Interpersonally incomparable responses pose a significant problem for survey researchers. If the manifest responses of individuals differ from their underlying true responses by monotonic transformations which vary from person to person, then the covariances of the manifest responses tools such as factor analysis may yield incorrect results. Satisfactory results for interpersonally incomparable ordinal responses can be obtained by assuming that rankings are based upon a set of multivariate normal latent variables which satisfy the factor or ideal point models of choice. Two statistical methods based upon these assumptions are described; their statistical properties are explored; and their computational feasibility is demonstrated in some simulations. We conclude that is possible to develop methods for factor and ideal point analysis of interpersonally incomparable ordinal data.

This paper studies the problem of scaling ordinal categorical data observed over two or more sets of categories measuring a single characteristic. Scaling is obtained by solving a constrained entropy model which finds the most probable values of the scales given the data. A Kullback-Leibler statistic is generated which operationalizes a measure for the strength of consistency among the sets of categories. A variety of data of two and three sets of categories are analyzed using the entropy approach.

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the “standard” (Two-Way) MDS model. In this Extended Two-Way Euclidean model the n stimuli (or other objects) are assumed to be characterized by coordinates on R common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between object i and object j then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity, sj, can be thought of as the sum of squares of coordinates on those dimensions specific to object i, all of which have nonzero coordinates only for object i. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)

We further assume that δij =F(dij) +eij where δij is the proximity value (e.g., similarity or dissimilarity) of objects i and j, dij is the extended Euclidean distance defined above, while eij is an error term assumed i.i.d. N(0, σ2). F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition to R common) dimensions.

This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.

Centering a matrix row-wise and rescaling it column-wise to a unit sum of squares requires an iterative procedure. It is shown that this procedure converges to a stable solution. This solution need not be centered row-wise if the limiting point of the interations is a matrix of rank one. The results of the present paper bear directly on several types of preprocessing methods in Parafac/Candecomp.

A maximin model for IRT-based test design is proposed. In the model only the relative shape of the target test information function is specified. It serves as a constraint subject to which a linear programming algorithm maximizes the information in the test. In the practice of test construction, several demands as linear constraints in the model. A worked example of a text construction problem with practical constraints is presented. The paper concludes with a discussion of some alternative models of test construction.

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.

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.

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.

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.

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.

Harshman's DEDICOM model providesa framework for analyzing square but asymmetric materices of directional relationships among n objects or persons in terms of a small number of components. One version of DEDICOM ignores the diagonal entries of the matrices. A straight-forward computational solution for this model is offered in the present paper. The solution can be interpreted as a generalized Minres procedure suitable for handing asymmetric matrices.

In FACTALS an alternating least squares algorithm is utilized. Mooijaart (1984) has shown that this algorithm is based upon an erroneous assumption. This paper gives a proper solution for the loss function used in FACTALS.

An algorithm is presented for computing a test statistic for comparing medians based on a small-sample size estimate of the variance of the usual sample median.