The validity conditions for univariate repeated measures designs are described. Attention is focused on the sphericity requirement. For a v degree of freedom family of comparisons among the repeated measures, sphericity exists when all contrasts contained in the v dimensional space have equal variances. Under nonsphericity, upper and lower bounds on test size and power of a priori, repeated measures, F tests are derived. The effects of nonsphericity are illustrated by means of a set of charts. The charts reveal that small departures from sphericity (.97 ≤ ε < 1.00) can seriously affect test size and power. It is recommended that separate rather than pooled error term procedures be routinely used to test a priori hypotheses.

For assigning subjects to treatments the point of intersection of within-group regression lines is ordinarily used as the critical point. This decision rule is critized and, for several utility functions and any number of treatments, replaced by optimal monotone, nonrandomized (Bayes) rules. Both treatments with and without mastery scores are considered. Moreover, the effect of unreliable criterion scores on the optimal decision rule is examined, and it is illustrated how qualitative information can be combined with aptitude measurements to improve treatment assignment decisions. Although the models in this paper are presented with special reference to the aptitude-treatment interaction problem in education, it is indicated that they apply to a variety of situations in which subjects are assigned to treatments on the basis of some predictor score, as long as there are no allocation quota considerations.

Asymptotic distribution theory of Brogden's form of biserial correlation coefficient is derived and large sample estimates of its standard error obtained. Its efficiency relative to the biserial correlation coefficient is examined. Other modifications of the statistic are evaluated, and on the basis of these results, recommendations for choice of estimator of biserial correlation are presented.

A linear utility model is introduced for optimal selection when several subpopulations of applicants are to be distinguished. Using this model, procedures are described for obtaining optimal cutting scores in subpopulations in quota-free as well as quota-restricted selection situations. The cutting scores are optimal in the sense that they maximize the overall expected utility of the selection process. The procedures are demonstrated with empirical data.

Procedures are described for the analysis of profiles of means in repeated measures designs under order restriction for patterns of mean change. The exact likelihood ratio is derived for the case of two groups of subjects, and a computationally simple alternative to the exact likelihood ratio test is provided for designs involving more than two groups. Tables of critical values are provided for the case of simple-order alternatives.

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation between linear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.

A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.

Commonality components have been defined as a method of partitioning squared multiple correlations. In this paper, the asymptotic joint distribution of all 2k − 1 squared multiple correlations is derived. The asymptotic joint distribution of linear combinations of squared multiple correlations is obtained as a corollary. In particular, the asymptotic joint distribution of commonality components are derived as a special case. Simultaneous and nonsimultaneous asymptotic confidence intervals for commonality components can be obtained from this distribution.

An expression is given for weighted least squares estimators of oblique common factors, constrained to have the same covariance matrix as the factors they estimate. It is shown that if as in exploratory factor analysis, the common factors are obtained by oblique transformation from the Lawley-Rao basis, the constrained estimators are given by the same transformation. Finally a proof of uniqueness is given.

The author provides a full-fledged matrix derivation of Sherin's matrix formulation of Kaiser's varimax criterion. He uses matrix differential calculus in conjunction with the Hadamard (or Schur) matrix product. Two results on Hadamard products are presented.

The method proposed by Harman and Fukuda to treat the so-called Heywood case in the minres method in factor analysis i.e., the case where the resulting communalities are greater than one, involves the frequent solution of eigenvalue problems. A simple method to treat this problem requiring less computing time and enjoying higher numerical stability is described in this paper.

Milligan presented the conditions that are required for a hierarchical clustering strategy to be monotonic, based on a formula by Lance and Williams. In the present paper, the statement of the conditions is improved and shown to provide necessary and sufficient conditions.