A quantitative system is presented to permit the determination of separate multidimensional perceptual spaces for individuals having different viewpoints about stimulus interrelationships. The structure of individual differences in the perception of stimulus relationships is also determined to provide a framework for ascertaining the varieties of consistent individual viewpoints and their relationships with other variables.

Forty-two choice models, each representing stimuli by one-dimensional probability distributions, are obtained by relaxing the assumptions of Thurstone's Case V Law of Comparative Judgment. The models which imply or fail to imply each of nine testable probabilistic conditions are determined. Stochastic transitivity is vulnerable in most of these models. The results suggest discarding weak stochastic transitivity, and in its place using the conjunction of weak stochastic transitivity and the triangular condition. However, unless it is possible to predict which stimuli will produce violations of the conditions, none of the conditions can be rejected on the basis of too frequent intransitive triads of choices.

An axiomatization of mental test theory is presented. The presentation deals explicitly with the following concepts: true score, parallel forms, reliability, validity, and combined test score.

Moses has proposed a nonparametric test of the null hypothesis that two samples, say a treatment and a control group, have been drawn from the same population; his test is designed to be sensitive to the alternative hypothesis that the treatment group is extreme in either or both directions as compared with the controls. This paper presents a test of similar character, except that rather than being based on a range of the control group ranks, it depends upon the variance of those ranks. This has the advantage of making ties less troublesome. Tables of significance values are presented for some small sample sizes. A normal approximation is developed and seen to be useful.

A two-variable generating function is described which yields the sampling probabilities of the Shannon-Wiener information measure. Expansion and collection of terms in like powers of the first variable imposes the restriction that the sum of the k category frequencies equal n; collection of terms in like powers of the second variable then produces terms whose coefficients are the required probabilities. The method may be used with either equal or unequal category probabilities for any finite n and k, and thus represents a general solution to the small sample problem. Tables of sampling probabilities are presented.

The powered vector method of factor analysis yields directly and without rotation factors satisfying objectives of parsimony, orthogonality, and meaningfulness. The method is objective, computationally efficient, and easily programmed for digital computers. The computational procedure is described. Illustrative analyses are presented. Results of applications of the powered vector method are compared with results obtained using the principal axes solution followed by orthogonal rotation.

On the assumption that a partitioning can be found such that three mutually exclusive test vector configurations span the same factor space, a procedure is derived whereby symmetric parts of the correlation matrix are estimated from functions of asymmetric parts treated symmetrically. This yields an explicit matrix formula for communality estimation which generalizes earlier work by Albert. Conventional factoring methods, with all their computational and fitting advantages, can be applied once the symmetric portions of the correlation matrix have been estimated. Extension to four subgroups of test vectors allows for a matrix generalization of the old tetrad difference criterion to the multiple-factor case.