Formulas are derived for unbiased sample estimators of any raw or central moment of the frequency distribution of true scores. A general method is developed for obtaining from each examinee's observed score a least squares estimate of his true score.

Under the assumption that responses to different dichotomous items are statistically independent, exact distributions for coefficients of reproducibility, CR, are derived. These distributions are useful in testing whether an observed coefficient of reproducibility differs significantly from chance expectations and whether further scaling manipulations are warranted. The effects of purification on sampling distributions of errors are calculated to demonstrate a relationship between scaling operations and expected chance values of the CR.

Simple statistical methods are developed to test whether coefficients of reproducibility, of homogeneity, or of consistency differ significantly from what can be expected if responses to different items are statistically independent. Simple methods are also developed for estimating the variance of coefficients of reproducibility when it is not assumed that responses to different items are independent. These estimates are used to test whether a coefficient differs significantly from any prescribed value, and also to obtain confidence intervals for these coefficients. The rationale for the measurement of reproducibility is also discussed.

A method is presented for securing a hierarchical factor solution which achieves simple structure at each hierarchical level without rotation or even preliminary arbitrary orthogonal or oblique solutions. The method is based upon the assumption that if overlap is removed from clusters the remaining specifics will achieve simple structure automatically. The problem presented earlier by Schmid and Leiman, using oblique simple structural rotation as a basis, is reworked by this new approach.

A generalization of the Bush-Mosteller learning model is proposed in connection with two-alternative learning situations with continuous reinforcement. The problem is to test the hypothesis that reward and non-reward are equally effective in promoting learning. Statistically this reduces to testing a hypothesis about the value of a single parameter θ, while a set of other parameters remains unspecified. The test presented has the property of being asymptotically locally most powerful among all tests of the same size and asymptotically similar. The application of the test is illustrated.

An exposition of a calculus for empirical classes (CEC), one of the few attempts by logicians to deal with the problem of constructs and indicators, is presented. The CEC provides the groundwork for a formal structure for the situation in which individuals have a degree of membership in various classes rather than having either membership or nonmembership—a situation nearly always true in empirical research. The CEC is presented and its relation to various social science concepts is mentioned. An application of the CEC model to latent structure analysis (LSA) suggests alternatives to the local independence assumption including one called the local scale assumption, which has a close relation to a Guttman scale.

Formulas are given for a descriptive statistic related to the fourfold-point correlation but having always-attainable limits ±1.