Classical item analysis procedures were developed for dichotomously scored items and do not apply to items allowing multiple correct responses. Maximum likelihood procedures analogous to those employed in polychotomous bio-assay are presented which yield estimates of the sets of parameters for items having multiple nonordered responses. Expressions for the estimates of the asymptotic variances of the item parameters and on overall chi-square goodness of fit test are also provided.

Statistical properties of several methods for obtaining estimates of factor loadings and procedures for estimating the number of factors are compared by means of random sampling experiments. The effect of increasing the ratio of the number of observed variables to the number of factors, and of increasing sample size, is examined. A description is given of a procedure which makes use of the Bartlett decomposition of a Wishart matrix to generate random correlation matrices.

An alpha-O coefficient of internal consistency is defined for an observed score composite. Maximizing alpha-O leads to a system of psychometric (vs. statistical) factor analysis in which successive factors describe dimensions of successively less internal-consistency. Factoring stops when alpha-O is zero or less. In contrast to Kaiser-Caffrey's alpha-C analysis, when the factored matrix is rank 1, alpha-O does not reach unity; it can approach unity only as the number of variables reach infinity. The relative usefulness and domains of generalization of alpha-C and alpha-O are compared. Basically, alpha-C analysis is concerned with the representativeness of factors while alpha-O analysis is concerned with the assessibility of factors. Consequently, either system of factoring can and should be summarized by both the alpha-C and alpha-O coefficients. Not surprisingly, alpha-O analysis is computationally analogous to Rao's canonical factor analysis.

Previous results of the application of Lawley's selection theorem to the common factor analysis model are extended to a revision of Tucker's three-mode principal components model. If the regression of the three-mode manifest variates on variates used to select subpopulations is both linear and homoscedastic, the two factor pattern matrices, the core matrix, and the residual variance-covariance matrix in the three-mode model can all be assumed to be invariant across subpopulations. The implication of this finding for simple structure is discussed.

A general procedure is described for obtaining weighted linear combinations of variables. This includes as special cases, multiple regression weights, canonical variate analysis, principal components, maximizing composite reliability, canonical factor analysis, and certain other well-known methods. The general procedure is shown to yield certain desirable invariance properties, with respect to transformations of the variables.

Necessary and sufficient conditions for the symmetry of the limiting distributions of response probabilities are derived for the Bush and Mosteller two experimenter-controlled events learning model [Bush and Mosteller, 1955].