"Cost-Utility" as a Measure of the Efficiency of a Test

Abstract

Frequently a test result or measure x applying to an individual is used to “predict” not a quantity y of the individual, but instead, whether the individual belongs to A or B of two mutually exclusive categories. The question discussed is how to measure the effectiveness of the test x in achieving the dichotomization. Measures such as the biserial or tetrachoric r are considered open to objection on conceptual and operational grounds. A method is advanced that consists in determining (1) the utility, defined as the fraction of individuals actually belonging to category A which the test correctly designates as A and (2) the cost defined as the fraction of individuals actually belonging to category B which the test incorrectly designates as A. Two tests are to be compared on the basis of their comparative costs for the same utility. When a test can be used at any utility, an over-all measure of effectiveness is the mean-cost (M.C.) for all utilities. Derived from this is an index, the mean-cost-rating (M.C.R. = 1 — 2 M.C.) which like the correlation coefficient extends from zero to unity. The equation giving cost for utility is written for the case where x is normally distributed. The M.C. and M.C.R. are tabled for the case where the standard deviation of x is equal for A and B. Cost-utility relationship for such tests is depicted on a prob.-prob, scale. Comparison of normally distributed tests in terms of such representation is discussed.

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