As an extension of Lastovicka's four-mode components analysis an n-mode components analysis is developed. Using a convenient notation, both a canonical and a least squares solution are derived. The relation between both solutions and their computational aspects are discussed.

In this—partly—expository paper the parameter identifiability and estimation of a general dynamic structural model under indirect observation will be considered from a system theoretic perspective. The general dynamic model covers (dynamic) factor analytic models, (dynamic) MIMIC models and Jöreskog's linear structural model as special cases. Its reduced form is—under a slightly different specification—known in system theory and econometrics as the stochastic, stationary version of the state-space model. By using concepts and methods from system theory, such as the observability and controllability concept, the (steady-state) Kalman filter and a general nonlinear ML-estimation procedure known as prediction-error estimation the general dynamic model will be identified. It will be shown that Jöreskog's LISREL-procedure is a special case of the prediction-error estimation procedure.

The field of linear structural equation modeling with continuous variables is reviewed. Trends in psychometric theory and data analysis across the five decades of publication of Psychometrika are discussed, especially the clarification of concepts of population and sample, explication of the parametric structure of models, delineation of concepts of exploratory and confirmatory data analysis, expansion of statistical theory in psychometrics, estimation via optimization of an explicit objective function, and implementation of general function minimization methods. Developments in the ideas of factor analysis, latent variables, as well as structural and causal modeling are noted. Some major conceptual achievements involving general covariance structure representations, multiple population models, and moment structures are reviewed. The major statistical achievements of normal theory generalized least squares estimation, elliptical and distribution-free estimation, and higher-moment estimation are discussed. Computer programs that implement some of the theoretical developments are described.

A closed form estimator of the uniqueness (unique variance) in factor analysis is proposed. It has analytically desirable properties—consistency, asymptotic normality and scale invariance. The estimation procedure is given through the application to the two sets of Emmett's data and Holzinger and Swineford's data. The new estimator is shown to lead to values rather close to the maximum likelihood estimator.

Proximity data can be represented by an extended tree, which generalizes traditional trees by including marked segments that correspond to overlapping clusters. An extended tree is a graphical representation of the distinctive features model. A computer program (EXTREE) that constructs extended trees is described and applied to several sets of conceptual and perceptual proximity data.

It is shown that the scoring algorithm for maximum likelihood estimation in exploratory factor analysis can be developed in a way that is many times more efficient than a direct development based on information matrices and score vectors. The algorithm offers a simple alternative to current algorithms and when used in one-step mode provides the simplest and fastest method presently available for moving from consistent to efficient estimates. Perhaps of greater importance is its potential for extension to the confirmatory model. The algorithm is developed as a Gauss-Newton algorithm to facilitate its application to generalized least squares and to maximum likelihood estimation.

Approximately 15% of all papers appearing in Psychometrika during the period 1960–1984 have been statistical or data analytic in nature. While these papers cover a very wide range of issues and are technically quite similar to those appearing in other major statistical journals, collectively they possess features which make them identifiably Psychometrika.

A number of models for categorical item response data have been proposed in recent years. The models appear to be quite different. However, they may usefully be organized as members of only three distinct classes, within which the models are distinguished only by assumptions and constraints on their parameters. “Difference models” are appropriate for ordered responses, “divide-by-total” models may be used for either ordered or nominal responses, and “left-side added” models are used for multiple-choice responses with guessing. The details of the taxonomy and the models are described in this paper.

Progress over the past twenty-five years in the development and improvement of models for representation of similarity data is reviewed. The discussion includes comments on class, dimensional, ideal type, and dichotomous attribute representations of underlying similarity structures. Most of the theoretical research in the area appeared, and continues to appear, in Psychometrika.

Experiments are considered where each of a sample of subjects is assigned to one of C categories separately by each of a fixed or varying group of observers. Building on earlier publications, general procedures are proposed to analyze agreements and disagreements among observers. In the case of a varying group of observers, it is shown that it is not necessary to demand a constant number of observers per subject. In the case of a fixed group of observers, the problem of missing data is considered.

