This paper attempts to review and expand upon the relationship between graph theory and the clustering of a set of objects. Several graphtheoretic criteria are proposed for use within a general clustering paradigm as a means of developing procedures “in between” the extremes of complete-link and single-link hierarchical partitioning; these same ideas are then extended to include the more general problem of constructing subsets of objects with overlap. Finally, a number of related topics are surveyed within the general context of reinterpreting and justifying methods of clustering either through standard concepts in graph theory or their simple extensions.

A simple procedure for testing heterogeneity of variance is developed which generalizes readily to complex, multi-factor experimental designs. Monte Carlo Studies indicate that the Z-variance test statistic presented here yields results equivalent to other familiar tests for heterogeneity of variance in simple one-way designs where comparisons are feasible. The primary advantage of the Z-variance test is in the analysis of factorial effects on sample variances in more complex designs. An example involving a three-way factorial design is presented.

The procedures recently developed by Jöreskog for studying similarities and differences in factor structures between different groups were applied to data from a study designed by Thurstone to investigate the sampling stability of a hypothesized isolated configuration. The hypothesis of an isolated configuration was rejected but not by much. The results represent a restricted and unique simple structure solution with the factor loadings and uniqueness values invariant over the two samples.

A probabilistic, multidimensional version of Coombs' unfolding model is obtained by assuming that the projections of each stimulus and each individual on each axis are normally distributed. Exact equations are developed for the single dimensional case and an approximate one for the multidimensional case. Both types of equations are expressed solely in terms of univariate normal distribution functions and are therefore easy to evaluate. A Monte Carlo experiment, involving 9 stimuli and 3 subjects in a 2 dimensional space, was run to determine the degree of accuracy of the multidimensional equation and the feasibility of using iterative methods to obtain maximum likelihood estimates of the stimulus and subject coordinates. The results reported here are gratifying in both respects.

A new formula is developed for the relative efficiency of two tests measuring the same trait. The formula expresses relative efficiency solely in terms of the standard errors of measurement and, surprisingly, the frequency distributions of true scores. Approximate methods for estimating relative efficiency may make this function routinely available. A numerical illustration compares new and old estimates of relative efficiency for subtests from the Scholastic Aptitude Test.

Various writers have suggested the use of projective transformations in psychological theories. Among them are Herbst [t970], Atkin [1972] and Levine [in press]. In general these suggestions relate to scaling problems in psychophysics and in social psychology. The purpose of the present note is to extend these applications to test theory. One of the models proposed by Rasch [1960] can be seen as a special case of the projective transformation and this fact has certain implications for the use of the model.

On the basis of the idea of common elements to account for similarity, Restle [1959] developed a set-theoretical model on the generation of similarity judgments. Restle differentiates between his “more qualitative discussion” (p. 207), regarding stimuli as sets of elements and using no geometric concepts or assumptions, and metric quantitative developments of similarity, regarding stimuli as points in a geometric space. To the latter we can count the additive difference model of Beals, Krantz, & Tversky [1968] and Tversky & Krantz [1970].

Despite these differences the two models lead to the same conclusions on the characteristics of similarity judgments.

Recently, D’Agostino and Cureton {1972] discussed a test of normality against skewed alternatives; the test was based upon g1, the third standardized sample moment. In an earlier paper, D’Agostino [1970] suggested that an experimenter might be interested in testing the underlying assumption of normality associated with an analysis of variance. For such a test, one would be required to decide if k independent random samples were drawn from k normal populations. The purpose of this note is to suggest a simple extension of the test based upon g1, to the case of k independent random samples.

In articles by Homn [1969] and Carroll and Chang [1970] and in a memorandum by Bloxom [1968], a model was proposed for individual differences in multidimensional scaling. The distance in a p-dimensional space between stimuli j and k for person i is assumed to be given by

for N persons and for all pairs of n stimuli, where wim, is the “weight” given to axis m by person i and ajm, is the projection of stimulus j on that axis.