Problems in applying the analysis of variance are discussed. Emphasis is placed on using the technique to understand the data. The scale of the dependent variable is important for the analysis. Crossed and nested categories must be recognized. The error terms in the analysis depend on whether the classes of each independent variable are (1) all out of a few or (2) a few out of many. To simplify the analysis, mean squares should be aggregated with their error term when they are less than twice its size. An illustrative example is discussed in detail.

Given the preference ordering of each of a number of individuals over a set of stimuli, it is proposed that if the preference orderings are generated in a Euclidean space of r dimensions which can be recovered by unfolding the preference orderings, then a factor analysis of the correlations between individual's preference orderings will yield a space of r + 1 dimensions with the original r-space embedded in it, and the additional dimension will be one of social utility. The proposition is clearly shown to be satisfied by means of the Monte Carlo technique for both random and lattice stimuli in three dimensions and for two other examples with random stimuli in one and two dimensions.

I am honored to be able to speak on the 25th anniversary of the founding of the Psychometric Society. It would be appropriate for your president on this occasion to review the 25-year history of our society and of mathematical methods in psychology. Fortunately our committee headed by Charles Wrigley has taken care of this need, and speakers better qualified than I will speak to these issues. I shall turn to a less demanding problem.

In limiting my topic I was faced with certain constraints arising from my training, background, and interests. One constraint arises from a misspent youth during which too many years were spent studying disciplines other than mathematics. A second constraint, which probably stems from my training in experimental psychology, is a tendency to be more interested in data than theory. Finally, there is the Air Force personnel research experience which developed in me an abiding interest in applied prediction problems. If you consider this combination to be an odd one for the President of the Psychometric Society, don't blame me. You elected me!

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.

Formulas are derived for using the available item statistics and score statistics on a test to estimate the moments of the score distribution of a lengthened (or shortened) form of the same test. Other formulas are derived for estimating the bivariate moments of the scatterplot between two parallel test forms using only the data available on either form alone. An empirical study is made showing in each case satisfactory agreement between the theoretical values predicted from the formulas and the values actually observed. These results suggest the utility of the true-score model used in deriving the formulas.

This paper studies asymptotic properties of Luce's beta model. Asymptotic results are given for the two-operator and four-operator cases of contingent and noncontingent reinforcement.

The varimax solution for Thurstone's classic Primary Mental Abilities study is presented. Comparisons between the factors of Thurstone's original subjectively rotated factor pattern, Zimmerman's subjectively revised solution, Wrigley, Saunders, and Neuhaus' quartimax results, and the present varimax factor matrix are made by finding correlations between factors defined by these four solutions. It is pointed out that any possible ultimate merit of the varimax solution should be based on its psychological meaningfulness and on the rationale of the varimax criterion—not on its relationship to the other studies.

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 new paired comparison method, based upon choices between lotteries, is developed for the measurement of utilities of objects with respect to the utility of receiving nothing, i.e., the status quo. The method is used to estimate the utilities of four birthday gifts. These objects had also been studied in an earlier experiment which used choices between single objects and pairs of objects to determine a rational origin. A comparison of the results of the two experiments indicates that both methods scale objects with respect to the same rational origin and unit of measurement.

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.

Likelihood ratio tests have been proposed by Wilks for testing the hypothesis of equal means, variances, and covariances (Hmvc) and the hypothesis of equal variances and covariances (Hvc) in a p-variate normal distribution. Using exact distributions of the appropriate likelihood ratio statistics, tables of the .05 and .01 points of these distributions are constructed for p = 4, 5, 6, 7 and sample size n = 25 (5) 60 (10) 100. A correction factor is recommended for larger n. Two numerical examples illustrate use of the tables. A nonparametric test is proposed for Hmvc when the multivariate parent population is known to be non-normal.

A quantitative model for the behavior of albino rats in choice-making situations is presented. The model, which is based upon a cognitive conceptualization of the learning process, is shown to yield predictions which are equivalent to those produced by the linear operator stochastic models at the asymptotic limit but which differ from these during early trials in the learning situation.

Applying a Spearman formula for factor loadings to a variant of the diagonal method, the Guttman simplex model is factored algebraically into n/2 additive factors. The finding that communalities can be discovered such that the rank of a simplex becomes n/2 is contradictory to Guttman's contention that the minimal rank is n − 2. Certain matrices of 4 and 5 variables presented by Guttman as simplexes, can, in general, be considered 2-factor matrices, easily analyzed to simple structure without rotation. One example of 6 variables is factored by the method described to a 3-factor structure.

A generalization of the customary model underlying the Law of Comparative Judgment is established. Methods for estimating location parameters and testing a certain hypothesis are discussed. The problem of the influence of the shape of the curve used to grade responses is raised and a possible approach to its solution is indicated. Two examples are provided.

In the two-choice situation, the Wald sequential probability ratio decision procedure is applied to relate the mean and variance of the decision times, for each alternative separately, to the error rates and the ratio of the frequencies of presentation of the alternatives. For situations involving more than two choices, a fixed sample decision procedure (selection of the alternative with highest likelihood) is examined, and the relation is found between the decision time (or size of sample), the error rate, and the number of alternatives.

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.

Some extensions of the existing determinantal methods for solving the accounting equations in latent class analysis are presented. These extensions cover more cases than previous methods, give rise to new sufficient conditions for identifiability of the latent class model, and give insight into the necessity of various sufficient conditions for identifiability. These implications to the identifiability problem are discussed.

A distinction is made between reliability data and decision data. Each of these sets of data may be matched or independent, depending on whether the same instruments (tests, judges, etc.) are applied to every individual in the group or the instruments to be applied to each individual are selected independently for him. Reliability formulas are developed (for both single observations and for composites of k observations) for the case where reliability data are matched but decision data are independent. Formulas previously reported in the literature are inappropriate for this case.

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.

When multiple significance tests are computed, a certain number of “significant” findings will emerge simply because of chance fluctuations. In the present paper, some factors affecting the number of nominally significant results are elaborated and a general method is suggested which permits unbiased inference as to the significance of a set of findings, as a set. The method advocated employs a high speed computer to generate empirically a sampling distribution tailormade to a particular data matrix. The method is illustrated in the case of dichotomous response to inventory items, where it is found that the statistical model still often used as a basis for estimation is overly conservative. Some problems in the application of the method are discussed.