Techniques for partitioning objects into optimally homogeneous groups on the basis of empirical measures of similarity among those objects have received increasing attention in several different fields. This paper develops a useful correspondence between any hierarchical system of such clusters, and a particular type of distance measure. The correspondence gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data. In an explicitly defined sense, one method forms clusters that are optimally “connected,” while the other forms clusters that are optimally “compact.”

The Thurstone and the Bradley-Terry models, both initially advanced on intuitive grounds, have proved useful in the analysis of paired comparisons. The psychological meaning of these models and their relation to one another is unclear, but they fit data. Stevens has observed that there may be two basic mechanisms of discrimination 1) additive and 2) substitutive. We advance two corresponding mathematical models: that experienced sensation is 1) the sum of a large number of independent signals and 2) the maximum of a large number of independent signals. These assumptions yield 1) Thurstone's model and 2) the model of Bradley-Terry. Psychological interpretations of the various parameters, in terms of sensation, present themselves in a natural manner. Thus this paper presents a theory which unifies and interprets two paired comparison models that have proved useful in fitting experimental data.

Formulas are derived for the asymptotic variances and covariances of the maximum likelihood estimators for oblique simple structure models which are identified by prior specification of zero elements in the factor loading matrix. The formulas are expressed in terms of the various submatrices of the inverse of the required variance-covariance matrix. A numerical example using artificial data is given and problems in the application of the formulas discussed.

In an earlier paper [Psychometrika, 31, 1966, p. 147], Srivastava obtained a test for the Hypothesis H0 : Σ =α0Σ0 + ... +αlΣl, where Σi are known matrices,αi are unknown constants and Σ is the unknown (p × p) covariance matrix of a random variablex (withp components) having ap-variate normal distribution. The test therein was obtained under (p × p) covariance matrix of a random variablex (with p components) the condition that Σ0, Σ1, ..., Σl form a commutative linear associative algebra and a certain vector θ, dependent on these, has non-negative elements. In this paper it is shown that this last condition is always satisfied in the special situation (of importance in structural analysis in psychometrics) where Σ0, Σ1, ..., Σl are the association matrices of a partially balanced association scheme.

A signed graph,S, is colorable if its point set can be partitioned into subsets such that all positive lines join points of the same subset and all negative lines join points of different subsets.S is uniquely colorable if there is only one such partition. Developed in this note is a new matrix, called the type matrix of S, which provides a classification of the way pairs of points are joined in S. Such a classification yields a criterion for colorability and unique colorability.

In this paper a method is developed for analyzing data resulting from the use of a modification of the usual paired comparisons procedure which allows for ratings of size of difference. The scaling model is essentially an extension of the Thurstonian successive intervals model. The method is applied to scaling of a set of nine “immoral” actions and the results agree rather well with those from conventional successive intervals scaling of the same stimuli.

A Monte Carlo approach is employed in determining whether or not certain variables produce systematic effects on the sampling variability of individual factor loadings. A number of sample correlation matrices were generated from a specified population, factored, and transformed to a least-squares fit to the population values. Influences of factor strength, communality and loading size are discussed in relation to the statistics summarizing the results of the above procedures. Influences producing biased estimators of the population values are also discussed.

For the tests in which the score on an item is not restricted to 0 and 1, but is any number on a continuous scale, a procedure for estimating an examinee's true score is given. For the case of 0, 1 item scoring this problem was considered by Lord [1959]. Following Lord, the least squares estimation procedure is used and the regression coefficient is obtained, which is compared with the generalized KR(20) and KR(21) formulas. Also, results are discussed using analysis of variance models.

Estimates of test size (probability of Type I error) were obtained for several specific repeated measures designs. Estimates were presented for configurations where the underlying covariance matrices exhibited varying degrees of heterogeneity. Conventional variance ratios were employed as basic statistics in order to produce estimates of size for a conventional test, an ∊-adjusted test, and ∊-adjusted test and a conservative test. Indices for empirical distributions of two estimators of ∊j, a measure of covariance heterogeneity, were also provided.