Asymptotic Analysis of Mixed Effects Models: Theory, Applications, and Open Problems

Large sample theory plays a critical role in the development and application of statistical methodology. Asymptotic Analysis of Mixed Effects Models, by Jiming Jiang, provides a comprehensive account on asymptotic analyses of mixed effects models. Jiang is the author of two previous textbooks, Linear and Generalized Linear Mixed Models and Their Applications (Jiang, 2007) and Large Sample Techniques for Statistics (Jiang, 2010). Ten years later, he has written an even more impressive monograph that covers asymptotic theories in mixed models. This book is partly related to the author’s own research and covers a combination of asymptotic theory and real-life examples. With five chapters and about 250 pages, the book is compact, but rich with information.

Mixed models have a wide range of applications in the physical, biological, and social sciences, and there are many textbooks devoted to the practical and applied aspects of these models. Books devoted to the theoretical aspects of these models, however, are harder to find, and I am always looking for a good one. Theoretical books are sometimes hard to follow, but I found this book to be the opposite. Jiang does a great job in using his story writing skills to keep readers engaged in the topics with motivations, counterexamples, case studies, and open questions. The book starts with a preview of the asymptotic theory to come later, using generalized estimating equations (GEE), restricted maximum likelihood (REML), and generalized linear mixed models (GLMM) as examples. The author keeps this chapter short with a focus on sketching out the main ideas behind the proofs while providing specific citations (including section and page numbers) for readers who want to dive into the technical details. This strategy greatly enhances the readability by presenting complex theory in a straightforward manner and makes the material much more approachable, especially for readers who are just beginning to research large sample theory.

Chapter 2 covers the asymptotic analysis of linear mixed models, which are based on maximum likelihood (ML) and REML, and includes a discussion of model prediction, model diagnosis, model selection, and misspecification along with its treatment of parameter estimation. With the advance of technology, big data are becoming widespread in scientific experiments. The dimensionality of these experiments presents a challenge to traditional theory, in which the number of variables can increase faster than the sample size. The theories presented in this section clearly show the asymptotic superiority of REML over ML for big data with high dimensionality. That is, ML requires additional conditions to ensure that the sample size accommodates the increasing dimensionality. In contrast, REML estimators are consistent under much less restrictive conditions. This chapter also introduces martingale limit theory to assess asymptotic properties in the balanced mixed ANOVA model and the general non-Gaussian mixed model. Jiang uses high-dimensional genetics data to illustrate model misspecifications, which could be very useful for biostatisticians and bioinformaticians. At last, this chapter discusses some open questions, including unbalanced designs and challenges in obtaining standard errors in variance component estimation.

The elegant writing continues in Chapter 3, where Jiang lays out details for the asymptotic properties of generalized linear mixed models (GLMM). There are several existing books that cover GLMM concepts, methods, and application, so the author focuses here on asymptotic properties, and covers a variety of estimators. The chapter contains a detailed comparison between GLMM and GEE, which helps readers better understand the differences in asymptotic properties between these models. Other estimation methods, including the maximum likelihood estimation and conditional inference, are then covered thoroughly. At the end of the chapter, Jiang points out some open questions with a special focus on challenges in GLMM diagnostics.

I feel that Chapters 4 and 5, which cover some applications of mixed effect models that are often left out of textbooks, are hidden gems. Small area estimation (SAE), covered in Chapter 4, has become a very active area in survey methodology research. Chapter 4 will serve as a great addition to Rao’s classic book, Small Area Estimation, now in its second edition (Rao and Molina, 2015). Since small area estimation largely relies on mixed effect modeling, many properties mentioned in other chapters (such as model prediction covered in Chapter 2) can be tailored to small area estimation. The comparison of different estimators shows that ML and REML asymptotically perform better than other frequentist approaches. In recent years, the Bayesian approach has become widely applied in small area estimation. Thus, Jiang also covers hierarchical Bayes estimation for the Fay–Herriot model. Due to the popularity of Bayesian estimation, I wish a subsection could be devoted to assessing the Bayesian approach in the more general models. Another challenging topic in small area estimation is that, since the data become sparser as the geographic units get smaller (e.g., census tracts, zip code), in some areas it may be completely missing. Data sparsity is briefly mentioned in this chapter but the book does not address how to deal with the problem of areas in which data are missing entirely. I also would have liked the book to provide some insights about how to account for the survey sampling design (including sampling weight, strata, and cluster) in small area estimation since ignoring survey sampling design can lead to estimation bias in both mean and variance parameters. However, the chapter does extensively cover resampling methods in SAE, including jackknifing and bootstrap. The newly developed Monte-Carlo jackknife method is widely applicable in different scenarios.

Chapter 5 covers asymptotic analyses in other mixed effects models, focusing on nonlinear/semiparametric/nonparametric mixed effects models, frailty models for survival analysis and joint modeling. There are several applied books on frailty and joint models, which are focused on the concept, modeling, programming, and applications, but very little on the asymptotic properties of these models, so I was excited to see Jiang address that gap in this chapter. Although more complex structures are widely used in practice, the large sample theory in Chapter 5 is primarily limited to the clustering structure with random intercepts. For complex models, the likelihood functions often do not have a closed form. Many algorithms have been developed for other mixed effects models with varying variance and covariance structures, but theoretical work is critically needed to better understand the relative efficiency of the different estimators.

This book is not for entry-level students in statistics. To fully grasp the proofs or apply these techniques to other scenarios, a reader will first need to master the basic skills of asymptotic theory, perhaps by reading Jiang’s earlier books. However, I highly recommend this book for Ph.D. students with research interests in large sample theory, mixed models and its extensions, including frailty modeling, and small area modeling. As a monograph, the book does not provide exercise questions and it might be difficult to use as a stand-alone textbook for graduate level courses, but is a great reading assignment for Ph.D.-level students. Students can follow Jiang’s discussions on open topics and further extend one of the open questions as their own research topic.

Let me conclude this review by citing Jiang’s statement on page (22) of the book, “An asymptotic theory is not very useful unless it can be fully understood by a practitioner.” Jiang has completely followed this principle by developing complex asymptotic theories during his 20+ years of research in this field. With this book, he has now presented us with a succinct and clear overview of asymptotic theories in mixed effects models.