The gain from selection (GS) is defined as the standardized average performance of a group of subjects selected in a future sample using a regression equation derived on an earlier sample. Expressions for the expected value, density, and distribution function (DF) of GS are derived and studied in terms of sample size, number of predictors, and the prior distribution assigned to the population multiple correlation. The DF of GS is further used to determine how large sample sizes must be so that with probability .90 (.95), the expected GS will be within 90 percent of its maximum possible value. An approximately unbiased estimator of the expected GS is also derived.

For random-model, fully-crossed, two- and three-facet experimental designs the following two problems were considered. First, equations were developed for determining the optimal number of conditions of a facet for maximizing the coefficient of generalizability under the constraint that the total number of observations per subject is constant. Second, the problem of determining the minimum number of observations per subject for a specified generalizability coefficient is solved for the two-facet crossed design.

The use of one-way analysis of variance tables for obtaining unbiased estimates of true score variance and error score variance in the classical test theory model is discussed. Attention is paid to both balanced (equal numbers of observations on each person) and unbalanced designs, and estimates provided for both homoscedastic (common error variance for all persons) and heteroscedastic cases.

It is noted that optimality properties (minimum variance) can be claimed for estimates derived from analysis of variance tables only in the balanced, homoscedastic case, and that there they are essentially a reflection of the symmetry inherent in the situation. Estimates which might be preferable in other cases are discussed. An example is given where a natural analysis of variance table leads to estimates which cannot be derived from the set of statistics which is sufficient under normality assumptions. Reference is made to Bayesian studies which shed light on the difficulties encountered.

In line with the latent trait model, the continuous response level is defined and considered, in contrast to the discrete response levels, which have already been explored by the author. Discussions are mainly focused on the homogeneous case and the open response situation. The operating density characteristic of the continuous item score is defined. Also the basic function, information functions and the positive-exponent family are discussed on the continuous response level, in connection with the sufficient condition that a unique maximum estimate is provided for the response pattern, which consists of the continuous item scores.

Birnbaum's three-parameter logistic model for the multiple-choice item in the latent trait theory is considered with respect to the item response information function and the unique maximum condition. It is clarified that with models of knowledge or random guessing nature, which include the three-parameter logistic model, the unique maximum condition is not satisfied for the correct answer, and the item response information function is negative for the interval (− ∞, θg). It is suggested that we should use θg as a criterion in selecting optimal items for a specified group of examinees, so that we can practically avoid the possibility of non-unique maxima of the likelihood function on the response pattern given by an examinee in the group.

A probabilistic unfolding model is presented which makes it possible to scale stimuli and subjects simultaneously, without assumptions concerning the distribution of subjects or of stimuli. The applicability of the model is demonstrated in areas of attitude measurement where the assumption of normality is inadequate.

A test statistic is introduced which allows one to test the hypothesis of agreement of several judges on the ranking of items within each of two groups and between the two groups. The groups of judges may be unequal in size. A normal approximation for the test statistic is developed. The relationship to existing techniques given by Kendall, Friedman, Page, Spearman, and Lyerly is discussed. A generalization of the coefficient of concordance is presented and the extension of the method to multi-group problems is suggested.

In this paper a rapid and reliable method is found for estimating the value of the Bivariate Normal Correlation Coefficient, ρ, given values of the joint probability and the normal deviates, h and k, or the related areas. This technique finds useful application in the computational approximation of the tetrachoric correlation coefficient, r, when the underlying distributions may be assumed to be normal.

In Coombs' unidimensional unfolding theory each individual ranks all stimuli, often in terms of preference judgments. These ranks, called I scales, are used to infer the latent continuum called a J scale which is presumed to have generated the I scale rankings. A major problem concerns the inference of the J scale given a set of I scales because an I scale is not unique to any one J scale. This paper presents a procedure for estimating J scale probabilities given a set of I scales. Models for three and four or more stimuli are presented. Results of computer tests are reported.