This paper is a presentation of the statistical sampling theory of stepped-up reliability coefficients when a test has been divided into any number of equivalent parts. Maximum-likelihood estimators of the reliability are obtained and shown to be biased. Their sampling distributions are derived and form the basis of the definition of new unbiased estimators with known sampling distributions. These unbiased estimators have a smaller sampling variance than the maximum-likelihood estimators and are, because of this and some other favorable properties, recommended for general use. On the basis of the variances of the unbiased estimators the gain in accuracy in estimating reliability connected with further division of a test can be expressed explicitly. The limits of these variances and thus the limits of accuracy of estimation are derived. Finally, statistical small sample tests of the reliability coefficient are outlined. This paper also covers the sampling distribution of Cronbach's coefficient alpha.

Criterion measures are frequently obtained by averaging ratings, but the number and kind of ratings available may differ from individual to individual. This raises issues as to the appropriateness of any single regression equation, about the relation of variance about regression to number and kind of criterion observations, and about the preferred estimate of regression parameters. It is shown that if criterion ratings all have the same true score the regression equation for predicting the average is independent of the number and kind of criterion scores averaged.

Two cases are distinguished, one where criterion measures are assumed to have the same true score, and the other where criterion measures have the same magnitude of error of measurement as well. It is further shown that the variance about regression is a function of the number and kind of criterion ratings averaged, generally decreasing as the number of measures averaged increases. Maximum likelihood estimates for the regression parameters are derived for the two cases, assuming a joint normal distribution for predictors and criterion average within each subpopulation of persons for whom the same type of criterion average is available.

Proof is given for a theorem stated but not proved by Eckart and Young in 1936, which has assumed considerable importance in the theory of lower-rank approximations to matrices, particularly in factor analysis.

Three models for preference behavior are developed within the framework of Coombs' theory of data, an absolute difference model, a ratio model, and a two-stage model. Each of these models describes a mechanism by which unilateral preferences may be determined on a unidimensional J scale. Differential implications of the models for response latencies are derived, and some early data employing an application of the unfolding technique are presented in support of the two-stage model.

The purpose of this paper is to develop mathematical formulations of test reliability when the estimation of reliability requires successive observations and these observations are not independent of the process (or act) of measurement. The specific application of concern arises in performance testing where performance varies as a function of practice and where practice accrues as a result of measurement. Repeated measures are experimentally nonindependent in such cases. Formulations for the estimation of reliability are developed for each of two cases. The first of these assumes a linear model for acquisition; the second assumes a nonlinear (quadratic) model. Illustrative data are presented whereby insights may be gained as to the inadequacy of the models.

The Fisher-Yates “Exact Test” has become widely known to researchers in psychology as the significance test of choice for comparing pairs of proportions in 2 × 2 contingency tables, particularly when the expected frequency of one or more of the cells is small. Less widely recognized are some readily available tabular aids for calculating the “Exact Test” in a substantial number of the cases in which it might be applied. Thus, Finney [1] has tabled probability levels at or beyond the .05 level for those 2 × 2 tables in which neither of the two row marginals exceeds 15. Since it is a matter of arbitrary choice as to which pair of marginals of an obtained table is designated the row pair and which the column pair, Finney’s table has broad applicability. It has been incorporated into the Biometrika Tables for Statisticians [3], which also present extensive discussion of its applications and extensions. Finney’s table has been extended by Latscha [2] to accommodate those contingency tables in which neither of the two row marginals exceeds 20. The Biometrika Tables provide additional, if less direct, help in their tables of n! for values of n up to 250, and of log10n! for values of n up to 1000.