Generalizability theory concerns the adequacy with which a “universe” score can be inferred from a set of observations. In this paper the theory is applied to a universe in which observations are classifiable according to two independent variable aspects of the measuring procedure. Several types of universe scores are developed and the variance components ascertained for each type. The composition of expected observed-score variance and the adequacy of inference to a particular type of universe score is a function of the procedure used in gathering data. A generalizability study provides estimates of variance components which can be used in designing an efficient procedure for a particular decision purpose.

There have been a number of important developments by several authors which are based on the assumption that test taking behavior is stochastic. That is, it is assumed that the probability that an individual will “endorse” or “pass” a dichotomous item can be represented by some function of an underlying, “latent” attribute (possibly multivariate) called the item trace line or item characteristic curve. Various forms for trace lines have been postulated such as polynomial [9, 11], step function [9], normal ogive [13, 19], and logistic [1, 2]. An interesting model due to Rasch [18] utilizes only a subject parameter and item parameter to determine the probability of passing an item. The only model which does not make some formal assumptions about trace lines is Lazarsfeld's latent class model. All of these models for test behavior are unified by two basic assumptions. One is the stochastic assumption of a probability of passing or endorsing an item.

The present model treats the scaling of pair-comparison preference judgments among a unidimensional set of stimuli across a population of individuals. Given a set S of n stimuli, S = {S1, S2, …, Sn}, the model yields a partially ordered metric on the interstimulus distances which may be used to construct an interval scale of values forS. Obtained also are a set of predictions P = {P1, P2, …, Pn} wherePi is the proportion of individuals in the population whose first choice among the elements of S is Si. A numerical illustration is offered and comparisons are drawn with Coombs' unfolding technique.

Inequalities relating the communalities to the multiple-correlation coefficients are derived. They are stronger than the well-known inequalities which have played an important role in factor analysis for the past thirty years. Necessary and sufficient conditions for equality are obtained.

A set of inequalities on communalities developed by Darroch [1] is used to provide some rules for determining admissible sets of communalities in factor analysis.

By taking advantage of an experimental design which increases the number of observations per subject, the sensitivity of the proposed test is increased. In the first of a series of subexperiments, the score of a randomly selected experimentalS is ranked with the scores of the remaining (control) Ss, and the experimental S is discarded. This procedure is repeated until one S remains. The test criterion, Rn, is the sum of ranks of the experimental Ss; its exact sampling distribution, a normal approximation, and the sensitivity of the test relative to the Wilcoxon T, are described. Because of the large number of scores obtained, the test is particularly useful when preparing Ss is more costly than testing them, e.g., in tests following involved conditioning procedures. Since each S is discarded after experimental treatment, the test fills a real need in studies involving radiation, ablation, or other irreversible treatments.

A reliability study is assumed to be carried out with each of a number of observers making a dichotomous judgment concerning each of a sample of subjects. A nonparametric model is proposed for the errors underlying such judgments, and conditions are given under which Cochran's Q statistic is valid for testing the hypothesis of no systematic differences among the judgments of the different observers. Inferences concerning the probabilities of error are shown to be possible in terms of the intraclass correlation coefficient. A numerical example is given.

Two common methods of measuring personality variables are examined with respect to the ways in which they control respondent biases. A form of the mental-test theory model is developed which makes explicit differences in score components that result from different methodological constraints imposed by the two methods. From the model, it is possible to specify the operations necessary to provide equivalent information from the two sets of data. Performance of these operations in an empirical test supports the model by producing generally higher correlations between the variables having score components which are more closely matched.

Explicit solutions are obtained for a sequence of limiting distributions of response probabilities for the two experimenter-controlled events learning model of Bush and Mosteller [2]. A generalization to the s experimenter-controlled events model is found.

When the purpose of the experiment is to compare treatments, the Sequences × Positions Latin Square has been employed to control unwanted effects attributable to individuals, position, and sequence. This particular Latin Square has been subjected to criticism on the grounds there is confounding due to structure, random variables, and subject interactions. Special Latin Square, a subclass of the Sequences × Positions Latin Square, is basically a p ×p factorial design in blocks of size p. The two factors are treatments (T) and positions (P). Sequence is one component of the TP interaction, and square uniqueness is the sum of the remaining components. This completely replicated factorial design has no structural or random variable confounding; if subject interactions are present, square uniqueness may be used as the error term and the bias in the test of treatments will be conservative.

An objective method for the orthogonal rotation of factors which gives results closer to the graphic method is proposed. First, the fact that the varimax method does not always satisfy simple-structure criteria, e.g., the positive manifold and the level contributions of all factors, is pointed out. Next, the principles of our method which are based on “geometric vector” are discussed, and the computational procedures for this method are explained using Harman and Holzinger's eight physical variables. Finally, six numerical examples by our method are presented, and it is shown that they are very close to the factors obtained from empirical studies both in values and in signs.