Standard procedures for estimating item parameters in item response theory (IRT) ignore collateral information that may be available about examinees, such as their standing on demographic and educational variables. This paper describes circumstances under which collateral information about examinees may be used to make inferences about item parameters more precise, and circumstances under which it must be used to obtain correct inferences.

Zegers' (1986) chance-corrected coefficients of association are derived by alternative methods. A different definition of chance correction is used. It is shown that our correction and that of Zegers are identical for large samples. Three possible assumptions for the derivation of metric coefficients are examined. The first, variable reflection, formulated by Zegers and ten Berge (1985), leads to coefficients that require chance-correction. Two other assumptions, zero covariance and covariance reflection, are proposed and it is shown that the latter two assumptions lead directly to coefficients of identity and proportionality that do not require chance correction (i.e., are identical to the Zegers, 1986, corrected coefficients).

In the course of the medical program at the University of Limburg, students complete a total of 24 progress tests, consisting of items drawn from a constant itembank. A model is presented for the growth of knowledge reflected by these results. The Rasch model is used as a starting point, but both ability and difficulty parameters are taken to be random, and moreover the logistic distribution is replaced by the normal. Both individual and group abilities are estimated and explained through simple linear regression. Application to real data shows that the model fits very well.

A Bayesian approach to the testing of competing covariance structures is developed. The method provides approximate posterior probablities for each model under consideration without prior specification of individual parameter distributions. The method is based on ayesian updating using cross-validated pseudo-likelihoods. Given that the observed variables are the samefor all competing models, the approximate posterior probabilities may be obtained easily from the chi square values and other known constants, using only a hand calculator. The approach is illustrated using and example which illustrates how the prior probabilities can alter the results concerning which model specification is preferred.

A method is proposed for the detection of item bias with respect to observed or unobserved subgroups. The method uses quasi-loglinear models for the incomplete subgroup × test score × Item 1 × ... × item k contingency table. If subgroup membership is unknown the models are Haberman's incomplete-latent-class models.

The (conditional) Rasch model is formulated as a quasi-loglinear model. The parameters in this loglinear model, that correspond to the main effects of the item responses, are the conditional estimates of the parameters in the Rasch model. Item bias can then be tested by comparing the quasi-loglinear-Rasch model with models that contain parameters for the interaction of item responses and the subgroups.

Millsap and Meredith (1988) have developed a generalization of principal components analysis for the simultaneous analysis of a number of variables observed in several populations or on several occasions. The algorithm they provide has some disadvantages. The present paper offers two alternating least squares algorithms for their method, suitable for small and large data sets, respectively. Lower and upper bounds are given for the loss function to be minimized in the Millsap and Meredith method. These can serve to indicate whether or not a global optimum for the simultaneous components analysis problem has been attained.

For multiple populatios, a longtidinal factor analytic model which is entirely exploratory, that is, no explicit identification constraints, is proposed. Factorial collapse and period/practice effects are allowed. An invariant and/or stationary factor pattern is permitted. This model is formulated stochastically. To implement this model a stagewise EM algorithm is developed. Finally a numerical illustration utilizing Nesselroade and Baltes' data is presented.

This paper presents a new stochastic multidimensional scaling vector threshold model designed to analyze “pick any/n” choice data (e.g., consumers rendering buy/no buy decisions concerning a number of actual products). A maximum likelihood procedure is formulated to estimate a joint space of both individuals (represented as vectors) and stimuli (represented as points). The relevant psychometric literature concerning the spatial treatment of such binary choice data is reviewed. The nonlinear probit type model is described, as well as the conjugate gradient procedure used to estimate parameters. Results of Monte Carlo analyses investigating the performance of this methodology with synthetic choice data sets are presented. An application concerning consumer choices for eleven competitive brands of soft drinks is discussed. Finally, directions for future research are presented in terms of further applications and generalizing the model to accommodate three-way choice data.

We propose a method to reduce many categorical variables to one variable with k categories, or stated otherwise, to classify n objects into k groups. Objects are measured on a set of nominal, ordinal or numerical variables or any mix of these, and they are represented as n points in p-dimensional Euclidean space. Starting from homogeneity analysis, also called multiple correspondence analysis, the essential feature of our approach is that these object points are restricted to lie at only one of k locations. It follows that these k locations must be equal to the centroids of all objects belonging to the same group, which corresponds to a sum of squared distances clustering criterion. The problem is not only to estimate the group allocation, but also to obtain an optimal transformation of the data matrix. An alternating least squares algorithm and an example are given.

In the context of covariance structure analysis, a unified approach to the asymptotic theory of alternative test criteria for testing parametric restrictions is provided. The discussion develops within a general framework that distinguishes whether or not the fitting function is asymptotically optimal, and allows the null and alternative hypothesis to be only approximations of the true model. Also, the equivalent of the information matrix, and the asymptotic covariance matrix of the vector of summary statistics, are allowed to be singular. When the fitting function is not asymptotically optimal, test statistics which have asymptotically a chi-square distribution are developed as a natural generalization of more classical ones. Issues relevant for power analysis, and the asymptotic theory of a testing related statistic, are also investigated.

