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.

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.

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.

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.

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.

A method is presented for converting the scores on one form of a test to those on another form of the same test. The method is particularly applicable to the case where each form has been administered to a different group and the only link between the two forms is a subset of items common to both. The proposed method, called the item method of conversion, has been applied to several tests for which other methods of conversion are available for comparison. The necessary data are limited to tests for which the total score is the criterion for item analyses. The method gives highly satisfactory results for all the tests to which it has been applied, particularly when the two groups are rather different, in which case the delta method (a different item method) is inappropriate.

In contrast to Thurstone's Law of Comparative Judgment, a model in which a comparison pair and its complement are assumed to give rise to two different distributions of differences is considered. The consequences of this revised model on scaling problems is developed.

A method of rotating a set of orthogonal axes into a reference frame on which loadings are as nearly proportional to a predetermined set of weights as possible is presented. The method, an approximate algebraic solution, often requires some additional graphical refinement but eliminates most of the rotations involved in usual graphical solutions. Its primary value is speed and ease of calculation, involving only one matrix multiplication and solution of a simple formula to determine the rotation cosines.