The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals.

Existing analytic oblique rotation schemes proceed by optimizing a simplicity function applied to the reference structure. This article suggests optimizing a simplicity function applied to primary loadings directly. The feasibility of the suggestion is demonstrated using the quartimin criterion. An algorithm to implement the optimization is derived and the existence of an admissible solution proved. Practical comparisons with the biquartimin method are made using Thurstone's Box Problem and Holzinger and Swineford's Twenty-Four Psychological Tests Problem.

A logistic model developed by Birnbaum was tested in two ways. First, plots of proportions of subjects in different score categories were examined for consistency with the assumption of a logistic trace line, and especially for departures from the logistic which seemed due to guessing in multiple choice items. The results showed that guessing seemed to have little effect. Second, an attempt was made to predict the obtained score distributions of samples of subjects on six tests from item parameters estimated on independent samples. The fits were good in all cases, despite considerable differences between the tests, and some extremely odd distributions.

A model is presented for evaluating potential effectiveness of a Bayesian classification system using the expected value of the posterior probability for true classifications as an evaluation metric. For a given set of input parameters, the value of this complex metric is predictable from a simply computed row variance metric. Prediction equations are given for several representative sets of input parameters.

This paper is addressed to the classical problem of estimating factor loadings under the condition that the sum of squares of off-diagonal residuals be minimized. Communalities consistent with this criterion are produced as a by-product. The experimental work included several alternative algorithms before a highly efficient method was developed. The final procedure is illustrated with a numerical example. Some relationships of minres to principal-factor analysis and maximum-likelihood factor estimates are discussed, and several unresolved problems are pointed out.

Given a set of items (predictors) suppose one wishes to predict another set of items (predictands) in asimultaneous way. Such a situation may occur when the predictands are different measurable aspects of the same phenomenon. Alternatively one might wish to predict the success of an event (say a successfully performed task) which has many correlated or uncorrelated failure modes (say a set of possible mental or physical disabilities each of them by itself precluding the achievement of the said task.) In such a case a unidimensional prediction is of value only if prediction is simultaneous for all possible failure modes. A linear summarization of the predictors is suggested, which is unique and has “maximum” predictability value for all predictands simultaneously. Other summarizations or scores are found that give “maximum” explanation of residual measures on the predictands and that are uncorrelated. The set of those simultaneous linear predictions is compared to the set of the individual multiple regression predictions as used, for instance, in the same context by Horst [4] for each predictand given the original predictors. We suggest that this technique can be applied in particular to the summarization of a subset of items when the whole set of items constitutes the set of predictands.

Various theorems are obtained for N-trial sample sequences from the general two-state, first-order Markov chain with stationary transition probabilities. Four lemmas which facilitate the derivations are given. A brief discussion of applications to binary data, estimation, and evaluation is given, including a maximum-likelihood procedure for estimating transition probabilities which are restricted by inequalities.

A method of design symbolization (DS) is given for a class of designs in which all factors are related by complete crossing or by nesting. Characterization of acceptable DS's, and logical relational properties are given. It is shown how the design model can be obtained from the DS.

Given several Gramian matrices, a least-square fit to all the matrices by one factor matrix, with a predetermined number of factors, is shown to be the principal axes solution of the average of the matrices.

In this paper graphs are given for estimating a correlation ρXY from a double-entry table. 2 Hr, twice the statistic Hr, “relative uncertainty” (the average of the ratios of the conditional uncertainty of each variable to its uncertainty) was computed. Mean 2 Hr's were computed from 100 samples for each of several sample sizes and numbers of categories. Graphs relating mean 2 Hr to \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat r_H$$\end{document}, the estimated correlation, for 3–17 categories to each variable and 3 sample sizes—50, 100, and 500—were constructed. 2.5 per cent and 97.5 per cent fiducial limits curves were constructed for establishing confidence limits on ρXY and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hat r_H$$\end{document}.