Many well-known equations in classical test theory are mathematical identities in populations of individuals but not in random samples from those populations. First, test scores are subject to the same sampling error that is familiar in statistical estimation and hypothesis testing. Second, the assumptions made in derivation of formulas in test…

Results of this study indicate that the correlation between half-test scores over repeated splits, over persons, and over repeated testings resulting in different sets of observed scores, is given by Kuder-Richardson Formula 21. (RF)

Research supported by the National Research Council of Canada, Grant APA-252-2057-B.

The reliability of differences between pretest and posttest measures with intervening experimental treatment can be highly reliable under some conditions. If the error components of pretest and posttest measures are correlated, the classical formula for the reliability of differences underestimates the actual reliability of gains. (Author/CM)

The correlation between bias components of test scores and unbiased observed scores is shown to be an effective predictor of changes in reliability and validity resulting from elimination of bias. Plausible assumptions about value of correlation and size of related variance components indicate that reducation in reliability and validity is a…

Classical test theory findings can be derived from the concepts of conditional expectation, conditional independence, and related notions. It is shown that these concepts provide precisely the formalism needed to obtain the classical results with minimal assumptions and with greatest economy in the methods of proof. (RC)

The usual formulas for the reliability of differences between two test scores are based on the assumption that the error scores are uncorrelated. Formulas are presented for the general case where this assumption is unnecessary. (Author/JKS)

Although a great deal of attention has been devoted over a period of years to the estimation of reliability from item statistics, there are still gaps in the mathematical derivation of the Kuder-Richardson results. The main purpose of this paper is to fill some of these gaps, using language consistent with modern probability theory. (Author)

Formulas for the standard error of measurement of three measures of change (simple differences; residualized difference scores; and a measure introduced by Tucker, Damarin, and Messick) are derived. A practical guide for determining the relative error of the three measures is developed. (Author/JKS)

A mathematical link between test reliability and test validity is derived, taking into account the correlation between error scores on a test and error scores on a criterion measure. When this correlation is positive, the "paradoxical" nonmonotonic relation between test reliability and test validity occurs universally. (Author/BW)

This paper provides a list of 10 salient features of the standard error of measurement, contrasting it to the reliability coefficient. It is concluded that the standard error of measurement should be regarded as a primary characteristic of a mental test. (Author/DWH)

It is suggested that error of measurement cannot be routinely incorporated into the "error term" in statistical tests, and that the reliability of test scores does not have the simple relationship to statistical inference that one might expect. (Author/GK)

Coefficient alpha was examined through computer simulation as an estimate of test reliability under violation of two assumptions. Coefficient alpha underestimated reliability under violation of the assumption of essential tau-equivalence of subtest scores and overestimated it under violation of the assumption of uncorrelated subtest error scores.…

Reliability coefficients of linear combinations of observed scores have anomalous properties which have led to difficulties in the investigation of difference scores and gain scores in test theory. Discrepancies between classical results and correct results obtained from more general formulas, which allow for correlated errors, are examined…