An approximate statistical test is derived for the hypothesis that the reliability coefficients (Cronbach's α) associated with two measurement procedures are equal. Control of Type I error is investigated by comparing empirical sampling distributions of the test statistic with the theoretical model derived for it. The effect of platykurtosis in the test-score distribution on the test statistic is also considered.

The inter-battery method of factor analysis was devised to provide information relevant to the stability of factors over different selections of tests. Two batteries of tests, postulated to depend on the same common factors, but not parallel tests, are given to one sample of individuals. Factors are determined from the correlation of the tests in one battery with the tests in the other battery. These factors are only those that are common to the two batteries. No communality estimates are required. A statistical test is provided for judging the minimum number of factors involved. Rotation of axes is carried out independently for the two batteries. A final step provides the correlation between factors determined by scores on the tests in the two batteries. The correlations between corresponding factors are taken as factor reliability coefficients.

By using the concepts of antimetry and n-chain it is possible to define and to investigate some properties of connectivity in a sociometric group. It is shown that the number of elements in a group, the number of antimetries, and the degree of connectivity must satisfy certain inequalities. Using the ideas of connectivity, a generalized concept of clique, called an n-clique, is introduced. n-cliques are shown to have a very close relationship to the existence of cliques in an artificial structure defined on the same set of elements, thus permitting the determination of n-cliques by means of the same simple matrix procedures used to obtain the clique structures. The presence of two or more m-cliques, where m is the number of elements in the group, is proved to mean an almost complete splitting of the group.

Regularized generalized canonical correlation analysis (RGCCA) is a generalization of regularized canonical correlation analysis to three or more sets of variables. It constitutes a general framework for many multi-block data analysis methods. It combines the power of multi-block data analysis methods (maximization of well identified criteria) and the flexibility of PLS path modeling (the researcher decides which blocks are connected and which are not). Searching for a fixed point of the stationary equations related to RGCCA, a new monotonically convergent algorithm, very similar to the PLS algorithm proposed by Herman Wold, is obtained. Finally, a practical example is discussed.

A Metropolis–Hastings Robbins–Monro (MH-RM) algorithm for high-dimensional maximum marginal likelihood exploratory item factor analysis is proposed. The sequence of estimates from the MH-RM algorithm converges with probability one to the maximum likelihood solution. Details on the computer implementation of this algorithm are provided. The accuracy of the proposed algorithm is demonstrated with simulations. As an illustration, the proposed algorithm is applied to explore the factor structure underlying a new quality of life scale for children. It is shown that when the dimensionality is high, MH-RM has advantages over existing methods such as numerical quadrature based EM algorithm. Extensions of the algorithm to other modeling frameworks are discussed.

A distinction is drawn between the method of principal components developed by Hotelling and the common factor analysis discussed in psychological literature both from the point of view of stochastic models involved and problems of statistical inference. The appropriate statistical techniques are briefly reviewed in the first case and detailed in the second. A new method of analysis called the canonical factor analysis, explaining the correlations between rather than the variances of the measurements, is developed. This analysis furnishes one out of a number of possible solutions to the maximum likelihood equations of Lawley. It admits an iterative procedure for estimating the factor loadings and also for constructing the likelihood criterion useful in testing a specified hypothesis on the number of factors and in determining a lower confidence limit to the number of factors.

To assess the reliability of congeneric tests, specifically designed reliability measures have been proposed. This paper emphasizes that such measures rely on a unidimensionality hypothesis, which can neither be confirmed nor rejected when there are only three test parts, and will invariably be rejected when there are more than three test parts. Jackson and Agunwamba's (1977) greatest lower bound to reliability is proposed instead. Although this bound has a reputation for overestimating the population value when the sample size is small, this is no reason to prefer the unidimensionality-based reliability. Firstly, the sampling bias problem of the glb does not play a role when the number of test parts is small, as is often the case with congeneric measures. Secondly, glb and unidimensionality based reliability are often equal when there are three test parts, and when there are more test parts, their numerical values are still very similar. To the extent that the bias problem of the greatest lower bound does play a role, unidimensionality-based reliability is equally affected. Although unidimensionality and reliability are often thought of as unrelated, this paper shows that, from at least two perspectives, they act as antagonistic concepts. A measure, based on the same framework that led to the greatest lower bound, is discussed for assessing how close is a set of variables to unidimensionality. It is the percentage of common variance that can be explained by a single factor. An empirical example is given to demonstrate the main points of the paper.

