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Book Review: Constrained Statistical Inference: Inequality, Order, and Shape Restrictions

One way that sociologists might view the material in this book is for its potential to increase the power of their statistical tests. This view is appropriate, given the book’s focus on inference as opposed to estimation, as I return to when summarizing its limitations. In principle, the book can be thought of as a guide to transforming two-tail tests into one-tail tests with extensions. The bibliography, for example, contains a page and a half of references to the work over the past 15 years of Cohen and Sackrowitz and colleagues, one title of which is ‘‘Directional Tests for One-Sided Alternatives in Multivariate Models’’ (Annals of Statistics 1998), and another is ‘‘Inadmissibility of Some Tests for Order-Restricted Alternatives’’ (Statistics and Probability Letters 1995). Order restrictions are the core of the book, which builds on earlier statistical texts by Barlow et al. (1972) and Robertson, Wright, and Dykstra (1988) specifically on this subject. The level of presentation is described as ‘‘intermediate . . . for the convenience of upper-undergraduate and lower-graduate-level [statistics] students as well as researchers in applied areas’’ (p. 23). The orientation is toward experimental, biostatistics applications, though not exclusively. In the sixth of its nine chapters, the first sociological application appears: the social stratification classic of the relationship of father’s to son’s occupational status (Goodman’s [1991] ‘‘Measures, Models, and Graphical Displays of Cross-Classified Data,’’ in the Journal of the American Statistical Association is cited). Emphasizing the biostatistical perspective, however, we are asked to consider the ‘‘father’s occupational status as a treatment and son’s occupational status as a response’’ (p. 318). This is not to say that the material of the book is generally not relevant to sociological applications, but instead that the sociological reader will not be shown their relevance. He or she will have to find them for himself or herself. The organization of the book is based on the development of the statistical theory, with areas of application interspersed as examples. The first three substantive chapters (2, 3, and 4) are described as being based on likelihood principles. Of most interest for sociologists may be the development of the statistical theory of isotonic regression, in Chapter 2 (‘‘Comparison of Population Means and Isotonic Regression’’). In the words of the authors, ‘‘an isotonic regression is a weighted least squares estimate subject to a set of pairwise constraints’’ (p. 43). If the population is divided into k-ordered subgroups on a given dimension, then under a simple order, the hypothesis of no relationship to a given outcome variable may be tested against the alternative hypothesis that the outcome for at least one of the Sociological Methods & Research Volume 35 Number 3 February 2007 447-451 2007 Sage Publications http://smr.sagepub.com hosted at http://online.sagepub.com