Applied Statistics for the Six Sigma Green Belt

is incorporated when sample sizes are unequal. Because least squares is mentioned only briefly much later in the book and solving normal equations is not covered, this is of little consequence; restrictions are used in a few places to explain degrees of freedom. The Bonferroni method for simultaneous comparisons is derived from the Sidak inequality, requiring an unmentioned, unneeded distributional assumption. Analysis is presented well before the model, an ordering maintained for later analyses/models. For μT , the grand mean parameter in the one-way ANOVA model, the authors (p. 163) define, both in words and in symbols, the quantity σ 2 total = E[(Yij − μT )2]; as defined, this is a function of the treatment mean μj . The intent is clearly to average also over treatments, but with weights (related to sample sizes?) unknown. σ 2 total is a component of the effect size, a summary measure of treatment effect differences relative to error variability not often found in texts aimed at the physical sciences. Power and sample size calculations are presented in terms of effect size and using tables; here (unlike in most of the book) the presentation is more confusing than it need be. Simple, transparent calculations with noncentral F would be preferable. The early chapters on single-factor experiments serve quite naturally, as is commonly done in design texts, as a foundation for the succeeding material. Power calculations, multiple comparison adjustments, and other topics appear in consistent formats throughout. Chapters 10–15 (also composing about 30% of the text) focus first on two crossed factors, evolve into blocking, then move to error control through the use of covariates. There is a very careful, detailed, and consequently effective development of simple effects, main effects, interactions, and their interplay and interpretation. Sum-to-zero constraints are a part of the two-factor model. The impact of unequal cell sizes is presented in terms of how means are formed and the concept of what one wishes to estimate, motivating least squares means and type III sums of squares without reference to the underlying model (Sec. 14.3). The treatment of blocking includes a discussion of post hoc blocking (pp. 231–232) that is a bit generous in its evaluation, failing to tie in earlier caveats for nonexperimental design. However, both the potential gains and problems with analysis of covariance, and its effectiveness relative to blocking, are given a fair accounting (Chap. 15, especially Sects. 15.6 and 15.7). Setting aside depth and breadth of coverage, all of the techniques thus far encountered are found in virtually every introductory design text. It is the remaining 40% of this text that differentiates it from many others, wherein the authors explore a topic of considerable importance in behavioral science research: within-subjects designs. More commonly termed repeated measures in the statistical literature, here the goal is to increase efficiency by assigning more than one experimental treatment to each subject. Chapter 16 offers a thorough accounting of the basic analysis for the simplest of within-subject experiments, with all treatments assigned once to each subject. This provides a springboard to Chapter 17, which is a series of considerations regarding strengths and weaknesses of within-subjects experimentation, presented with substantial depth although with no more mathematical complexity than earlier material. Among the topics covered are (a) problems arising from carryover effects, carryover effects with interaction, and incidental and context effects; (b) univariate and multivariate models; (c) compound symmetry, sphericity, and corrective actions; (d) minimizing the influence of incidental and carryover effects through randomization and through systematic designs (namely, row-balanced Latin squares); and (e) handling missing values, including the caveat “when the proportion of data that has been lost is great, you will need the assistance of a statistician” (p. 393). This is truly an impressive list for a text at this level. The coverage in Chapters 18–20 of within-subjects (or split-plot) alternatives to the two-factor experiment, allowing all levels of either one or both factors to be assigned within subjects, are on the whole quite thorough. Analysis details are abundant, with careful exposition of all the varieties of contrasts and details for effect size and sample size calculations. There are two points on which some discussion could have served readers well, surprising in their omission in that the authors have been so careful not to leave conceptual gaps. The first of these is design choice: For those situations where an experimenter can plausibly assign a factor either within or between subjects, on what considerations is this choice based? The second is model motivation: Why, with two within-subject factors does the authors’ model [eq. (18.5), p. 417] contain separate, random effects for subject interaction with level of factor A, level of factor B , and A × B combination, and does the appropriateness of this model depend on the nature of the treatments and the experimental procedure? Natural questions for any good student, these should not have been ignored. Incidentally, the model just cited is induced by randomization theory for the split-block (also known as split-plot in strips) design. As a modeling choice, it implies within-subject correlation as a function of treatment assignment. Chapters 21 and 22 present three-way (and by implication, higher-way) factorials for between-subjects experiments. Depth and coverage are comparable to that in earlier chapters. The within-subjects coverage in Chapter 23 is not as thorough, omitting, for example, effect size and sample size calculations, in addition to the issues raised in the previous paragraph. Chapters 24 and 25, “Random Factors and Generalization” and “Nested Factors,” will give students a general sense of these topics, but no more than that. For instance, the “steps for assigning error terms in designs with random factors” in Table 24.3 (p. 535) and the subsequent text do outline how F-ratios and quasi-F’s are constructed, but are certainly not sufficient to impart competence to students at this level. The authors point out (p. 537) that, “You need to be aware of these principles if you are going to use these analyses. The packaged computer programs usually do not perform them correctly without some intervention. . . .” At this juncture, an instructor’s best bet is to very carefully show students how to correctly code for a package of choice. The authors have done a fine job addressing their target audience as identified earlier in this review. The technical quibbles listed here will not hinder that group. It is no small accomplishment to bring this much material to this level, and the authors are to be commended for maintaining a clear conceptual focus that allows those with little statistical preparation to understand what is going on. For students studying this text, I say, “Good luck, work hard, and if you want to advance further, prepare yourself for a statistics course in applied linear models.”