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# A Generalization of the One-Sided Two-Sample Kolmogorov-Smirnov Statistic for Evaluating Diagnostic Tests

Mitchell H. Gail and Sylvan B. Green
Biometrics
Vol. 32, No. 3 (Sep., 1976), pp. 561-570
DOI: 10.2307/2529745
Stable URL: http://www.jstor.org/stable/2529745
Page Count: 10
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## Abstract

Suppose a continuous diagnostic measurement is used to classify patients, and suppose E$_1$ false negative errors and E$_2$ false positive errors result. The quantities E$_1$ and E$_2$, and the total number of misclassifications, L = E$_1$ + E$_2$, depend on the choice of cut-off value. We have determined the null distribution of min L, where minimization is over all possible cut-off values. The statistic, min L, can be used as a quick one-sided two-sample test, and min L is also useful for evaluating publications which present only a 2 x 2 table of false positives, false negatives, true positives and true negatives. In such cases, one can use min L to assess the usefulness of the diagnostic measurement, even if one suspects that the authors chose that particular cut-off value which minimized L after looking at the data. We extend these results to a more general weighted loss L = $\nu$E$_1$ + $\mu$E$_2$ where $\nu$ and $\mu$ are positive integers, and we show that min L is a generalization of the one-sided two-sample Kolmogorov-Smirnov statistic, and, indeed, exactly equivalent to that statistic for appropriate choices of $\nu$ and $\mu$.

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