Suppose Pi(i) (i = 1, 2, ..., m, j = 1, 2, ..., n) give the locations of mn points in p-dimensional space. Collectively these may be regarded as m configurations, or scalings, each of n points in p-dimensions. The problem is investigated of translating, rotating, reflecting and scaling the m configurations to minimize the goodness-of-fit criterion Σi=1m Σi=1n Δ2(Pj(i)Gi), where Gi is the centroid of the m points Pi(i) (i = 1, 2, ..., m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special case m = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.

The mathematical problem of approximating one matrix by another of lower rank is closely related to the fundamental postulate of factor-theory. When formulated as a least-squares problem, the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another. The solution of the problem is simplified by first expressing the matrices in a canonic form. It is found that the problem always has a solution which is usually unique. Several conclusions can be drawn from the form of this solution.

A hypothetical interpretation of the canonic components of a score matrix is discussed.

A rating response mechanism for ordered categories, which is related to the traditional threshold formulation but distinctively different from it, is formulated. In addition to the subject and item parameters two other sets of parameters, which can be interpreted in terms of thresholds on a latent continuum and discriminations at the thresholds, are obtained. These parameters are identified with the category coefficients and the scoring function of the Rasch model for polychotomous responses in which the latent trait is assumed uni-dimensional. In the case where the threshold discriminations are equal, the scoring of successive categories by the familiar assignment of successive integers is justified. In the case where distances between thresholds are also equal, a simple pattern of category coefficients is shown to follow.

A review of model-selection criteria is presented, with a view toward showing their similarities. It is suggested that some problems treated by sequences of hypothesis tests may be more expeditiously treated by the application of model-selection criteria. Consideration is given to application of model-selection criteria to some problems of multivariate analysis, especially the clustering of variables, factor analysis and, more generally, describing a complex of variables.

This discussion paper argues that both the use of Cronbach’s alpha as a reliability estimate and as a measure of internal consistency suffer from major problems. First, alpha always has a value, which cannot be equal to the test score’s reliability given the interitem covariance matrix and the usual assumptions about measurement error. Second, in practice, alpha is used more often as a measure of the test’s internal consistency than as an estimate of reliability. However, it can be shown easily that alpha is unrelated to the internal structure of the test. It is further discussed that statistics based on a single test administration do not convey much information about the accuracy of individuals’ test performance. The paper ends with a list of conclusions about the usefulness of alpha.

Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used. By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided. The Em procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability. This includes models with more than one latent dimension.

A common problem for both principal component analysis and image component analysis is determining how many components to retain. A number of solutions have been proposed, none of which is totally satisfactory. An alternative solution which employs a matrix of partial correlations is considered. No components are extracted after the average squared partial correlation reaches a minimum. This approach gives an exact stopping point, has a direct operational interpretation, and can be applied to any type of component analysis. The method is most appropriate when component analysis is employed as an alternative to, or a first-stage solution for, factor analysis.

A computer program is described that is designed to reconstruct the metric configuration of a set of points in Euclidean space on the basis of essentially nonmetric information about that configuration. A minimum set of Cartesian coordinates for the points is determined when the only available information specifies for each pair of those points—not the distance between them—but some unknown, fixed monotonic function of that distance. The program is proposed as a tool for reductively analyzing several types of psychological data, particularly measures of interstimulus similarity or confusability, by making explicit the multidimensional structure underlying such data.

We describe a general procedure by which any number of parameters of the factor analytic model can be held fixed at any values and the remaining free parameters estimated by the maximum likelihood method. The generality of the approach makes it possible to deal with all kinds of solutions: orthogonal, oblique and various mixtures of these. By choosing the fixed parameters appropriately, factors can be defined to have desired properties and make subsequent rotation unnecessary. The goodness of fit of the maximum likelihood solution under the hypothesis represented by the fixed parameters is tested by a large sample x2 test based on the likelihood ratio technique. A by-product of the procedure is an estimate of the variance-covariance matrix of the estimated parameters. From this, approximate confidence intervals for the parameters can be obtained. Several examples illustrating the usefulness of the procedure are given.

Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.

A structural equation model is proposed with a generalized measurement part, allowing for dichotomous and ordered categorical variables (indicators) in addition to continuous ones. A computationally feasible three-stage estimator is proposed for any combination of observed variable types. This approach provides large-sample chi-square tests of fit and standard errors of estimates for situations not previously covered. Two multiple-indicator modeling examples are given. One is a simultaneous analysis of two groups with a structural equation model underlying skewed Likert variables. The second is a longitudinal model with a structural model for multivariate probit regressions.

