The method of deriving the second derivatives of the goodness-of-fit functions of maximum likelihood and least-squares confirmatory factor analysis is discussed. The full set of second derivatives is reported.

Considerations of factor score estimates have concentrated on internal characteristics. This report considers external characteristics of four methods for determining factor score estimates; that is, relations of these estimates to measures on attributes not entered into the factor analysis. These external characteristics are important for many uses of factor score estimates. Findings are that different ones of the methods are appropriate for different uses.

A general one-way analysis of variance components with unequal replication numbers is used to provide unbiased estimates of the true and error score variance of classical test theory. The inadequacy of the ANOVA theory is noted and the foundations for a Bayesian approach are detailed. The choice of prior distribution is discussed and a justification for the Tiao-Tan prior is found in the particular context of the “n-split” technique. The posterior distributions of reliability, error score variance, observed score variance and true score variance are presented with some extensions of the original work of Tiao and Tan. Special attention is given to simple approximations that are available in important cases and also to the problems that arise when the ANOVA estimate of true score variance is negative. Bayesian methods derived by Box and Tiao and by Lindley are studied numerically in relation to the problem of estimating true score. Each is found to be useful and the advantages and disadvantages of each are discussed and related to the classical test-theoretic methods. Finally, some general relationships between Bayesian inference and classical test theory are discussed.

The oblimax, promax, maxplane, and Harris-Kaiser techniques are compared. For five data sets, of varying reliability and factorial complexity, each having a graphic oblique solution (used as criterion), solutions obtained using the four methods are evaluated on (1) hyperplane-counts, (2) agreement of obtained with graphic within-method primary factor correlations and angular separations, (3) angular separations between obtained and corresponding graphic primary axes. The methods are discussed and ranked (descending order): Harris-Kaiser, promax, oblimax, maxplane. The Harris-Kaiser procedure—independent cluster version for factorially simple data, P'P proportional to Φ, with equamax rotations, for complex—is recommended.

This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matrices S, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements of S. A comparable and more easily proven theorem for the vector model is also introduced. At most n-2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.

A simple Bayesian solution for the joint estimation of several parallel profile differentials for multivariate normal populations is obtained. This generalizes a previous result for a single profile differential.

The problem of whether a precise connection exists between stochastic processes of the type considered in mathematical learning theory and the Guttman simplex is investigated. The approach is to consider a class of models that characterize the sequential properties of discrete data, and to derive a set of conditions which a model must satisfy in order to generate inter-trial correlations rij with the ‘perfect simplex’ property: rik = rijrjk for all trials i < j < k. It is shown that the Chapman-Kolomogorov Equations provide a necessary and sufficient condition for this property to hold. It follows that a process which is Markovian in the errors and successes will have this property. It is also shown that, if certain reliability assumptions are introduced, then the all-or-none model has the simplex property if the appropriate correlation coefficients are corrected for attenuation.

Recent studies of sequential decision making performance have utilized optimal dynamic programming solutions as the criteria for evaluating the quality of human decisions. The difficulties incumbent in increasing the dimensionality of dynamic programming formulations effectively prohibit their extension beyond the case of a single decision variable. In the following article an alternative optimization formulation for a two-decision-variable case is presented, followed by the development of a corresponding one-decision-variable problem. In addition, related methodological problems limiting the comparability of results among previous studies are presented. It is suggested that the proposed problem formulation and its accompanying interpretation for the decision maker largely eliminate these previous methodological problems. Furthermore, the problem formulation permits quantification of the relative weights that should optimally be placed on future decision periods in the multiple-stage decision problem.