The class of symmetric path-independent models with experimenter-controlled events is considered in conjunction with two-choice probability learning experiments. Various refinements of the notion of probability matching are defined, and the incidence of these properties within this class is studied. It is shown that the linear models are the only models of this class that predict a certain phenomenon that we call stationary probability matching. It is also shown that models within this class that possess an additional property called marginal constancy predict approximate probability matching.

A model for multiple regression was developed which allows individual differences to emerge empirically. The model encompasses as special cases several of the previous attempts to improve psychological prediction by deviating from the usual linear multiple regression model. The model is tested with both artificial and real data. The results indicate that the model effectively reduces the variance of the error of prediction, and that the weights obtained are stable over different samples, and, to some extent, over different sets of predictors.

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.

A Markov learning model may be stated in the form of a transition matrix, starting vector, and response probability vector. Utilizing these and some general properties of absorbing Markov chains, general expressions are derived for several statistics of the learning process which can be applied to any model of this form. Included are derivations for the mean learning curve, number of total errors, trial numbers of the first success and the last error, and the number of error runs. As an illustration, all derivations are worked out for the simple two-state one-element model.

A general method is presented for estimating variance components when the experimental design has one random way of classification and a possibly unbalanced fixed classification. The procedure operates on a sample covariance matrix in which the fixed classes play the role of variables and the random classes correspond to observations. Cases are considered which assume (i) homogeneous and (ii) nonhomogeneous error variance, and (iii) arbitrary scale factors in the measurements and homogeneous error variance. The results include maximum-likelihood estimations of the variance components and scale factors, likelihood-ratio tests of the goodness-of-fit of the model assumed for the design, and large-sample variances and covariances of the estimates. Applications to mental test data are presented. In these applications the subjects constitute the random dimension of the design, and a classification of the mental tests according to objective features of format or content constitute the fixed dimensions.

The solution is presented to the problem of finding that orthonormal matrix closest in the least-square sense to a given matrix of full rank. An application is shown to the problem in multiple regression analysis of determining the “importance” of each independent variable.

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.

Kaiser has given the varimax criterion for the solution of the rotation problem in factor analysis as well as a practical computational procedure for maximizing this criterion. In the present paper, the maximization condition is shown as a matrix equation involving only the unknown orthogonal rotation matrix. This matrix equation can be solved iteratively as a sequence of symmetric eigenproblems.

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.

This paper presents briefly the rationale of the tetrachoric correlation coefficient. Pearson's results are outlined and several estimates of the coefficient are given. These estimates are compared with Pearson's expressions to determine the relative accuracy of the various approximations in determining the tetrachoric correlation coefficient.

The derivative of Kaiser's Varimax criterion, if set to zero, yields a set of equations which are quite similar to those obtained for a least-squares problem of the “Procrustes” type. This similarity suggested an iterative technique for orthogonal rotation, dubbed “Varisim,” which was programmed for the IBM 7094 in FORTRAN. An empirical comparison between Varimax and Varisim, which was based on a number of data sets taken from the literature yielded three major results so far: (i) Varisim is slower than Varimax, roughly by a factor of 3, (ii) Varisim yields factors which in general contribute more evenly to the common test variance than Varimax factors, and which (iii) line up more closely with oblique configurations obtained with Binormamin than Varimax factors.

This paper shows how the concept of an incidence matrix of communications can be used to define the entropy of a finite scheme. The properties of the entropy function are examined and the function is found to be best interpreted as a total expected participation index. Data is presented showing the relationship between structural centrality and the new total expected participation index. In general, as the network becomes more centralized the smaller the value of the participation index and as the network becomes more structurally decentralized the greater the participation index.

Certain properties and consequences of factor indeterminancy are deduced from an orthogonal transformation derived by Ledermann. The first deduction is the matrix of cosines between maximally distinct factor axes which represent the same factor solution. This result is used to investigate the indeterminancy of factor extensions.

Scales constructed from paired-comparison designs under Thurstone's Case V model are discussed in relation to those derived from similarity data by means of the Unilateral Law of Comparative Judgment [3]. Factors inherent in the triadic similarity task are then considered with respect to scale invariance across experimental designs. Illustrative data, although revealing the influence of these factors upon similarity response consistency, indicate the similarity scale to be somewhat robust to their biasing effects.

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 order to determine the robustness of the maximum-likelihood estimation procedure, random variates from six distributions were generated independently on a high speed computer and then used to represent the common and specific factors in a factor analysis model in which the coefficients of these factors had been specified. Using Lawley's approximate x2 statistic in evaluating the estimates obtained, the estimation procedure is found to be insensitive to changes in the distributions considered.

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}.

The relationship between the factor pattern, F, derived from fallible (containing measurement error) observations on variables and the factor pattern, F*, derived from infallible observations on variables is investigated. A widely believed relationship between F and F*, viz.,F* = AF where A is a diagonal matrix containing the inverses of the square roots of the reliabilities of the variables, is shown to be false for several factor analytic techniques. Under suitable assumptions, it is shown that for Kaiser and Caffrey's “alpha factor analysis”F* and F are related by F* = AF. Empirical examples for which the corresponding elements of F* and AF are equal to two decimal places are presented. The implications of the equality of F* and AF for alpha factor analysis are discussed.

Tables are given for the calculation of variances of maximum-likelihood estimates in a multiple biserial model. For the estimates of individual correlations between the dichotomized variable and each of the graduated variables additional calculation is necessary; variances for estimates of the multiple correlation and the point of dichotomy are available directly. The required formulas and notation are those of a recent paper by Hannan and Tate.