The problem of determining linear functions for two sets of variables so as to maximize the correlation between the two functions has been solved by Hotelling. This article presents a more efficient computational solution for the case of two sets of variables and a generalized solution for any number of sets. Applications are discussed and a numerical example is included to demonstrate the solution for more than two sets.

The selection of one of several stimuli as most similar to a reference stimulus is assumed to satisfy a choice axiom that permits assigning ratio scale values to each variable-reference stimuli pair. The logarithm of this scale is treated as a distance measure, leading to the following testable conclusions about the pairwise choice probabilities as the reference stimulus is varied. First, the plot is a symmetrically truncated ogive with horizontal tails. Second, if two pairs of choice stimuli have the same midpoint, the ogive of one pair is part of the ogive of the other. In terms of this model, the hysteresis and midpoint displacement effects in the method of bisection are discussed, and relations with Coombs' unfolding techniques are explored.

By virtue of certain modifications in the Law of Comparative Judgment, equations are developed which (i) permit the construction of a joint scale of individuals and items, as in the case of attitude measurement, directly from their pair-comparison preferences, and (ii) take into account the variable of laterality which is significant for the construction of group preference scales.

From a theoretical point of view, paired comparisons and the law of comparative judgment provide an excellent approach to the problem of psychological measurement. However, if a reasonably large number of stimuli are to be investigated, paired comparisons become extremely time-consuming and fatiguing to the subjects. A balanced incomplete block design, requiring multiple rank order judgments for each subject, provides an efficient experimental method for obtaining paired comparisons judgments. Features of the analysis proposed for this design are discussed in detail. A program for the analysis is available for the IBM 650 electronic computer.

A stochastic model is proposed to account for the behavior of subjects in recognition tasks in which stimuli are presented, one at a time, in a protracted sequence. The basic assumption is that the memory trace resulting from the presentation of a particular stimulus not only fades away during the presentation of subsequent stimuli but also “diffuses” in such a way as to become decreasingly stimulus specific. An account is thereby provided for both (a) the increase in the probability of false recognition with the total number of stimulus presentations and (b) the departure of curves of forgetting from the previously proposed simple exponential decay functions. An expression for the amount of information carried along when the number of stimulus presentations becomes large is then derived for subjects who conform with the model.

The seven analysis-of-variance mean squares for an unreplicated three-way classification may be written as linear combinations of a mean variance and three mean covariances. Formulas are presented for computing the mean variances and mean covariances from linear combinations of mean squares. The relevance of these formulas for assessing rater biases and trait independence is discussed, a numerical example is provided, and proposed extensions are briefly noted.

Within the model of isotonic space, a principle is presented which generalizes the unfolding technique to the multidimensional case. The availability of exhaustive configurational solutions given complete data is pointed out. Finally three criteria are suggested for the choice of a particular solution from among the set of all solutions, which are applicable in the case either of complete or incomplete data.

Maximum likelihood estimates of item parameters of a scholastic aptitude test were computed using the normal and logistic models. The goodness of fit of ogives specified by the pairs of item parameters to the observed data was determined for all items. While negligible differences in the limen values were found, differences in item discrimination indices indicated that interpretation of these indices requires separate frames of reference. The empirical results showed the logistic model to be a useful alternative to the normal model in item analysis.