Rasch proposed an exact conditional inference approach to testing his model but never implemented it because it involves the calculation of a complicated probability. This paper furthers Rasch’s approach by (1) providing an efficient Monte Carlo methodology for accurately approximating the required probability and (2) illustrating the usefulness of Rasch’s approach for several important testing problems through simulation studies. Our Monte Carlo methodology is shown to compare favorably to other Monte Carlo methods proposed for this problem in two respects: it is considerably faster and it provides more reliable estimates of the Monte Carlo standard error.

The assumptions underlying item response theory (IRT) models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model. It is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which, however, are necessary but not sufficient. In this paper, we describe how the theory for likelihood-based inference under equality and inequality constraints may be used to test the underlying assumptions of IRT models. It turns out that the analysis based directly on the latent structure is simpler and more flexible. In particular, we indicate how several interesting extensions of the Rasch model may be obtained by partial relaxation of the basic constraints. An application to a data set provided by Educational Testing Service is used to illustrate the approach.

In this paper, we reconsider the merits of unfolding solutions based on loss functions involving a normalization on the variance per subject. In the literature, solutions based on Stress-2 are often diagnosed to be degenerate in the majority of cases. Here, the focus lies on two frequently occurring types of degeneracies. The first type typically locates some subject points far away from a compact cluster of the other points. In the second type of solution, the object points lie on a circle. In this paper, we argue that these degenerate solutions are well fitting and informative. To reveal the information, we introduce mixtures of plots based on the ideal point model of unfolding, the vector model, and on the signed distance model. In addition to a different representation, we provide a new iterative majorization algorithm to optimize the average squared correlation between the distances in the configuration and the transformed data per individual. It is shown that this approach is equivalent to minimizing Kruskal’s Stress-2.

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.

Although RC(M)-association models have become a generally useful tool for the analysis of cross-classified data, the graphical representation resulting from such an analysis can at times be misleading. The relationships present between row category points and column category points cannot be interpreted by inter point distances but only through projection. In order to avoid incorrect interpretation by a distance rule, joint plots should be made that either represent the row categories or the column categories as vectors. In contrast, the present study proposes models in which the distances between row and column points can be interpreted directly, with a large (small) distance corresponding to a small (large) value for the association. The models provide expressions for the odds ratios in terms of distances, which is a feature that makes the proposed models attractive reparametrizations to the usual RC(M)-parametrization. Comparisons to existing data analysis techniques plus an overview of related models and their connections are also provided.

We make theoretical comparisons among five coefficients—Cronbach’s α, Revelle’s β, McDonald’s ωh, and two alternative conceptualizations of reliability. Though many end users and psychometricians alike may not distinguish among these five coefficients, we demonstrate formally their nonequivalence. Specifically, whereas there are conditions under which α, β, and ωh are equivalent to each other and to one of the two conceptualizations of reliability considered here, we show that equality with this conceptualization of reliability and between α and ωh holds only under a highly restrictive set of conditions and that the conditions under which β equals ωh are only somewhat more general. The nonequivalence of α, β, and ωh suggests that important information about the psychometric properties of a scale may be missing when scale developers and users only report α as is almost always the case.

The rater agreement literature is complicated by the fact that it must accommodate at east two different properties of rating data: the number of raters (two versus more than two) and the rating scale level (nominal versus metric). While kappa statistics are most widely used for nominal scales, intraclass correlation coefficients have been preferred for metric scales. In this paper, we suggest a dispersion-weighted kappa framework for multiple raters that integrates some important agreement statistics by using familiar dispersion indices as weights for expressing disagreement. These weights are applied to ratings identifying cells in the traditional inter-judge contingency table. Novel agreement statistics can be obtained by applying less familiar indices of dispersion in the same way.

Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures that take into account the dependence of such observations have been developed. Among procedures for 2-level covariance structure analysis, Muthén’s maximum likelihood (MUML) has the advantage of easier computation and faster convergence. When data are balanced, MUML is equivalent to the maximum likelihood procedure. Simulation results in the literature endorse the MUML procedure also for unbalanced data. This paper studies the analytical properties of the MUML procedure in general. The results indicate that the MUML procedure leads to correct model inference asymptotically when level-2 sample size goes to infinity and the coefficient of variation of the level-1 sample sizes goes to zero. The study clearly identifies the impact of level-1 and level-2 sample sizes on the standard errors and test statistic of the MUML procedure. Analytical results explain previous simulation results and will guide the design or data collection for the future applications of MUML.

Fully and partially ranked data arise in a variety of contexts. From a Bayesian perspective, attention has focused on distance-based models; in particular, the Mallows model and extensions thereof. In this paper, a class of prior distributions, the Binary Tree, is developed on the symmetric group. The attractive features of the class are: it provides a closed-form solution to the posterior distribution; and a simple way to interpret the parameters of the prior distribution. The advantages of the proposed method are illustrated by comparing it to metric-based models using data analyzed by other researchers in this context.

The collection of repeated measures in psychological research is one of the most common data collection formats employed in survey and experimental research. The behavioral decision theory literature documents the existence of the dynamic evolution of preferences that occur over time and experience due to learning, exposure to additional information, fatigue, cognitive storage limitations, etc. We introduce a Bayesian dynamic linear methodology employing an empirical Bayes estimation framework that permits the detection and modeling of such potential changes to the underlying preference utility structure of the respondent. An illustration of revealed stated preference analysis (i.e., conjoint analysis) is given involving students’ preferences for apartments and their underlying attributes and features. We also present the results of several simulations demonstrating the ability of the proposed procedure to recover a variety of different sources of dynamics that may surface with preference elicitation over repeated sequential measurement. Finally, directions for future research are discussed.

In the signal detection paradigm, the non-parametric index of sensitivity A′, as first introduced by Pollack and Norman (1964), is a popular alternative to the more traditional d′ measure of sensitivity. Smith (1995) clarified a confusion about the interpretation of A′ in relation to the area beneath proper receiver operating characteristic (ROC) curves, and provided a formula (which he called A′′) for this commonly held interpretation. However, he made an error in his calculations. Here, we rectify this error by providing the correct formula (which we call A) and compare the discrepancy that would have resulted. The corresponding measure for bias b is also provided. Since all such calculations apply to “proper” ROC curves with non-decreasing slopes, we also prove, as a separate result, the slope-monotonicity of ROC curves generated by likelihood-ratio criterion.

The purpose of this note is twofold: (a) to present the formula for the item information function (IIF) in any direction for the Multidimensional 3-Parameter Logistic (M3-PL) model and (b) to give the equation for the location of maximum item information (θmax) in the direction of the item discrimination vector. Several corollaries are given. Implications for future research are discussed.