Logit and Probit: Ordered and Multinomial Models

focuses on item response theory (IRT), the aim here is different, and this text nicely complements many of the standard books on IRT. IRT is commonly presented by focusing on the analysis of large-scale standardized tests. The results of such an analysis include precise descriptions of item characteristics and point estimates of examinee abilities with standard errors. This is the focus of the text of Hambleton et al. (1991), which discusses parametric IRT models. In these models, each item is represented by a parametric (often logistic) item response function (IRF) whose parameters describe such properties of the items as dif culty, ability to discriminate between ability levels, and susceptibility to guessing. Ramsay (1997), for example, extended this methodology to nonparametric IRT by using nonparametric regression techniques to estimate the shape of the IRFs. Following the work of Mokken (1971), the current volume takes an approach to nonparametric IRT that leaves the form of the IRFs unspeci ed except for a few fundamental properties (i.e., unidimensionality, local independence, and monotonicity) common to most parametric IRT models. This is because the authors’ focus is not on obtaining precise point estimates, but instead on constructing scales (sets of either dichotomous or polytomous/Likert-type items) that will do a fairly accurate job of ranking the examinees in order of ability, ranking the items in terms of “dif culty,” and constructing measures of item and scale quality. This alternate focus allows for analysis of sample sizes (one example in the text has 12 dichotomous items and 425 examinees) that are on the small end of those usually considered in the study of standard IRT methods. Although the book is aimed at social scientists who develop their own tests, the methods described are  rmly grounded in the most fundamental IRT assumptions and are still applicable to large-scale testing. Chapters 1 and 2 give an easy-to-follow overview of IRT and the goals of the procedures described in the text. In particular, they introduce Mokken’s monotone homogeneity model (MHM) and double monotonicity model (DMM), which are generalizations of the most commonly used parametric IRT models. Like the other chapters, these contain a selection of upto-date references and a brief set of exercises (with answers) to reinforce the concepts. Chapter 3 continues the introduction to the topic by providing an example applying the MHM-based methodology to a real dataset. This analysis is carried out using the computer package Mokken Scale Analysis for Polytomous Items (Molenaar and Sijtsma 2000), abbreviated as MSP. Because the theory in Chapter 3 is presented at only a heuristic level (the details coming in subsequent chapters) the reader might be left with the (not entirely incorrect) feeling that the methods are somewhat ad hoc, although effective. The remaining chapters provide an overview of the theory behind the MHM and DMM with examples (again using the MSP package). Chapter 4 discusses the basic properties, including the scalability coef cients, and Chapter 5 discusses using these basic properties to form an automated procedure for re ning a set of items into one or more appropriate scales. Chapter 6 then discusses what bene ts can be gained by using the more restrictive DMM instead of the MHM assumptions. Finally, Chapters 7 and 8 expand the earlier methodology to polytomous items. One of the topics in Chapter 7 (the ordering of latent trait by sum score) is one of a few examples in which the book’s introductory and friendly feeling may lead the reader to think that the particular issue being discussed is  rmly resolved. In several places the proposed resolutions are discussed further only in unpublished or non-English language articles. Introduction to Nonparametric Item Response Theory is an accessible introduction to constructing tests/scales using nonparametric IRT. It should be of great use to social scientists who construct their own measurement instruments, as well as those who provide them statistical support. For those who focus on larger-scale standardized tests, the volume provides a refreshing step back to look at some important issues in the  eld and some alternative methods of analysis. Finally, the MSP software designed to carry out the procedures seems to be priced comparably to other measurement packages, and is much more user-friendly (even without consulting the manual! ) than most.