Hierarchical Linear Models: Applications and Data Analysis Methods

to the confusion, not only is “constraint” sometimes used when “restriction” is meant, but notational expressions like P ¿i D 0 (a restriction) are invoked as constraints ( P ¿ 0 i D 0) for solving the normal equations. Incidentally, whereas the generalized inverse is used for the development of Gauss–Markov theory in Chapter 4, the normal equations for speci c applications of overparameterized models are solved almost exclusively by the constraint method. The book includes four more chapters (8–11): “Multiple Regression Models,” “Fixed-Effects Linear Models,” “Random-Effects and Mixed-Effects Models,” and “Special Topics. ” The  rst two contain mostly the standard results one would expect, and so will not be discussed here. The random/mixed effects treatment is rather short compared to that in any of the books cited earlier, but does touch on the ANOVA method, restricted maximum likelihood, and likelihood estimation. What is interesting is how the authors have answered question (4) in my introductory paragraph; all four chapters cover topics that you will not  nd in other texts aiming for the same audience. These include, among others, sections on nonparametric analysis of one-way and balanced two-way layouts (Kruskal–Wallis and Friedman procedures), robust regression (least absolute deviations and M regression), nonparametric regression (projection pursuit, wavelet methods, and others), Bayesian estimation, and Kalman  ltering. This coverage is not deep; one would not, for instance, want a student’s sole exposure to the Kruskal–Wallis analysis to be what is found here. From an instructor’s standpoint, the book’s presentation of these topics provides adequate foundations on which to build more expansive classroom lectures. The main ideas are stated with a few basic results and, as is true throughout the book, adequate references are given for those who wish to dig deeper. Useful exercises are provided at the end of each chapter, although, as with any graduate text, instructors will likely need to supplement these to highlight points of individual emphasis. Don’t expect data; in most chapters, all of the exercises are nonnumeric. Selected brief solutions are collected following an appendix on basic probability distributions. In the Preface the authors state that “our book requires some knowledge of statistics . basic estimation theory and hypothesis testing at an undergraduate level is de nitely required” (p. x). In addition, familiarity with the use and application of standard linear models (regression, balanced oneand two-way classi cations) will greatly enhance understanding and accessibility. Examples are included, but not to the extent found in either of the recent books mentioned earlier, and there is little attempt at motivation, for instance, for parameterizations of classi cation models. Many students would  nd this an unfriendly  rst meeting with classi cation models. The book’s limitation is the brevity with which numerous topics, and not just the nonstandard ones, are handled. To cite two more examples, in the battery of normality tests presented, most are accorded a single paragraph, with no accompanying plots or numerical demonstrations. For the two-way crossclassi cation with interaction and unbalanced data, not addressed are the test for interaction, comparison of row and column tests arising from nested model building with the usually preferable unweighted row and column tests, and the issues of what can/should be tested if some cells are empty. Still, it is this sometimes brevity, with depth yielding to breadth, that is the book’s strength. The authors give us a thorough accounting of the core of linear model theory while having it  ow into more modern extensions in many directions. It is an excellent reference book, cast at a bit higher level than either Hocking (1996) or Rencher (2000). As a text for that  rst graduate course in linear model theory, it should serve the suf ciently mature student well.