A mathematical model for two-choice behavior in situations where both choices are desirable is discussed. According to the model, one or the other choice is ultimately preferred, and a functional equation is given for the fraction of the population ultimately preferring a given choice. The solution depends upon the learning rates and upon the initial probabilities of the choices. Several techniques for approximating the solution of this functional equation are described. One of these leads to an explicit formula that gives good accuracy. This solution can be generalized to the two-armed bandit problem with partial reinforcement in each arm, or the equivalent T-maze problem. Another suggests good ways to program the calculations for a high-speed computer.

Tucker's inter-battery method of factor analysis is shown to assume implicitly that, within the factor space of overlap between the vector configurations of the two test batteries involved, the locations of the two sets of principal axes and the sizes of the associated characteristic roots are identical. A suggested modification of the inter-battery method to avoid these restrictions is considered.

A model is proposed which treats rankings given by a group of judges as representing regions in an isotonic space of dimensionality r. Three possible criteria for estimating lower bound dimensionality are discussed: mutual boundary, cardinality, and the occurrence of transposition groups. Problems associated with each criterion are mentioned.

Methods are given for deciding whether to use some or no predictor variables in a regression analysis. Previously obtained results on the more general problem, whether to use k or k − r predictor variables are reviewed with emphasis on applications.

The Doolittle, Wherry-Doolittle, and Summerfield-Lubin methods of multiple correlation are compared theoretically as well as by an application in which a set of predictors is selected. Wherry's method and the Summerfield-Lubin method are shown to be equivalent; the relationship of these methods to the Doolittle method is indicated. The Summerfield-Lubin method, because of its compactness and ease of computation, and because of the meaningfulness of the interim computational values, is recommended as a convenient least squares method of multiple correlation and predictor selection.

The comparative effects of reward and nonreward on learning are considered in connection with a two-alternative learning situation. Conditions are more general in nature than those discussed in an earlier article. In the statistical model proposed, the question of whether reward and nonreward are equivalent in their effects on learning reduces to testing a composite hypothesis on a multivariate probability distribution. An asymptotic test of this hypothesis is described, and its use is illustrated with data from psychological experiments.

An empirical study of test scores shows the variance of the errors of measurement to be significantly associated with true score in each of four groups studied; it also shows the distribution of the errors of measurement to be significantly skewed in three of these four groups. The mathematical rationale underlying the statistical treatment is presented. Standard error formulas are given for making the necessary significance tests.