Coefficients for Tests from a Decision Theoretic Point of View

From a decision theoretic point of view a general coefficient for tests, d, is derived. The coefficient is applied to three kinds of decision situations. First, the situation is considered in which a true score is estimated by a function of the observed score of a subject on a test (point estimation). Using the squared error loss function and Kelley's formula for estimating the true score, it is shown that d equals the reliability coefficient from classical test theory. Second, the situation is considered in which the ob served scores are split into more than two cate gories and different decisions are made for the categories (multiple decision). The general form of the coefficient is derived, and two loss functions suited to multiple decision situations are described. It is shown that for the loss function specifying con stant losses for the various combinations of cate gories on the true and on the observed scores, the coefficient can be computed under the assumptions of the beta-binomial model. Third, the situation is considered in which the observed scores are split into only two categories and different decisions are made for each category (dichotomous decisions). Using a loss function that specifies constant losses for combinations of categories on the true and ob served score and the assumption of an increasing regression function of t on x, it is shown that coeffi cient d equals Loevinger's coefficient H between true and observed scores. The coefficient can be com puted under the assumption of the beta-binomial model. Finally, it is shown that for a linear loss function and Kelley's formula for the regression of the true score on the observed score, the coefficient equals the reliability coefficient of classical test theory.