applied statistics program. This book is organized into 24 chapters. Excluding the  rst two and the last two chapters, it may also be divided into two parts. In the  rst part, comprising Chapters 3–13, emphasis is on the formulation and the  tting of, as well as on inference and diagnostics for, mixed models in general. In the second part, comprising Chapters 14–22, the problem of missing data is discussed in full detail, with emphasis on how to obtain valid inferences from observed longitudinal data and how to perform sensitivity analyses with respect to assumptions made about the dropout process. A more detailed structure follows. Chapter 1 introduces the scope, while Chapter 2 presents the examples used throughout the book. Chapters 3–9 provide the core about the linear mixed model. Chapters 10–13 discuss extensions to the original model and more advanced tools for model exploration and in uence diagnostics. Chapters 14–16 introduce the reader to basic incomplete longitudinal data concepts, such as dropout, which refers to the case in which all observations on a subject are obtained until a certain point in time. Chapters 17 and 18 discuss strategies to model incomplete longitudinal data, based on the linear mixed model. The sensitivity of such strategies to parametric assumptions is investigated in Chapters 19 and 20 (more technical material is deferred until Appendix B). Some additional missing data topics are presented in Chapters 21 and 22. Chapter 23 is devoted to design considerations, such as designing experiments with minimal risk of high losses in ef ciency due to dropout. Building on the methodology developed in the book, Chapter 24 presents  ve case studies. Appendix A reviews a number of software tools for  tting mixed models. The balanced mix of real data examples, modeling software, and theory makes this book a useful reference for practitioners using mixed models in their data analysis. Researchers will also  nd this book appealing for its extensive literature review, for the presentation of novel methodologies, and for its discussion about needed research. In this topic (e.g., p. 237: specialized software for  tting nonrandom dropout models; p. 296: methods that investigate the sensitivity of the results with respect to the model assumptions; p. 374: other sensitivity analysis approaches in the pattern-mixture content). It is mentioned in the Preface that selected macros (and programs) for tools discussed in the text, as well as publicly available datasets, can be found at Springer-Verlag’s URL: http://www.springer-ny.com/. The given URL is too general, and it took some time to access the information for this book. A more ef cient approach is referring the reader to the following URL: http://www.luc. ac.be/censtat/members/geertmpub.html, where the datasets can be found. At this same site, macros, errata, and updates of the materials in the book should be made available. Adding the following to the Preface would have been useful to the prospective reader: prerequisites for the technical material in the book, recommended outlines for undergraduateand graduate-level courses, and typographical conventions. Some minimum prerequisites for the technical material in the book, in my opinion, include a knowledge of calculus and linear algebra and a working knowledge of probability and statistics such as provided in advanced undergraduate or graduate courses in statistics, mathematics, and related  elds. Some knowledge of the SAS language is de nitely desirable but not a prerequisite for following the material in the book. Great care has been taken in presenting the data analyses in a software-independent fashion. The format of this book consists of (1) presenting a research question, (2) translating it into a statistical model by means of algebraic notation, and (3) implementing such a model (for most cases) using a SAS code. Although most analyses were done with the M IXED procedure of the SAS software package (discussed in detail in Chap. 8), some other commercially available packages are discussed as well. Appendix A.3, for example, describes the built-in function lme() for analyzing linear mixed models in S and S-PLUS. (The companion function for nonlinear mixed models, nlme(), is also mentioned. ) On pages 493–494, the reader should not be confused with mle() and nmle(), which should have read lme() and nlme(), respectively. In summary, the increasing popularity of mixed models is explained by the  exibility they offer in modeling complex data, by the handling of balanced and unbalanced data in a uni ed framework, and by the availability of reliable and ef cient software for  tting them. This book provides an overview of the theory and application of linear mixed models in the analysis of correlated data. The scope of this book is restricted to linear mixed models for continuous outcomes. This book provides guidance on using the available methodology more effectively to practitioners from a wide variety of areas. Because of its discussion and detailed reviews, advanced students and researchers may bene t from it as well.