How critical is the concept of the latent trait to modern test theory ? The appeal to some unobservable characteristic modulating response probability can lead to some confusion and misunderstanding among users of psychometric technology. This paper looks at a geometric formulation of item response theory that avoids the need to appeal to unobservables. It draws on concepts in differential geometry to represent the trait being measured as a differentiable manifold within the space of possible joint item response probabilities given conditional independence. The result is a manifest and in principle observable representation of the trait that is invariant under one-to-one transformations of trait scores. These concepts are illustrated by analyses of an actual test.

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In this paper, mathematical modeling is treated as distinct from curve fitting. Considerations of psychological reality behind our data are emphasized, and criteria such as additivity in a model, its natural generalization to a continuous response mode], satisfaction of the unique maximum condition and orderliness of the modal points of the operating characteristics of the ordered polychotomous responses are proposed. Strengths and weaknesses of mathematical models for ordered polychotomous responses that include the normal ogive model, the logistic model, the acceleration model and the family of ordered polychotomous models developed from Bock's nominal model are observed and discussed in terms of such criteria. It was concluded that it will be better to leave Bock's model as a nominal model as he intended it to be, without expanding it to ordered polychotomous models.

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Three closely related latent space dimensionality assessment procedures are surveyed. They are DIMTEST (called POLY-DIMTEST in the polytomous case), HCA/ PROX, and DETECT. Each procedure works for both dichotomous and polytomous scoring. These procedures form a conceptual unity because they all estimate item pair conditional covariances given latent ability. As such, they are nonparametric and weak local independence based procedures. DIMTEST assesses for dichotomously scored items whether a test's latent space is unidimensional or not. POLY-DIMTEST does the same for polytomously scored items. HCA/PROX does a hierarchical cluster analysis of items based on the Roussos dimensionality-sensitive proximity measure, thereby searching for item clusters that are dimensionally homogeneous within cluster and heterogeneous between clusters. When such approximate simple structure holds, DIMTEST is used in conjunction with HCA/PROX to identify which level of the HCA/ PROX cluster partition hierarchy best describes the multidimensional simple structure driving the data. DETECT, with the aide of Zhang's genetic algorithm based optimization procedure, counts the number of dominant dimensions, measures the size of the departure from unidimensionality, and searches for dimensionally homogeneous item clusters when approximate simple structure holds. All three procedures are described and judged effective based on a blend of simulation studies and real data analyses, using studies in the literature as well as some studies reported herein. Finally, an overview of some work in progress is given. This includes enhancements of the above procedures as well as some new procedures.

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A multidimensional model of differential item functioning (DIF) developed by Shealy and Stout is presented. It explains how individual items combine to produce differential test functioning (DTF) at the test level. Recent developments based on this approach, including development of the DIF/DTF detection procedure SIBTEST, are surveyed. The Shealy-Stout model not only offers insight into how DIF can occur, but suggests methods for investigating the root causes of DIF, which are useful to both substantive and statistical researchers of DIF. This new modeling paradigm offers the possibility of the proactive reduction of DIF in tests at the item manufacturing stage. SIBTEST is a nonparametric procedure that both tests for and estimates the amount of DIF in an item or set of items while controlling inflated Type I error by using a regression correction technique. It is shown to perform as well as and in many realistic situations better than other popular DIF assessment procedures. Recent modifications to the procedure demonstrate that it can be an effective tool for examining both crossing DIF and, through the use of kernel smoothing, local DIF. The procedure has been extended for use with tests containing polytomous items as well as tests that are intentionally multidimensional. Many real and simulated data analyses are presented.

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In this paper, the marginal posterior distributions for both of item parameters and their hyperparameters of the item response model were derived numerically using the Gibbs Sampler. In order to make use of the Gibbs Sampler, the latent variables were introduced and generated by pseudo random numbers. Simulaton study showed that the proposed method provided more precise estimation of item parameters. Then, this method was extended to the case where the items or the subjects are clasiified into several groups and therefore the item parameters or the ability parameters are partially exchangeable. The application of this method to the real data resulted in reasonable estimates.

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This paper is concerned with the derivation and the application of the maximization of the likelihood which is the product of binomial or multinomial densities under the linear constraints among the cell probabilities. This type of problems arises when we wish to smooth the histogram, to test the linear trend of proportions, or to compare the means/variances of number-right test scores across several groups without normality assumption. In the applications to the latent variable models, it is shown that the same method can be used to estimate the distribution of the latent variables.

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