Tables for Transmutation of orders of Merit into Normal Equivalents

Reliability coefficients of linear combinations of observed scores have anomalous properties which have led to persistent difficulties in the investigation of difference scores and gain scores in test theory. Interpretation of these test scores is further complicated by effects of correlated errors of measurement which are likely to appear in difference scores and gain scores in practice. In this paper the discrepancies between classical results and correct results obtained from more general formulas, which allow for correlated errors, are examined systematically. These discrepancies depend strongly on the reliability coefficients of the respective tests and are smallest when the influence of the variables related by the formulas is least. A vector representation of difference scores reveals that these anomalies arise from simple geometric relations among observed scores, true scores, and error scores inherent in the test-theory model. In this context, doubts as to the usefulness of difference scores and gain scores in testing practice expressed by previous authors appear to be justified.