The factor analysis model and Lazarsfeld's latent structure scheme for analyzing dichotomous attributes are derived to show how the latter model avoids three knotty problems in factor analysis: communality estimation, rotation, and curvilinearity. Then the latent structure model is generalized into latent profile analysis for the study of interrelations among quantitative measures. Four latent profile examples are presented and discussed in terms of their limitations and the problems of latent metric and dimensionality thereby raised. The possibility of treating higher order empirical relations in a manner paralleling their various uses in the latent structure model is indicated.

Coefficient κ is generally defined in terms of procedures of computation rather than in terms of a population. Here a population definition is proposed. On this basis, the interpretation of κ as a measure of diagnostic reliability in characterizing an individual, and the effect of reliability, as measured by κ, on estimation bias, precision, and test power are examined. Factors influencing the magnitude of κ are identified. Strategies to improve reliability are proposed, including that of combining multiple unreliable diagnoses.

Lawley's selection theorem is applied to subpopulations derived from a parent in which the classical factor model holds for a specified set of variables. The results show that there exists an invariant factor pattern matrix that describes the regression of observed on factor variables in every subpopulation derivable by selection from the parent, given that selection does not occur directly on the observable variables and does not reduce the rank of the system. However, such a factor pattern matrix is not unique, which in turn implies that if a simple structure factor pattern matrix can be satisfactorily determined in one such subpopulation the same simple structure can be found in any subpopulation derivable by selection. The implications of these results for “parallel proportional profiles” and “factor matching” techniques are discussed.

A procedure is derived for obtaining an orthogonal transformation which most nearly transforms one given matrix into another given matrix, according to some least-squares criterion of fit. From this procedure, three analytic methods are derived for obtaining an orthogonal factor matrix which closely approximates a given oblique factor matrix. The case is considered of approximating a specified subset of oblique vectors by orthogonal vectors.

If d is a measure of dissimilarity on a finite set with n elements, the smallest positive constant c* such that d + c has an euclidean representation for all c ≥c* is shown to be the largest eigen-value of a matrix of size 2n.

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower’s method of generalized Procrustes analysis are suggested.

The general use of coefficient alpha to assess reliability should be discouraged on a number of grounds. The assumptions underlying coefficient alpha are unlikely to hold in practice, and violation of these assumptions can result in nontrivial negative or positive bias. Structural equation modeling was discussed as an informative process both to assess the assumptions underlying coefficient alpha and to estimate reliability

A study by Shaw (7) some twenty years ago is frequently cited by social scientists to support the generalization that groups are superior to individuals in problem-solving. Shaw suggests that personal interaction within the group is responsible for the superior performance of groups. This article re-examines her data in the light of two models which propose that the difference in quality of solution between group and individual performance is solely a matter of ability. It is shown that Shaw's data may be considered to have been an outcome of behavior postulated by the models. Since Shaw's observations relate to a special population and to special kinds of problems, the proposed models may not be appropriate under differing experimental conditions. In fact, Lorge et al. (4) have indicated that experimental demonstration of the superiority of groups over individuals in problem-solving depends not only on the kind of group but also on the kind of problem to be solved. In addition, the diversity of transfer of training for groups and for individuals is considered.

The polyserial and point polyserial correlations are discussed as generalizations of the biserial and point biserial correlations. The relationship between the polyserial and point polyserial correlation is derived. The maximum likelihood estimator of the polyserial correlation is compared with a two-step estimator and with a computationally convenient ad hoc estimator. All three estimators perform reasonably well in a Monte Carlo simulation. Some practical applications of the polyserial correlation are described.

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order r. They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when r is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson’s X2. For this model in nonsparse situations, the proposed statistics are also more powerful than X2. In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.

A universe of infinitely many quantitative variables is considered, from which a sample of n variables is arbitrarily selected. Only linear least-squares regressions are considered, based on an infinitely large population of individuals or respondents. In the sample of variables, the predicted value of a variable x from the remaining n − 1 variables is called the partial image of x, and the error of prediction is called the partial anti-image of x. The predicted value of x from the entire universe, or the limit of its partial images as n → ∞, is called the total image of x, and the corresponding error is called the total anti-image. Images and anti-images can be used to explain “why” any two variables xj and xk are correlated with each other, or to reveal the structure of the intercorrelations of the sample and of the universe. It is demonstrated that image theory is related to common-factor theory but has greater generality than common-factor theory, being able to deal with structures other than those describable in a Spearman-Thurstone factor space. A universal computing procedure is suggested, based upon the inverse of the correlation matrix.

A quantitative system is presented to permit the determination of separate multidimensional perceptual spaces for individuals having different viewpoints about stimulus interrelationships. The structure of individual differences in the perception of stimulus relationships is also determined to provide a framework for ascertaining the varieties of consistent individual viewpoints and their relationships with other variables.