A basis for analyzing test-retest reliability.

Three sources of variation in experimental results for a test are distinguished: trials, persons, and items. Unreliability is defined only in terms of variation over trials. This definition leads to a more complet~ analysis than does the conventional one; Spearman’s contention is verified that the conventional approach--which was formulated by Yule--introduces unnecessary hypotheses. It is emphasized that at least two trials are necessary to estimate the reliability coefficient. This paper is devoted largely t~ developing lowe.r baunds to the reliability coefficient that can be computed from but a single trial; these avoid the experimental difficulties of making two independent trials. Six different lower bounds are established, appropriate for different s~.tuations. Some of the bounds are easier to compute than are conventional fornmlas, and all the bounds assume less than do conventional formulas. The terminology used is that of psychological andsociological testing, but the discussicn actually provides a general analysis ofthe reliability of the sum of n variables.