Some of the methodologies that have become standard tools in psychometrics suffer from neglect. They are taken too much for granted and are not given the attention that seems appropriate to the important role they play in research advances. I propose to make some suggestions which may, in a modest way, assist in alleviating this difficulty. In a very general way I would like to suggest that effort expended in examining a variety of restatements of a methodological problem may lead to new methodologies of real value.

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.

The methods used in recent probabilistic learning models to generate mean curves of learning under random reinforcement are extended to the general case in which probability of reinforcement may vary in any specified manner as a function of trials and to cases in which probability of reinforcement on a given trial is contingent upon responses or outcomes of preceding trials.

The likelihood-ratio significance test is derived for the hypothesis that after correction for attenuation two variables have a perfect correlation in the population from which the sample is drawn.

A general function is derived describing the conditioning of a single stimulus component in a discriminative situation. This function, together with the combinatorial rules of statistical learning theory [5, 12], generates empirically testable formulas for learning of classical two-alternative discriminations, probabilistic discriminations, and discriminations based on the outcomes of preceding trials in partial reinforcement experiments.

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.

Consideration is given mathematical problems arising in two learning theories—one developed by Bush and Mosteller, the other developed by Estes. The theory of Bush and Mosteller leads to a class of Markov processes which have been studied in considerable detail (see [1] and [7]). The Estes model can be treated as a Markov chain, i.e., a Markov process with a finite number of states. For an important class of special cases, it is shown that the Bush-Mosteller model is, in a sense, a limiting form of the Estes model. The limiting probability distributions are derived for the cases treated in both models.

The concepts of differential prediction and multiple absolute prediction were developed in earlier papers [2, 3]. Methods for determining optimal distribution of testing time for each type of prediction are available [4, 5] and are appropriate for use provided that no altered time allotment approaches zero. In this article the methods developed in [4, 5] are extended to include cases where the altered time allotment for one or more tests may approach zero. The procedures developed are illustrated by numerical examples, after which the mathematical rationales are provided.

Solutions of the communality problem and of the problem of meaning of common and unique factors have been shown previously to depend intimately on certain relations with ordinary multiple correlation. To make these basic propositions more accessible, simple proofs of some of them are provided here, avoiding any matrix algebra. New results are also obtained, with no extra work, that extend the previously known propositions to a more general class of coefficients than that of communalities.

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 problem of estimation in the method of successive categories is reconsidered and a new least squares solution is obtained. An empirical comparison of this solution with Gulliksen's solution is presented.

A mathematical model is developed in an attempt to relate errors in multiple stimulus-response situations to psychological inter-stimulus and inter response distances. The fundamental assumptions are (a) that the stimulus and response confusions go on independently of each other, (b) that the probability of a stimulus confusion is an exponential decay function of the psychological distance between the stimuli, and (c) that the probability of a response confusion is an exponential decay function of the psychological distance between the responses. The problem of the operational definition of psychological distance is considered in some detail.

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.

The communality of a variable represents the degree of its generality across n − 1 behaviors. Domain-sampling principles provide a fundamental conception and definition of the communality. This definition may be alternatively stated in eight different ways. Three definitions lead to precise formulas that determine the true value of the communality: (i) from the k necessary and sufficient dimensions derived by iterated factoring, (ii) from the n − 1 remaining variable-domains, and (iii) from k′ multiple clusters of the n variables. Seven definitions provide approximation formulas: (i) one from the k dimensions as initially factored, (ii) one from the n − 1 remaining variables, and (iii) five from a single cluster. Rank of the matrix is not a desiratum in some definitions. Using an example designed by Guilford to illustrate multiple-factor analysis, applications of the formulas based on the three precise definitions recover the true communalities, and five approximation formulas each gives values closer than the ad hoc estimates usually employed in factor analysis.

A general least squares solution for successive intervals is presented, along with iterative procedures for obtaining stimulus scale values, discriminal dispersions, and category boundaries. Because provisions for weighting were incorporated into the derivation, the solution may be applied without loss of rigor to the typical experimental matrix of incomplete data, i.e., to a data matrix with missing entries, as well as to the rarely occurring matrix of complete data. The use of weights also permits adjustments for variations in the reliability of estimates obtained from the data. The computational steps involved in the solution are enumerated, the amount of labor required comparing favorably with other procedures. A quick, yet accurate, graphical approximation suggested by the least squares derivation is also described.

For continuous distributions associated with dichotomous item scores, the proportion of common-factor variance in the test, H2, may be expressed as a function of intercorrelations among items. H2 is somewhat larger than the coefficient a except when the items have only one common factor and its loadings are restricted in value. The dichotomous item scores themselves are shown not to have a factor structure, precluding direct interpretation of the Kuder-Richardson coefficient, rK-R, in terms of factorial properties. The value of rK-R is equal to that of a coefficient of equivalence, H2Φ, when the mean item variance associated with common factors equals the mean interitem covariance. An empirical study with synthetic test data from populations of varying factorial structure showed that the four parameters mentioned may be adequately estimated from dichotomous data.

Empirical and fictitious examples are described for investigating the applications of the absolute scaling method for item scaling and score scaling. A discrepancy between the correct values and the values estimated through the absolute scaling method is demonstrated. It is concluded that when the groups are different the assumption of an identity between test score conversion and item difficulty conversion is not met.

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.

The proposed quantitative description of maze learning rests on the assumption that two independent processes are involved: (i) a discovery process based on trial-and-error search for the correct response, (ii) a fixation process equivalent to that observed in serial learning. The model leads to predictions that are consistent with the available experimental data. In particular, the number of trials required for fixation is independent of the number of alternatives at each choice point (and hence independent of the number of bits of information contained in each correct response).

A method based on configural analysis has been given whereby test scoring techniques can be evaluated to see if they have optimal validity. Configural analysis has also been used to show how three well known item scoring techniques, multiple regression, total score, and multiple cut-off, imply (for optimal validity) certain conditions on the answer pattern means. The method is illustrated by a worked example.