The random effects ANOVA model plays an important role in many psychological studies, but the usual model suffers from at least two serious problems. The first is that even under normality, violating the assumption of equal variances can have serious consequences in terms of Type I errors or significance levels, and it can affect power as well. The second and perhaps more serious concern is that even slight departures from normality can result in a substantial loss of power when testing hypotheses. Jeyaratnam and Othman (1985) proposed a method for handling unequal variances, under the assumption of normality, but no results were given on how their procedure performs when distributions are nonnormal. A secondary goal in this paper is to address this issue via simulations. As will be seen, problems arise with both Type I errors and power. Another secondary goal is to provide new simulation results on the Rust-Fligner modification of the Kruskal-Wallis test. The primary goal is to propose a generalization of the usual random effects model based on trimmed means. The resulting test of no differences among J randomly sampled groups has certain advantages in terms of Type I errors, and it can yield substantial gains in power when distributions have heavy tails and outliers. This last feature is very important in applied work because recent investigations indicate that heavy-tailed distributions are common. Included is a suggestion for a heteroscedastic Winsorized analog of the usual intraclass correlation coefficient.

The test information function serves important roles in latent trait models and in their applications. Among others, it has been used as the measure of accuracy in ability estimation. A question arises, however, if the test information function is accurate enough for all meaningful levels of ability relative to the test, especially when the number of test items is relatively small (e.g., less than 50). In the present paper, using the constant information model and constant amounts of test information for a finite interval of ability, simulated data were produced for eight different levels of ability and for twenty different numbers of test items ranging between 10 and 200. Analyses of these data suggest that it is desirable to consider some modification of the test information function when it is used as the measure of accuracy in ability estimation.

We introduce two simple empirical approximate Bayes estimators (EABEs)— \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}_N (x)$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta _N (x)$$\end{document}—for estimating domain scores under binomial and hypergeometric distributions, respectively. Both EABEs (derived from corresponding marginal distributions of observed test score x without relying on knowledge of prior domain score distributions) have been proven to hold Δ-asymptotic optimality in Robbins' sense of convergence in mean. We found that, where \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} are the monotonized versions of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}_N$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta _N$$\end{document} under Van Houwelingen's monotonization method, respectively, the convergence rate of the overall expected loss of Bayes risk in either \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} or \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} depends on test length, sample size, and ratio of test length to size of domain items. In terms of conditional Bayes risk, \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} outperform their maximum likelihood counterparts over the middle range of domain scales. In terms of mean-squared error, we also found that: (a) given a unimodal prior distribution of domain scores, \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} performs better than both \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} and a linear EBE of the beta-binomial model when domain item size is small or when test items reflect a high degree of heterogeneity; (b) \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde{d}^* _N$$\end{document} performs as well as \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widetilde\delta ^* _N$$\end{document} when prior distribution is bimodal and test items are homogeneous; and (c) the linear EBE is extremely robust when a large pool of homogeneous items plus a unimodal prior distribution exists.

An asymptotic expression for the reliability of a linearly equated test is developed using normal theory. The reliability is expressed as the product of two terms, the reliability of the test before equating, and an adjustment term. This adjustment term is a function of the sample sizes used to estimate the linear equating transformation. The results of a simulation study indicate close agreement between the theoretical and simulated reliability values for samples greater than 200. Findings demonstrate that samples as small as 300 can be used in linear equating without an appreciable decrease in reliability.

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of order n × n as a linear combination of 1/2n(n + 1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetric n × n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n + 1) fixed binary matrices of rank one. The decomposition implies that an n × n × p array consisting of p symmetric n × n slabs has maximal rank 1/2n(n + 1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n + 1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n − 1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large as p, the number of matrices analyzed.

A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds. Computational aspects are also discussed.

The item response function (IRF) for a polytomously scored item is defined as a weighted sum of the item category response functions (ICRF, the probability of getting a particular score for a randomly sampled examinee of ability θ). This paper establishes the correspondence between an IRF and a unique set of ICRFs for two of the most commonly used polytomous IRT models (the partial credit models and the graded response model). Specifically, a proof of the following assertion is provided for these models: If two items have the same IRF, then they must have the same number of categories; moreover, they must consist of the same ICRFs. As a corollary, for the Rasch dichotomous model, if two tests have the same test characteristic function (TCF), then they must have the same number of items. Moreover, for each item in one of the tests, an item in the other test with an identical IRF must exist. Theoretical as well as practical implications of these results are discussed.

Hierarchical Bayes procedures for the two-parameter logistic item response model were compared for estimating item and ability parameters. Simulated data sets were analyzed via two joint and two marginal Bayesian estimation procedures. The marginal Bayesian estimation procedures yielded consistently smaller root mean square differences than the joint Bayesian estimation procedures for item and ability estimates. As the sample size and test length increased, the four Bayes procedures yielded essentially the same result.