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.

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters θ= (θ1, …, θn) and item parameters β=(β1, …, βk), where βj may be vector valued. With θ considered a random sample from a prior distribution with parameter ϕ, the estimation of (θ, β) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.

The separate questions on an essay test or the individual judges on a rater panel may constitute congeneric parts rather than tau-equivalent parts. Also, it may be necessary to infer the lengths of the congeneric parts from their variances and covariances, rather than from some obvious feature of each part, such as the range of possible scores. Cronbach's alpha coefficient applied to such part-tests data will underestimate total score reliability. Several reliability coefficients are developed for such instruments. They may be regarded as extensions of the coefficient developed by Kristof for a three-part test.

Factor analysis programs in SAS, BMDP, and SPSS are discussed and compared in terms of documentation, methods and options available, internal logic, computational accuracy, and results provided. Some problems with respect to logic and output are described. Based on these comparisons, recommendations are offered which include a clear overall preference for SAS, and advice against general use of SPSS for factor analysis.

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.

Consider a set of data consisting of measurements of n objects with respect to p variables displayed in an n × p matrix. A monotone transformation of the values in each column, represented as a linear combination of integrated basis splines, is assumed determined by a linear combination of a new set of values characterizing each row object. Two different models are used: one, an Eckart-Young decomposition model, and the other, a multivariate normal model. Examples for artificial and real data are presented. The results indicate that both methods are helpful in choosing dimensionality and that the Eckart-Young model is also helpful in displaying the relationships among the objects and the variables. Also, results suggest that the resulting transformations are themselves illuminating.

The problem of inferring the validity of a selection test (x) as a predictor of some criterion (y) when complete xy data are not available is investigated. The basic approach is to construct the predictive probability distribution of the unobserved y scores and then derive interval estimates of the least squares regression weights, the difference in average y scores for selected and unselected cases, and the residual variance in predicting y from x. Further, an approximation to the predictive distribution of the squared correlation between x and y in a future group is derived.

Given known item parameters, unbiased estimators are derived i) for an examinee's ability parameter θ and for his proportion-correct true score ς, ii) for the variances of θ and ς across examinees in the group tested, and iii) for the parallel-forms reliability of the maximum likelihood estimator\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat \theta$$\end{document}.

Rubin and Thayer recently presented equations to implement maximum likelihood (ML) estimation in factor analysis via the EM algorithm. They present an example to demonstrate the efficacy of the algorithm, and propose that their recovery of multiple local maxima of the ML function “certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.” It is shown here, in contrast, that these second derivatives verify that Rubin and Thayer did not find multiple local maxima as claimed. The only known maximum remains the one found by Jöreskog over a decade earlier. The standard errors obtained from the second derivatives and the Fisher information matrix thus remain appropriate where ML assumptions are met. The advantages of the EM algorithm over other algorithms for ML factor analysis remain to be demonstrated.

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].

The problem of characterizing the manifest probabilities of a latent trait model is considered. The item characteristic curve is transformed to the item passing-odds curve and a corresponding transformation is made on the distribution of ability. This results in a useful expression for the manifest probabilities of any latent trait model. The result is then applied to give a characterization of the Rasch model as a log-linear model for a 2J- contingency table. Partial results are also obtained for other models. The question of the identifiability of “guessing” parameters is also discussed.

A new algorithm to obtain the least-squares or MINRES solution in common factor analysis is presented. It is based on the up-and-down Marquardt algorithm developed by the present authors for a general nonlinear least-squares problem. Experiments with some numerical models and some empirical data sets showed that the algorithm worked nicely and that SMC (Squared Multiple Correlation) performed best among four sets of initial values for common variances but that the solution might sometimes be very sensitive to fluctuations in the sample covariance matrix.

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 logit vector model and a logit ideal point model are presented for external analysis of paired comparison preference judgments aggregated over a homogeneous group. The logit vector model is hierarchically nested within the logit ideal point model so that statistical tests are available to distinguish between these two models. Generalized least squares estimation procedures are developed to account for heteroscedastic sampling error variances and specification error variances. Two numerical illustrations deal with judgments concerning employee compensation plans and preferences for salt and sugar in the brine of canned green beans.

We address several issues that are raised by Bentler and Tanaka's [1983] discussion of Rubin and Thayer [1982]. Our conclusions are: standard methods do not completely monitor the possible existence of multiple local maxima; summarizing inferential precision by the standard output based on second derivatives of the log likelihood at a maximum can be inappropriate, even if there exists a unique local maximum; EM and LISREL can be viewed as complementary, albeit not entirely adequate, tools for factor analysis.

Interpretations regarding the effects of exogenous and endogenous variables on endogenous variables in linear structural equation systems depend upon the convergence of a matrix power series. Convergence depends upon the eigenvalues of the structural coefficient matrix. The test for convergence developed by Jöreskog and Sörbom is shown to be only sufficient, not necessary and sufficient.

A least squares algorithm for fitting additive trees to proximity data is described. The algorithm uses a penalty function to enforce the four point condition on the estimated path length distances. The algorithm is evaluated in a small Monte Carlo study. Finally, an illustrative application is presented.

The well-known Rasch model is generalized to a multicomponent model, so that observations of component events are not needed to apply the model. It is shown that the generalized model has retained the property of the specific objectivity of the Rasch model. For a restricted variant of the model, maximum likelihood estimates of its parameters and a statistical test of the model are given. The results of an application to a mathematics test involving six components are described.

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.