This monograph is a part of a more comprehensive treatment of estimation of latent traits, when the entire response pattern is used.

The fundamental structure of the whole theory comes from the latent trait model, which was initiated by Lazarsfeld as the latent structure analysis [Lazarsfeld, 1959], and also by Lord and others as a theory of mental test scores [Lord, 1952]. Similarities and differences in their mathematical structures and tendencies were discussed by Lazarsfeld [Lazarsfeld, 1960] and the recent book by Lord and Novick with contributions by Birnbaum [Lord & Novick, 1968] provides the dichotomous case of the latent trait model in the context of mental measurement.

I was sitting before my TV set, a while back, watching Captain Video and pondering the organizational problems of psychologists, psychometricians, psychodiagnosticians, psycho-somatists, psychosomnabulists, and psychoceramics (crack-pots to you). Wondering what I might do, in my small way, to help out, I decided to enlist Captain Video's help to bring me from the Black Planet that superogalactian hypermetrician, Dr. Idnozs Hcahscror-Tenib, cosmos-famous discoverer of Serutan.

Why delay? The Galaxy was on its way. and in half a light year Dr. Tenib was at my side prepared to devote his gargantuan talents to the task.

Seeing no point in confusing the good doctor by trying to describe to him the present administrative hodgepodge, I said, “Doctor, let's start from scratch. I want you to find out for me how these good people who are present at the annual meeting of the APA structure themselves? What families are represented? How many, or better, how few? And who belongs to each?”

A solutionT of the least-squares problem AT=B + E, given A and B so that trace (E′E)= minimum and T′T= I is presented. It is compared with a less general solution of the same problem which was given by Green [5]. The present solution, in contrast to Green's, is applicable to matrices A and B which are of less than full column rank. Some technical suggestions for the numerical computation of T and an illustrative example are given.

A scaled difference test statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\tilde{T}{}_{d}$\end{document} that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\tilde{T}_{d}$\end{document} is asymptotically equivalent to the scaled difference test statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\bar{T}_{d}$\end{document} introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\tilde{T}_{d}$\end{document} has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\bar{T}_{d}$\end{document} that avoids negative chi-square values.

This paper is concerned with the study of similarities and differences in factor structures between different groups. A common situation occurs when a battery of tests has been administered to samples of examinees from several populations.

A very general model is presented, in which any parameter in the factor analysis models (factor loadings, factor variances, factor covariances, and unique variances) for the different groups may be assigned an arbitrary value or constrained to be equal to some other parameter. Given such a specification, the model is estimated by the maximum likelihood method yielding a large sample x2 of goodness of fit. By computing several solutions under different specifications one can test various hypotheses.

The method is capable of dealing with any degree of invariance, from the one extreme, where nothing is invariant, to the other extreme, where everything is invariant. Neither the number of tests nor the number of common factors need to be the same for all groups, but to be at all interesting, it is assumed that there is a common core of tests in each battery that is the same or at least content-wise comparable.

Various models for sets of congeneric tests are considered, including models appropriate for the analysis of multitrait-multimethod data. All models are illustrated with real data. The special cases when two or more tests within a set are tau-equivalent or parallel are also considered. All data analyses are done within the framework of a general model by Jöreskog [1970].

The theoretically best estimate of the reliability coefficient is stated in terms of a precise definition of the equivalence of two forms of a test. Various approximations to this theoretical formula are derived, with reference to several degrees of completeness of information about the test and to special assumptions. The familiar Spearman-Brown Formula is shown to be a special case of the general formulation of the problem of reliability. Reliability coefficients computed in various ways are presented for comparative purposes.

A Monte Carlo study assessed the effect of sampling error and model characteristics on the occurrence of nonconvergent solutions, improper solutions and the distribution of goodness-of-fit indices in maximum likelihood confirmatory factor analysis. Nonconvergent and improper solutions occurred more frequently for smaller sample sizes and for models with fewer indicators of each factor. Effects of practical significance due to sample size, the number of indicators per factor and the number of factors were found for GFI, AGFI, and RMR, whereas no practical effects were found for the probability values associated with the chi-square likelihood ratio test.