A Note on Decision Theoretic Coefficients for Tests

Recently it was suggested that the Bayes risk might be used to characterize tests. To conform to common practices about indexes, a rescaling of the Bayes risk was proposed. The motivation for this new coefficient, d, was to provide an index that has a large value when the Bayes risk is small and that has a value in the closed interval [0, 1]. However, since d might have a value outside this range, a modification of d is described which yields an index that always has a value between zero and one. Recently, van der Linden and Mellenbergh (1978) argued in favor of using decision theoretic techniques for characterizing a test; they pointed out that the Bayes risk is a natural index to use from this point of view. Van der Linden and Mellenbergh also noted that the Bayes risk has two disadvan-tages. First, it is conventional in test theory to define indexes so that the scale has a direction opposite to that in which the Bayes risk is represented. Second, although in test theory indexes are nearly al-ways defined on the closed interval [0, 1], the range of the possible values of the Bayes risk can be dif-ferent. Let L(0(jc))0) be the loss function associated with a particular decision problem which is a func-tion of the decision rule, 9(x); the examinee’s true score, 0; and the examinee’s observed score, x. For a particular examinee, the risk or expected loss is R =EL(ê(x)18). The Bayes risk is RB = E.R, where Eo means expectation with respect to the probability density of 0. In an attempt to correct the two deficiencies described above, van der Linden and Mellenbergh suggest the index where R ~ and R&dquo; are the Bayes risks in the situations in which the test contains, respectively, complete and no information about the true scores. They point out, however, that the index d does not correct the second difficulty, since d might not be in the interval [0, 1]. The purpose of this note is to indicate a simple modification of d that solves this problem