Categories can be counted, rated, or ranked, but they cannot be measured. Likewise, persons or individuals can be counted, rated, or ranked, but they cannot be measured either. Nevertheless, psychology has realized early on that it can take an indirect road to measurement: What can be measured is the strength of association between categories in samples or populations, and what can be quantitatively compared are counts, ratings, or rankings made under different circumstances, or originating from different persons. The strong demand for quantitative analysis of categorical data has thus created a variety of statistical methods, with substantial contributions from psychometrics and sociometrics. What is the common basis of these methods dealing with categories? The basic element they share is that the sample space has a special geometry, in which categories (or persons) are point masses forming a simplex, while distributions of counts or profiles of ratings are centers of gravity, which are also point masses. Rankings form a discrete subset in the interior of the simplex, known as the permutation polytope, and paired comparisons form another subset on the edges of the simplex. Distances between point masses form the basic tool of analysis. The paper gives some history of major concepts, which naturally leads to a new concept: the shadow point. It is then shown how loglinear models, Luce and Rasch models, unfolding models, correspondence analysis and homogeneity analysis, forced classification and classification trees, as well as other models and methods, fit into this particular geometrical framework.

This manuscript introduces a new Bayesian finite mixture methodology for the joint clustering of row and column stimuli/objects associated with two-mode asymmetric proximity, dominance, or profile data. That is, common clusters are derived which partition both the row and column stimuli/objects simultaneously into the same derived set of clusters. In this manner, interrelationships between both sets of entities (rows and columns) are easily ascertained. We describe the technical details of the proposed two-mode clustering methodology including its Bayesian mixture formulation and a Bayes factor heuristic for model selection. We present a modest Monte Carlo analysis to investigate the performance of the proposed Bayesian two-mode clustering procedure with respect to synthetically created data whose structure and parameters are known. Next, a consumer psychology application is provided examining physician pharmaceutical prescription behavior for various brands of prescription drugs in the neuroscience health market. We conclude by discussing several fertile areas for future research.

By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis are computed via observations of the latent variables generated by the Metropolis-Hastings algorithm, while the diagnostic measures are obtained via the conformal normal curvature. Seven perturbation schemes, including some perturbation schemes on latent vectors, are investigated. The proposed procedure is illustrated by a simulation study and a real example.

The purpose of this study was to investigate and compare the performance of a stepwise variable selection algorithm to traditional exploratory factor analysis. The Monte Carlo study included six factors in the design; the number of common factors; the number of variables explained by the common factors; the magnitude of factor loadings; the number of variables not explained by the common factors; the type of anomaly evidenced by the poorly explained variables; and sample size. The performance of the methods was evaluated in terms of selection and pattern accuracy, and bias and root mean squared error of the structure coefficients. Results indicate that the stepwise algorithm was generally ineffective at excluding anomalous variables from the factor model. The poor selection accuracy of the stepwise approach suggests that it should be avoided.

To assess the reliability of congeneric tests, specifically designed reliability measures have been proposed. This paper emphasizes that such measures rely on a unidimensionality hypothesis, which can neither be confirmed nor rejected when there are only three test parts, and will invariably be rejected when there are more than three test parts. Jackson and Agunwamba's (1977) greatest lower bound to reliability is proposed instead. Although this bound has a reputation for overestimating the population value when the sample size is small, this is no reason to prefer the unidimensionality-based reliability. Firstly, the sampling bias problem of the glb does not play a role when the number of test parts is small, as is often the case with congeneric measures. Secondly, glb and unidimensionality based reliability are often equal when there are three test parts, and when there are more test parts, their numerical values are still very similar. To the extent that the bias problem of the greatest lower bound does play a role, unidimensionality-based reliability is equally affected. Although unidimensionality and reliability are often thought of as unrelated, this paper shows that, from at least two perspectives, they act as antagonistic concepts. A measure, based on the same framework that led to the greatest lower bound, is discussed for assessing how close is a set of variables to unidimensionality. It is the percentage of common variance that can be explained by a single factor. An empirical example is given to demonstrate the main points of the paper.

It is shown that in the context of the Model with Internal Restrictions on the Item Difficulties (MIRID), different componential theories about an item set may lead to equivalent models. Furthermore, we provide conditions for the identifiability of the MIRID model parameters, and it will be shown how the MIRID model relates to the Linear Logistic Test Model (LLTM). While it is known that the LLTM is a special case of the MIRID, we show that it is possible to construct an LLTM that encompasses the MIRID. The MIRID model places a bilinear restriction on the item parameters of the Rasch model. It is explained how this fact is used to simplify the results of Bechger, Verhelst, and Verstralen (2001) and Bechger, Verstralen, and Verhelst (2002), and extend their scope to a wider class of models.

This paper presents a row-column (RC) association model in which the estimated row and column scores are forced to be in agreement with an a priori specified ordering. Two efficient algorithms for finding the order-restricted maximum likelihood (ML) estimates are proposed and their reliability under different degrees of association is investigated by a simulation study. We propose testing order-restricted RC models using a parametric bootstrap procedure, which turns out to yield reliable p values, except for situations in which the association between the two variables is very weak. The use of order-restricted RC models is illustrated by means of an empirical example.

In many instances it is reasonable to assume that the population covariance matrix has positive elements. This assumption implies for the single factor analysis model that the loadings and regression weights for best linear factor prediction are positive. For the multiple factor analysis model where each variable loads on a single factor and a hierarchical factor model, it implies that the loadings and the factor correlations are positive. For the latter model it also implies that the regression weights for first- and second-order factor prediction are positive.