The earlier two-sample procedure of Feldt [1969] for comparing independent alpha reliability coefficients is extended to the case of K ≥ 2 independent samples. Details of a normalization of the statistic under consideration are presented, leading to computational procedures for the overall K-group significance test and accompanying multiple comparisons. Results based on computer simulation methods are presented, demonstrating that the procedures control Type I error adequately. The results of a power comparison of the case of K = 2 with Feldt’s [1969] F test are also presented. The differences in power were negligible. Some final observations, along with suggestions for further research, are noted.

Matching by Procrustes methods involves the transformation of one matrix to match with another. A special least squares criterion, the congruence coefficient, has advantages as a criterion for some factor analytic interpretations. A Procrustes method maximizing the congruence coefficient is given. This solution is identical to Mosier's [1939] approximate solution, but is an exact solution for maximum congruence.

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are

log (N − 1) and log {[n(n − 1)/2] − [(i − 1)n]},

where N is the number of observations; n, the number of variables; and i, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.

A numerical procedure is outlined for obtaining an interval estimate of the regression of true score on observed score. Only the frequency distribution of observed scores is needed for this. The procedure assumes that the conditional distribution of observed scores for fixed true score is binomial. The procedure is applied to several sets of test data.

A procedure for ordering object (stimulus) pairs based on individual preference ratings is described. The basic assumption is that individual responses are consistent with a nonmetric multidimensional unfolding model. The method requires data where a numerical response is independently generated for each individual-object pair. In conjunction with a nonmetric multidimensional scaling procedure, it provides a vehicle for recovering meaningful object configurations.

A large class of rank tests, which includes the familiar sign test and the Wilcoxon signed-ranks test, is described and discussed. This class of distribution-free tests provides a flexible basis for testing research hypotheses of various forms. Exact small sample and approximate large sample procedures are considered. Applications of these procedures are presented, including simple numerical examples.

In the last few years, a number of asymptotic results for the distribution of unrotated and rotated factor loadings have been given. This paper investigates the validity of some of these results based on simulation techniques. In particular, it looks at principal component extraction and quartimax rotation on a problem with 13 variables. The indication is that the asymptotic results are quite good.

A quadratic programming algorithm is presented for fitting Carroll’s weighted unfolding model for preferences to known multidimensional scale values. The algorithm can be applied directly to pairwise preferences; it permits nonnegativity constraints on subject weights; and it provides a means of testing various preference model hypotheses. While basically metric, it can be combined with Kruskal’s monotone regression to fit ordinal data. Monte Carlo results show that (a) adequacy of “true” preference recovery depends on the number of data points and the amount of error, and (b) the proportion of data variance accounted for by the model sometimes only approximately reflects “true” recovery.

By applying a general procedure for analyzing a correlation coefficient into components, the lens model equation is extended to a) analyze the effects of different types of variation and b) analyze the relations between judgmental systems that are not based on the same set of cues.

Generalized symmetric means are redefined in a way which allows them to be calculated for any matrix sampling design. It is proved that these sample gsm's are unbiased estimates of the analogous population gsm's. Illustrative examples are given.

The monotone regression function of Kruskal and the rank image function of Guttman and Lingoes were fitted to bivariate normal samples and their statistical properties contrasted.

Guttman’s index of indeterminacy (2ρ2 − 1) measures the potential amount of uncertainty in picking the right alternative interpretation for a factor. When alternative solutions for a factor are equally likely to be correct, then the squared multiple correlation ρ2 for predicting the factor from the observed variables is the average correlation ρAB between independently selected alternative solutions A and B, while var (ρAB) = (1 − ρ2)2/s, where s is the dimensionality of the space in which unpredicted components of alternative solutions are to be found. When alternative solutions for the factor are not equally likely to be chosen, ρ2 is the lower bound for E(ρAB); however, E(ρAB) need not be a modal value in the distribution of ρAB. Guttman’s index and E(ρAB) measure different aspects of the same indeterminacy problem.

The utility of the jackknife for constructing confidence intervals and testing hypotheses about the disattenuated correlation is evaluated for small samples. Computer simulations were used to generate the empirical sampling distributions of jackknife statistics for two sample sizes (30, 60), five values of the disattenuated correlation coefficient (1.00, .90, .80, .50, .00) and three pairs of reliabilities (.90, .80; .80, .80; .90, .50). The theoretical and cumulative proportions of jackknife statistics were compared at selected points in the appropriate t-distributions. The results obtained support the claim that the jackknife can be used to construct sensible confidence intervals. However, the jackknife possesses limited utility for testing hypotheses about the disattenuated correlation coefficient.

Relations between Tucker's three-mode multidimensional scaling and Carroll and Chang's INDSCAL are discussed. The possibility is raised that it may be profitable to attempt to transform a three-mode solution to the general form of an INDSCAL solution. Operationally, this involves transforming the three-mode core matrix so that each section is, as nearly as possible, a diagonal matrix. A technique is developed for accomplishing such a transformation, and is applied to two sets of data from the literature. Results indicate that the process is both feasible and valuable, providing useful information on the relative appropriateness of the two models.

A summary and interpretation of the recent literature on the indeterminacy of factor scores is given in simple terms. A good index of factor score determinacy is the squared multiple correlation of the factor with the observed variables.

This paper provides a generalization of the Procrustes problem in which the errors are weighted from the right, or the left, or both. The solution is achieved by having the orthogonality constraint on the transformation be in agreement with the norm of the least squares criterion. This general principle is discussed and illustrated by the mathematics of the weighted orthogonal Procrustes problem.

An alternative selection of subsample sizes for the Box-Scheffé test is compared to the single m used in Levy, “An empirical comparison of the Z-variance and Box-Scheffé tests for homogeneity of variance.” Use of the alternative subsample sizes is shown to suggest greater power for the Box-Scheffé test, although the test would still be less powerful than the Z-variance test. Because of the nonrobustness of the Z-variance test and the robustness of the Box-Scheffé test, the latter is recommended as a general technique unless an experimenter has assurance that all populations in the experiment are normal. Alternative techniques are also considered.

Examples are presented in which it is either desirable or necessary to transform two sets of orthogonal axes to simple structure positions by means of the same transformation matrix. A solution is then outlined which represents a two-matrix extension of the general “orthomax” orthogonal rotation criterion. In certain circumstances, oblique two-matrix solutions are possible using the procedure outlined and the Harris-Kaiser [1964] logic. Finally, an illustrative example is presented in which the preceding technique is applied in the context of an inter-battery factor analysis.

A direct proof is given for the generalized Spearman-Brown formula for any real multiple of test length.

Critical examination is made of the recent controversy over the value of Monte Carlo techniques in nonmetric multidimensional scaling procedures. The case is presented that the major relevance of Monte Carlo studies is not for the local minima problem but for the meaningfulness of the obtained solutions.