To capture the cognitive organization of a set of questions or problems pertaining to a body of information, Doignon and Falmagne have proposed, and analyzed in a number of papers, the concept of a knowledge space, that is, a distinguished collection of subsets of questions, representing the possible knowledge states. This collection of sets is assumed to satisfy a number of conditions. Since this concept is a deterministic one, the problem of empirical testing arises. A stochastic version of a knowledge space is developed in this paper, in which the knowledge states are considered as possible epochs in a subject's learning history. The knowledge space is decomposed as a union of a number of possible learning paths, called gradations. The model specifies how a subject is channelled through and progresses along a gradation. A probabilistic axiom of the “local indepencence” type relates the knowledge states to the observable responses. The predictions of this model are worked out in details in the case of parametric assumptions involving gamma distributions. An application of the model to artificial data is described, based on maximum likelihood methods. The statistical analysis is shown to be capable of revealing the combinatoric core of the model.

We propose a method for detecting influential observations in iterative principal factor analysis. For this purpose we derive the influence functions I(x; LLT) and I(x; Δ) for the common variance matrix T =LLT and the unique variance matrix Δ, respectively, in the common factor decomposition Σ =LLT + Δ. A numerical example is given for illustration.

In very simple test theory models such as the Rasch model, a single parameter is used to represent the ability of any examinee or the difficulty of any item. Simple models such as these provide very important points of departure for more detailed modeling when a substantial amount of data are available, and are themselves of real practical value for small or even medium samples. They can also serve a normative role in test design.

As an alternative to the Rasch model, or the Rasch model with a correction for guessing, a simple model is introduced which characterizes strength of response in terms of the ratio of ability and difficulty parameters rather than their difference. This model provides a natural account of guessing, and has other useful things to contribute as well. It also offers an alternative to the Rasch model with the usual correction for guessing. The three models are compared in terms of statistical properties and fits to actual data. The goal of the paper is to widen the range of “minimal” models available to test analysts.

Recently several new attempts have been made to find a robust method for comparing the variances of J dependent random variables. However, empirical studies have shown that all of these procedures can give unsatisfactory results. This paper examines several new procedures that are derived heuristically. One of these procedures was found to perform better than all of the robust procedures studied here, and so it is recommended for general use.

In many regression applications, users are often faced with difficulties due to nonlinear relationships, heterogeneous subjects, or time series which are best represented by splines. In such applications, two or more regression functions are often necessary to best summarize the underlying structure of the data. Unfortunately, in most cases, it is not known a priori which subset of observations should be approximated with which specific regression function. This paper presents a methodology which simultaneously clusters observations into a preset number of groups and estimates the corresponding regression functions' coefficients, all to optimize a common objective function. We describe the problem and discuss related procedures. A new simulated annealing-based methodology is described as well as program options to accommodate overlapping or nonoverlapping clustering, replications per subject, univariate or multivariate dependent variables, and constraints imposed on cluster membership. Extensive Monte Carlo analyses are reported which investigate the overall performance of the methodology. A consumer psychology application is provided concerning a conjoint analysis investigation of consumer satisfaction determinants. Finally, other applications and extensions of the methodology are discussed.

This research note proposes two reliability coefficients for tests with components of unknown functional lengths. The derived coefficients are extensions of the techniques devised by Kristof and Feldt and do not require a reduction of test components into parts. Simulation study indicates that the new coefficients yield reasonably stable reliability estimates when the number of test components is small.

A method for multidimensional scaling that is highly resistant to the effects of outliers is described. To illustrate the efficacy of the procedure, some Monte Carlo simulation results are presented. The method is shown to perform well when outliers are present, even in relatively large numbers, and also to perform comparably to other approaches when no outliers are present.

Jansen and Roskam (1986) discussed the compatibility of the unidimensional polytomous Rasch model with dichotomization of the response continuum. They derived a rather strict condition in which dichotomization of multicategory data that fit the unidimensional polytomous Rasch model, results in dichotomous data which fit the dichotomous Research model with effectively the same subject parameter. In this paper a more general dichotomization condition is derived for the polytomous Rasch model, which appears less restrictive, but upholds that the intrinsic logic of the unidimensional polytomous Rasch model defies dichotomization in general. The robustness of dichotomous analysis investigated in a simulation study. It shows a close relation with the two-parameters (Birnbaum) model. Theoretical and methodological implications are discussed.

Ordinal network representations are graph-theoretic representations of proximity data. They seek to provide parsimonious representations of the ordinal (nonmetric) information in observed proximity data by means of the minimum-path-length distance of a connected and weighted graph. In contrast to traditional tree-based graph-theoretic approaches, ordinal network representation is not limited to but includes the representation by trees. Asymmetry in the proximity data and violations of zero-minimality are allowed for. The paper explores fundamental representation and uniqueness results and discusses a method of constructing ordinal network representations. A simple strategy for handling the problem of errors in the data is described and illustrated.