Functional Approach to Optimal Experimental Design

Diagnostics are revisited, and Bartlett’s test is presented in detail. Finally, introductions are provided to logistic regression, M-estimation, and total least squares. A major feature of this book is the inclusion of optional linear algebra sections and problem sets, which are given for all but one of the chapters. In addition, there is a 16-page appendix that can provide a refresher or as an introduction to linear algebra. The goal of these sections is to demonstrate the power and flexibility of the linear algebra approach to linear models. Initially, this material is mechanical in nature. In later chapters (Chap. 7 and beyond), the scope, depth, and volume of the linear algebra sections and exercises increases sharply. This later material will be beyond the reach of the student without a prior linear algebra course, but will be a boon to the initiated undergraduate or possibly to the nonmajor graduate student. The linear algebra sections are fairly brief and do not attempt to recapitulate the entire chapter, but rather highlight two or three ideas previously introduced. These discussions advance to the point of introducing orthogonal projection operators and subspaces. A version of the Gauss–Markov theorem is subsequently presented and proven. Most arguments are coordinatized, and a full-rank design matrix is assumed. The problem sets are particularly strong. There are, of course, many dataoriented exercises that allow the student to work with the methods discussed. These problems are rarely purely mechanical and often ask the student to think about the analysis while proceeding. Each chapter also has several theoretical or mathematical exercises, which typically are very good. These problems often significantly expand the scope in the sense that they introduce new ideas and techniques. Occasionally students are asked to produce simulation-based solutions to problems. The exercises are characterized by both diversity and breadth. Although the authors promise to indicate which problems require calculus, which require simulation, and which require “just a little more persistence,” they do not seem to follow this practice religiously. For example, problems in the latter category are indicated only in the first two chapters. My criticisms of the text are minor. Given the vast nature of the topic area, the authors certainly needed to make some tough decisions regarding coverage. Having said that, I note that the discussions focus on regression models (quantitative covariates). The last chapter does discuss some one-way ANOVA and ANCOVA models, but goes no further. There is no discussion of two-way or higher models, or of experimental design models. These are models that the intended audience will encounter sooner rather than later. In addition, some may feel that the chapter on inference should be a bit broader. Unbiasedness is the only estimation criterion discussed carefully, and the introductions to confidence intervals and testing are done almost exclusively through examples. Testing is often a troublesome area for students, and the authors seem to break their own rule here regarding the provision of mathematical background. My review revealed a number of typographical and other errors commensurate with a first edition. In summary, Introduction to Linear Models and Statistical Inference reflects a strong appreciation of both theory and the realities of practical model building. The book is intended for a broad audience, and the availability of an electronic version should widen its appeal by expanding its accessibility. The authors’ many years of teaching experience are evident, and the beginning or intermediate statistics student should come away with an understanding that analyzing real data requires thoughtfulness and completeness. Truly linking methods and theory in an area like linear models is not trivial for most students, but tools like this one go a long way toward bridging that gap.