On a multiple-choice test in which each item has k alternative responses, the test taker is permitted to choose any subset which he believes contains the one correct answer. A scoring system is devised that depends on the size of the subset and on whether or not the correct answer is eliminated. The mean and variance of the score per item are obtained. Methods are derived for determining the total number of items that should be included on the test so that the average score on all items can be regarded as a good measure of the subject's knowledge. Efficiency comparisons between conventional and the subset selection scoring procedures are made. The analogous problem of r > 1 correct answers for each item (with r fixed and known) is also considered.

This paper details a Bayesian alternative to the use of least squares and equal weighting coefficients in regression. An equal weight prior distribution for the linear regression parameters is described with regard to the conditional normal regression model, and resulting posterior distributions for these parameters are detailed. Some interesting connections between this Bayesian procedure and several other methods for estimating optimal weighting coefficients are discussed. In addition, results are presented of a Monte Carlo investigation which compared the effectiveness of the Bayesian procedure relative to least squares, equal weight, ridge, and Bayesian exchangeability estimations.

Many authors have demonstrated for idealized item configurations that equal item weights are often virtually as good for a particular predictive purpose as the item weights that are theoretically optimal. What has not been heretofore clear, however, is what happens to the similarity between weighted and unweighted composites of the same items when the item configuration's variance structure is complex.

The jackknife by groups and modifications of the jackknife by groups are used to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis. Simulations are performed in which t-statistics based upon these jackknife estimates of the standard errors are computed. The validity of the t-statistics and their associated confidence intervals is assessed. Methods are given through which the computational efficiency of the jackknife may be greatly enhanced in the factor analysis model.

Guttman's assumption underlying his definition of “total images” is rejected: Partial images are not generally convergent everywhere. Even divergence everywhere is shown to be possible. The convergence type always found on partial images is convergence in quadratic mean; hence, total images are redefined as quadratic mean-limits. In determining the convergence type in special situations, the asymptotic properties of certain correlations are important, implying, in some cases, convergence almost everywhere, which is also effected by a countable population or multivariate normality or independent variables. The interpretations of a total image as a predictor, and a “common-factor score”, respectively, are made precise.

It is demonstrated that tests of homogeneity of independent correlation coefficients based on the simple forms of the normal and t approximations to the distribution of the correlation coefficients are comparable in terms of robustness, size and power.

Expressions involving optimal sign vectors are derived so as to yield two new applications. First, coefficient alpha for the sign-weighted composite is maximized in analogy to Lord's scale-independent solution with differential weights. Second, optimal sign vectors are used to define two groups of objects that are maximally distinct in terms of a function of the squared euclidean distances between groups. An efficient computing algorithm is described along with several examples.

Johnson has shown that the single linkage and the complete linkage hierarchical clustering algorithms induce a metric on the data known as the ultrametric. Through the use of the Lance and Williams recurrence formula, Johnson's proof is extended to four other common clustering algorithms. It is also noted that two additional methods produce hierarchical structures which can violate the ultrametric inequality.

An important relationship is given for two generalizations of coefficient alpha, Rajaratnam, Cronbach and Gleser's generalizability formula for stratified-parallel tests and Raju's coefficient beta.

Tucker, Damarin and Messick proposed a “base-free” measure of change which involves the computation of residual scores that are uncorrelated with true scores on the pretest. The present note discusses this change measure and demonstrates that, because of an incorrect derivation by the authors, properties they attribute to α (the coefficient for the regression of true scores from the second testing on true scores from the first testing) are, in fact, properties of b, the ordinary regression coefficient. A correct derivation of the initial position—change correlation in terms of α is obtained, and Tucker et al.'s discussion of the “law of initial values” is reconsidered.

Tucker's method of oblique congruence rotation is shown to be equivalent to a procedure by Meredith. This implies that Monte Carlo studies on congruence by Nesselroade, Baltes and Labouvie and by Korth and Tucker are highly comparable. The problem of rotating two matrices orthogonally to maximal congruence has not yet been solved. An approximate solution to this problem can be derived from Tucker's method. Even better results can be obtained from a Procrustes rotation followed by rotation to simple structure.