This book presents procedures for making statistical inferences on the basis of the classical linear statistical model, and discusses the various properties of those procedures. Supporting material on matrix algebra and statistical distributions is interspersed with a discussion of relevant inferential procedures and their properties. The coverage ranges considerably: some material is suitable for first-year graduate students, other material for advanced graduate students, and other portions would be of interest to postgraduate researchers. In particular, the material in Chapters 6 and 7 does not appear to be covered in an easily accessible manner in any other books, and generalizes the traditional normal-based linear regression model to the elliptical distributions and beyond. For the advanced reader, these chapters greatly elucidate just how far these familiar results can be extended. Refreshingly, the material also goes beyond the classical 20th century coverage to include some 21st century topics like microarray (big) data analysis, and control of false discovery rates in large scale experiments.
After setting the stage in the introductory Chapter 1 by demonstrating the versatility and universal applicability of the linear model, culminating in the sophistication of a hierarchical specification with random effects, the book begins in earnest in Chapter 2 by reviewing the necessary matrix theory. This covers all the usual prerequisite material that constitutes the algebraic foundation of the linear model and inference derived from it: orthogonality, idempotency, projections, generalized inverses, linear systems, quadratic forms, etc.
The coverage of preliminaries ends with the (fairly standard) material in Chapter 3, which introduces expected values, variances, and covariances of a random vector. A fair amount of attention is paid to a proper introduction of conditional moments, not forgetting the all-important variances and covariances conditional on a random vector that feature so prominently in optimal (mean squared error sense) estimation/prediction theory. This chapter ends, appropriately, with the multivariate normal distribution, which provides the theoretical underpinnings for so many of the inferential procedures for the linear model. The usual development of deriving the distribution through an affine transformation of a vector of independent standard normals is given, followed by discussions on symmetry, conditionals, independence, and higher-order moments.
The general linear model is the subject of Chapter 4. Every effort is made to keep the discussion general. The mean is an arbitrary function of covariates as long as it is linear in the coefficients β. The error vector covariance matrix Σ is treated in three increasing levels of complexity: Gauss–Markov model with homoscedastic and uncorrelated errors, Σ = σ2I; Aitken model with known correlation where the only unknown is the common scale parameter σ; and finally a general unknown (albeit parameterized) Σ. A variety of real examples richly illustrate the potential and usefulness of the classical linear model here. These range in intricacy from longitudinal setups with simple correlation structures like heteroscedastic errors and compound symmetry, to a variety of time series autocorrelation constructs, and finally to models with a spatial correlation context where the lack of a well-defined directional “flow” for the values necessitates substantial simplifying assumptions like isotropy.
Chapter 5 marks the transition from a (population) model-based discussion to sample-based inference. Following the classical approach, estimation and prediction are tackled first. The chapter starts with basic properties that define the usual concepts of optimality in point estimation: linearity, unbiasedness, equivariance, estimability, etc. After covering the prerequisite material on solution of linear systems, the discussion turns to least-squares and their optimality as best linear unbiased estimators (BLUEs). Maximum likelihood estimators (MLEs) are next up, culminating in the familiar result that BLUEs are MVUEs (minimum variance in the class of all estimators, linear or not) under a Gaussian likelihood. Greater generality is sought by spelling out what happens under the broader elliptical family. This attention to detail and offering of broader coverage (by answering the obvious “what if” questions about possible extensions) are in fact recurring themes throughout the book. A discussion of prediction and its subtle connection with estimation rounds out the chapter. Noteworthy is the fact that results are given (and proved) both from an algebraic as well as a geometric perspective, where optimal estimators are seen to be projections onto orthogonal subspaces.
In preparation for the discussion on confidence sets and testing, Chapter 6 lays the foundation for the prerequisite distribution theory. This includes not only a comprehensive coverage of the normal and derived families (χ2, t, F), but also gamma, beta, and its generalization to the Dirichlet. This nontraditional coverage of the latter distributions is needed in the discussion of generalized quadratic forms, which constitutes the next major section. Moment generating function results are given for quadratic forms, joint quadratic forms, and ratios thereof, in normal random variates. If you have ever wondered how much generalization is possible by embedding the ordinary (Gaussian) linear model in say, the elliptical family, wonder no more, as this is answered in painstaking detail. The chapter concludes with an equally general coverage of spectral decomposition results for symmetric matrices and matrix calculus.
Finally, Chapter 7 tackles the extensive and all-important topic of confidence sets and hypothesis tests, with the discussion naturally limited to estimable linear functions of β under the Gauss–Markov model, but focusing on simultaneous inference (possibly with infinitely many estimable functions). In preparation for testing subsets (of models), the chapter begins with some background on partitioning the model matrix. The classical F-test is then described, but avoiding normality assumptions whenever possible; in fact, as the reader has already come to expect, very general results can be given under a user-assumed distributional family. However, to recover optimality properties such as UMP/UMA (uniformly most powerful/accurate), specialization to the normal family is typically needed, and one section is devoted to this coverage. Section-long treatments are also given of one-sided inference, and inference on the nuisance scale parameter σ. The chapter concludes with two topics that do not get much attention in competing books. The first is an up-to-date discussion of multiple comparison procedures pertaining to large-scale inference, with a focus on controlling either familywise error rate (FWER) or false discovery rate (FDR). The final section broaches the topic of prediction, but from a confidence set or test perspective.
It is clear that the book is very flexible and extensive in its coverage of linear models. It could easily form the basis of the (typically required) linear models course taught in traditional statistics MS and PhD-level programs. Moreover, it is written in a way that the easier (MS-level) material is presented early on in the chapters (or at least it’s easy to find), with the harder PhD-level material following. So it could also be used as the basis for a more advanced PhD-level course as a follow-up to the first (MS-level) course. Complementing this, the end-of-chapter exercises are also of different levels of difficulty. However, some instructors may feel a little overwhelmed by the scope, and may therefore prefer a less ambitious text. On that note, it is probably safe to say that this comprehensive coverage will provide an appealing reference to advanced readers and PhD-level researchers. All in all, this is a very well-written book that provides an invaluable (and one is tempted to say, “definitive”) treatment of this classical subject.
Texas Tech University
Lubbock, TX
[email protected]