Nonlinear common factor models with polynomial regression functions, including interaction terms, are fitted by simultaneously estimating the factor loadings and common factor scores, using maximum-likelihood-ratio and ordinary-least-squares methods. A Monte Carlo study gives support to a conjecture about the form of the distribution of the likelihood-ratio criterion.

Procedures for assessing the invariance of factors found in data sets using different subjects and the same variables are often using the least squares criterion, which appears to be too restrictive for comparing factors.

Tucker's coefficient of congruence, on the other hand, is more closely related to the human interpretation of factorial invariance than the least squares criterion. A method maximizing simultaneously the sum of coefficients of congruence between two matrices of factor loadings, using orthogonal rotation of one matrix is presented. As shown in examples, the sum of coefficients of congruence obtained using the presented rotation procedure is slightly higher than the sum of coefficients of congruence using Orthogonal Procrustes Rotation based on the least squares criterion.

Test scores that are not perfectly reliable cannot be strictly equated unless they are strictly parallel [Lord, 1980]. This fact implies that tau-equivalence can be lost if an equipercentile equating is applied to observed scores that are not strictly parallel. Seventy-two simulated testing conditions are produced to simulate equating tests with different difficulties and discriminations. Number-correct and trait metrics are examined. When an equipercentile equating is applied to these data, locally biased (i.e., non-tau-equivalent) results are produced for tests of unequal difficulty. Differences between the criteria of tau-equivalence and equipercentile equivalence are discussed.

Green solved the problem of least-squares estimation of several criteria subject to the constraint that the estimates have an arbitrary fixed covariance or correlation matrix. In the present paper an omission in Green's proof is discussed and resolved. Furthermore, it is shown that some recently published solutions for estimating oblique factor scores are special cases of Green's solution for the case of fixed covariance matrices.

Under conditions that are commonly satisfied in optimal scaling problems, arbitrary sets of optimal weights can be obtained by choices of generalized inverse procedures. A simple relationship holds between these and the corresponding invariant item scores. The case of optimal scaling originally treated by Guttman [1941] yields a restricted form of multicategory factor analysis. It is suggested that the invariant parameters of optimal scaling should be interpreted, according to the principles of latent trait theory, rather than the arbitrary weights.

A method for joint analysis of reaction times and same-different judgments is discussed. A set of stimuli is assumed to have some parametric representation which uniquely defines dissimilarities between the stimuli. Those dissimilarities are then related to the observed reaction times and same-different judgments through a model of psychological processes. Three representation models of dissimilarities are considered, the Minkowski power distance model, the linear model, and Tversky's feature matching model. Maximum likelihood estimation procedures are developed and implemented in the form of a FORTRAN program. An example is given to illustrate the kind of analyses that can be performed by the proposed method.

Asymptotic formulas are derived for the bias in the maximum likelihood estimators of the item parameters in the logistic item response model when examinee abilities are known. Numerical results are given for a typical verbal test for college admission.

This paper concerns ordinal responses. An ordered Dirichlet distribution describes prior and posterior beliefs about the cumulative probabilities of response categories. Associating the response categories with intervals of a latent random variable then induces a distribution on the order statistics of that variable. The psychometrician can use the asymptotic theory of order statistics to learn how distributional assumptions about the latent variable effect inference. An example relates the skewness of a latent variable to the proportional odds and proportional hazards models of McCullagh [1980].

Repeated measures designs in psychology have traditionally been analyzed by the univariate mixed model approach, in which the repeated measures effect is tested against an error term based on the subject by treatment interaction. This paper considers the extension of this analysis to designs in which the individual repeated measures are multivariate. Sufficient conditions for a valid multivariate mixed model analysis are given, and a test is described to determine whether or not given data satisfy these conditions.

A method for generating multivariate nonnormal distributions with specified intercorrelations and marginal means, variances, skews, and kurtoses is proposed. As an example, the method is applied to the generation of simulated scores on three psychological tests administered to a single group of individuals.

The weighted euclidean distances model in multidimensional scaling (WMDS) represents individual differences as dimension saliences which can be interpreted as the orientations of vectors in a subject space. It has recently been suggested that the statistics of directions would be appropriate for carrying out tests of location with such data. The nature of the directional representation in WMDS is reviewed and it is argued that since dimension saliences are almost always positive, the directional representations will usually be confined to the positive orthant. Conventional statistical techniques are appropriate to angular representations of the individual differences which will yield angles in the interval (0, 90) so long as dimension saliences are nonnegative, a restriction which can be imposed. Ordinary statistical methods are also appropriate with several linear indices which can be derived from WMDS results. Directional statistics may be applied more fruitfully to vector representations of preferences.

A theoretical model is given for dealing with omitted responses. Two special cases are investigated.

Recent studies pertaining to an extended class of matched pairs tests based on powers of ranks are discussed. Previous questions regarding the asymptotic properties for this class of tests are clarified and a generalization of this class is described. This generalization raises a previously unanticipated concern about whether or not the analytic comparisons resulting from these tests correspond with an intuitive notion of what is being compared.