The experimental design considered in this paper is one in which each of a group of several subjects is observed in the presence of each of the other subjects of the group, the entire set of possible pairings being repeated or replicated on several occasions. Analysis of variance formulas are described for this somewhat unusual design. Both the model of constants and mixed model are considered. Reliability formulas growing out of the analysis of variance calculations are developed.

Observed rankings of objects can be treated as arising from a time dependent probability process. Under such circumstances, associations observed are an indication of the character of this underlying process. In the particular example treated in some detail here, a quantity related to Kendall's tau is found to have an important role and its properties are examined.

The representation of test scores as n-dimensional points leads directly to an estimate of error variance at a particular score level in the case of equivalent items. Approximations are suggested for the case of non-equivalent items. These approximations are compared, with satisfactory results, with empirical data prepared by Dr. Mollenkopf.

The detailed method of optimal regions is an extended form of the method of optimal regions which has been found effective in solving the personnel classification problem when the number of job categories is small. The automatic determination of the successive values of the vi, made possible by the more exact techniques of the detailed method, provide easier solutions for the more complex problems and provide solutions, which, for the most part, can be mechanized. In a sense the detailed method of optimal regions is more than a detailed form of the method of optimal regions. It is essentially a method of transformations by which the original matrix is reduced to a matrix from which the solution is easily obtained.

Although simple structure has proved to be a valuable principle for rotation of axes in factor analysis, an oblique factor solution often tends to confound the resulting interpretation. A model is presented here which transforms the oblique factor solution so as to preserve simple structure and, in addition, to provide orthogonal reference axes. Furthermore, this model makes explicit the hierarchical ordering of factors above the first-order domain.

A method of pattern analysis is presented for the case of dichotomous items and a quantitative criterion. This “configural scale” has maximum validity in the least squares sense. A technique for computing the configural scale as a polynomial function of the item scores is given. Tests of significance are outlined for such questions as: Is there a linear or non-linear relation between the quantitative criterion and the item scores? Does the addition of certain items to the test increase the validity of the configural scale? Are all the items in the configural scale fully effective?

An expression for the expected variance of the sampling errors for the validities of a set of correlated items that is computationally feasible when the number of items is large is developed. Since the item difficulties are assumed to be constant, the estimate must be applied to pools or sub-pools of items reasonably homogeneous with respect to difficulty.

For the purpose of extracting factors from matrices, it is proved that a certain formula is both necessary and sufficient. In factor analysis, the formula may be applied either to the correlation matrix, or directly to the score matrix (assuming the communality problem is solved). As many factors as desired can be extracted in one operation. Having such a compact formulation is useful for teaching as well as computing purposes, since it includes all techniques of factor extraction as special cases.

The use of binomial coefficients in place of factorials to shorten the calculation of exact probabilities for 2 × 2 and 2 × r contingency tables is discussed. A useful set of inequalities for estimating the cumulative probabilities in the tail of the distribution from the probability of a single table is given. A table of binomial coefficients with four significant places and n through 60 is provided.

A model for the acquisition of responses in an anticipatory rote serial learning situation is presented. The model is developed in detail for the case of a long intertrial interval and employed to fit data where the list length is varied from 8 to 18 words. Application of the model to the case of a short intertrial interval is considered; some predictions are derived and checked against experimental data.

The tau coefficient is defined, and a computational procedure for tied ranks is described. The procedure maintains continuous computational checks, saves labor, and particularly facilitates the use of tau with large samples. It is also shown how tau correlation may be applied to Q-sorts with any shape of forced distribution or with unforced distributions.

By interchanging persons and items, iterative inverse factor analysis provides a relatively inexpensive way of clustering persons according to their patterns of response to the items. In addition to permitting the clustering of large numbers of persons, the technique enables one to determine the bases for such clustering. The items of behavior used can be heterogeneous in content and form.