The procedures are illustrated within the context of two clinical diagnosis examples. In the first example it is investigated which categories are relatively hard to distinguish from one another; a new theorem is applied that shows a useful property of the statistic kappa. In the second example it is investigated if a subgroup of observers can be found with a significantly higher degree of interobserver agreement.

This paper describes a technique for principal components analysis of data consisting of n functions each observed at p argument values. This problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time. In such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful, as for example in the analysis of growth or learning curves. It is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis. The reproducing kernel for the Hilbert space of functions plays a central role, and defines the best interpolating functions, which are generalized spline functions. An example is offered of how sensitivity to derivative information can reveal interesting aspects of the data.

Two test statistics are proposed for testing the equality of two correlated proportions when some observations are missing on both responses. The performance of these tests in terms of size and power is compared with other tests by means of Monte Carlo simulations. The proposed tests are easily computed and compare favorably with other tests.

A joint Bayesian estimation procedure for the estimation of parameters in the three-parameter logistic model is developed in this paper. Procedures for specifying prior beliefs for the parameters are given. It is shown through simulation studies that the Bayesian procedure (i) ensures that the estimates stay in the parameter space, and (ii) produces better estimates than the joint maximum likelihood procedure as judged by such criteria as mean squared differences between estimates and true values.

Mathematical models have been represented in Psychometrika first in biophysics, then in stochastic models of behavior, and later in computer simulations of behavior. Other specialized journals now attract many articles in these areas, but Psychometrika's policy of broad interest in all aspects of quantitative psychology remains sound. This paper reviews Psychometrika's coverage of mathematical models and computation, and discusses some matters of editorial policy.

It is commonly held that even where questionnaire response is poor, correlational studies are affected only by loss of degrees of freedom or precision. We show that this supposition is not true. If the decision to respond is correlated with a substantive variable of interest, then regression or analysis of variance methods based upon the questionnaire results may be adversely affected by self-selection bias. Moreover such bias may arise even where response is 100%. The problem in both cases arises where selection information is passed to the score indirectly via the disturbance or individual effects, rather than entirely via the observable explanatory variables. We suggest tests for the ensuing self-selection bias and possible ways of handling the ensuing problems of inference.

A binomial test for hierarchical dependency is presented. The null hypothesis is that all members of a population who possess a certain skill are a subset of the members who possess another skill. This hypothesis is basic to the writings of several prominent theorists, such as Gagné and Piaget. The model, assumptions, formula derivations, and procedures for the test are explained. An illustrative example is also provided.

The present note contains a completing-the-squares type approach to the varimax rotation problem. This approach yields a direct proof of global optimality of a solution for optimal rotation in a plane. Because varimax rotation can be interpreted as diagonalization of a set of symmetric matrices, the present solution also applies to the diagonalization problem.

Under the assumption of normality, a formula is derived for the reliability of the maximum score. It is shown that the maximum score is more reliable than each of the single observations, but less reliable than their composite score.

This paper discusses the compatibility of the polychotomous Rasch model with dichotomization of the response continuum. It is argued that in the case of graded responses, the response categories presented to the subject are essentially an arbitrary polychotomization of the response continuum, ranging for example from total rejection or disagreement to total acceptance or agreement of an item or statement. Because of this arbitrariness, the measurement outcome should be independent of the specific polychotomization applied, for example, presenting a specific multicategory response format should not affect the measurement outcome. When such is the case, the original polychotomous model is called “compatible” with dichotomization.

A distinction is made between polychotomization or dichotomization “before the fact,” that is, in constructing the response format, and polycho- or dichotomization “after the fact,” for example in dichotomizing existing graded response data.

It is shown that, at least in case of dichotomization after-the-fact, the polychotomous Rasch model is not compatible with dichotomization, unless a rather special condition of the model parameters is met. Insofar as it may be argued that dichotomization before the fact is not essentially different from dichotomization after the fact, the value of the unidimensional polychotomous Rasch model is consequently questionable. The impact of our conclusion on related models is also discussed.

A recently suggested revision of Efron's bootstrap algorithm for small bivariate samples discards bootstrap samples having fewer than 3 distinct points in order to prevent degeneracies in Pearson's r. Examples are given showing that this approach may severely underestimate the variance of sample moments when sample sizes are very small.