It is customary in multivariate analysis to search out relational structure by partitioning measures of total variation or overall relatedness into components variously attributable to different sources. Although the multivariate systems most familiar today–notably, linear correlational analysis, analysis of variance, and Information theory–base their decompositions upon very different substantive quantities, they all embody the very same abstract mathematical structure which, moreover, is capable of generating potentially fruitful patterns of data analysis in many other substantive applications as well.

When knowledge of the response sequence permits one to specify the sequence of theoretical states that occurred, we propose to say that the theory has observable states. Our earlier usage of “identifiable” conflicts with established usage in econometrics. We also present a revised definition of implication between theories that avoids a technical flaw in the earlier definition, and we correct a mistake in an example in our earlier paper.

A model is considered for learning where there is an initial stage with only errors, an intermediate state with errors and correct responses, and an absorbing state with only correct responses. A model with observable states is constructed, and sufficient identifiable parameters are given for the original model. Distribution statistics and other properties of data are derived as functions of the identifiable parameters. Informal estimates and sufficient statistics are given for the identifiable parameters. A number of testable hypotheses about the theoretical parameters are described. Some identifying restrictions for the model are given, along with the estimates for the unrestricted free parameters which follow. An application is made to data from a pairedassociate learning experiment.

For the case of the one-element Markov learning model for which the guessing parameter p is assumed known the efficiency of Bower's method of moments estimator c+ for the learning parameter, c, is found. It seems that although c+ is not efficient, for some practical value of c it is not very inefficient. If p and c are small, ĉ, the maximum likelihood estimate, and c+ have approximately the same first term when expanded in powers of p.

It is shown that several variants of the Tucker-Messick points of view procedure are interpretable in terms of their “idealized individual” concepts. It is contended that the main function of the method is that it allows the computation of several vectors of mean judgments instead of only one and that each of these may be used to stand for the actual judgments of a subgroup of subjects. Vectors that do not correspond to the judgments of any subjects are to be avoided. It is alleged that Ross's interpretation of “points of view” is quite different from that intended by Tucker and Messick.

It is shown that the lower bounds for the number of common factors, established by Guttman [1954] and modified by Kaiser [1961], cannot decrease as the number of observed variates is increased. The result implies that the lower bounds cannot become weaker if the number of observed variates is increased and the number of factors remains constant.

When the regression of factored variates on variates defining subpopulations (or experimental groups) is (a) nonlinear or (b) heteroscedastic, an existing solution to the problem of factorial invariance does not apply. This paper presents an approach which can be used in place of the existing solution and derives a method for obtaining maximally similar orthogonal simple structure factor pattern matrices across subpopulations. The method is applied to data from three experimental groups.

A generalization of a cognitive probability model is proposed utilizing the serial position effect to account for decay in probability of recall. The general nature of the model permits the incorporation of almost any of the recent free-recall serial position curve results. Two examples, hypothesizing different serial position curves, are given.