A loglinear IRT model is proposed that relates polytomously scored item responses to a multidimensional latent space. The analyst may specify a response function for each response, indicating which latent abilities are necessary to arrive at that response. Each item may have a different number of response categories, so that free response items are more easily analyzed. Conditional maximum likelihood estimates are derived and the models may be tested generally or against alternative loglinear IRT models.

The partial credit model is considered under the assumption of a certain linear decomposition of the item × category parameters δih into “basic parameters” αj. This model is referred to as the “linear partial credit model”. A conditional maximum likelihood algorithm for estimation of the αj is presented, based on (a) recurrences for the combinatorial functions involved, and (b) using a “quasi-Newton” approach, the so-called Broyden-Fletcher-Goldfarb-Shanno (BFGS) method; (a) guarantees numerically stable results, (b) avoids the direct computation of the Hesse matrix, yet produces a sequence of certain positive definite matrices Bk, k = 1, 2, ..., converging to the asymptotic variance-covariance matrix of the \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat \alpha _j $$\end{document}. The practicality of these numerical methods is demonstrated both by means of simulations and of an empirical application to the measurement of treatment effects in patients with psychosomatic disorders.

Kroonenberg and de Leeuw have suggested fitting the IDIOSCAL model by the TUCKALS2 algorithm for three-way components analysis. In theory, this is problematic because TUCKALS2 produces two possibly different coordinate matrices, that are useless for IDIOSCAL unless they are equal. Kroonenberg has claimed that, when IDIOSCAL is fitted by TUCKALS2, the resulting coordinate matrices will be identical. In the present paper, this claim is proven valid when the data matrices are semidefinite. However, counterexamples for indefinite matrices are also constructed, by examining the global minimum in the case where the data matrices have the same eigenvectors. Similar counterexamples have been considered by ten Berge and Kiers in the related context of CANDECOMP/PARAFAC to fit the INDSCAL model.

We provide a unified, theoretical basis on which measures of data reliability may be derived or evaluated, for both quantitative and qualitative data. This approach evaluates reliability as the “proportional reduction in loss” (PRL) that is attained in a sample by an optimal estimator. The resulting measure is between 0 and 1, linearly related to expected loss, and provides a direct way of contrasting the measured reliability in the sample with the least reliable and most reliable data-generating cases. The PRL measure is a generalization of many of the commonly-used reliability measures.

We show how the quantitative measures from generalizability theory can be derived as PRL measures (including Cronbach's alpha and measures proposed by Winer). For categorical data, we develop a new measure for the general case in which each of N judges assigns a subject to one of K categories and show that it is equivalent to a measure proposed by Perreault and Leigh for the case where N is 2.

In this article, some additive models of behavioral measures are defined, and distributional tests of them are proposed. Major theoretical results include (a) conditions for additivity of components to predict the highest level of dominance in a model-free stochastic dominance hierarchy, (b) methods of identifying the shape of the hazard rate function of an added component from certain relationships among the observable density and distribution functions, and (c) effects of stochastic dependence between components on the distributional tests. Feasibility and usefulness of the methods are demonstrated by application to choice RT and ratings experiments.

In a recent article, Fagot proposed a generalized family of coefficients of relational agreement for multiple judges, focusing on the concept of empirically meaningful relationships. In this paper an ordinal coefficient of relational agreement, based on ranking data, is presented as a special case of the generalized family. It is shown that the proposed ordinal coefficient encompasses other ordinal coefficients, such as the Kendall coefficient of concordance, the average Spearman rank-order coefficient, and intraclass correlation based on ranks. It is also shown that the Kendall coefficient of concordance, corrected for chance agreement, is equivalent to the ordinal coefficient proposed in this paper.

The use of p-values in combining the results of independent studies often involves studies that are potentially aberrant either in quality or in actual values. A robust data analysis suggests the use of a statistic that takes these aberrations into account by trimming some of the largest and smallest p-values. We present a trimmed statistic based on an inverse cumulative normal transformation of the ordered p-values, together with a simple and convenient method for approximating the distribution and first two moments of this statistic.

De Vries (1993) discusses Pearson's product-moment correlation, Spearman's rank correlation, and Kendall's rank-correlation coefficient for assessing the association between the rows of two proximity matrices. For each of these he introduces a weighted average variant and a rowwise variant. In this note it is shown that for all three types, the absolute value of the first variant is greater than or equal to the absolute value of the second.