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Abstract:
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The classical empirical, area under the receiver operating characteristic (ROC) curve (AUC) is a two-sample U-statistic of order (1,1). Over the years, alternative methods have been proposed to estimate the variance of AUC. For example, Delong et al.have proposed a straightforward estimate that has attractive asymptotic properties. At small sample sizes, however, the DeLong method will be biased. In the early stage of investigation, researchers don't always have enough data; therefore, those asymptotic variance estimates such as DeLong's can be unreliable. In this article we propose a two-way random effects analysis of variance (ANOVA) method to compute an unbiased variance estimate of a two-sample U-statistic of order (1,1) in general, and of the AUC in particular. We prove that this variance estimate is equal to the fully U-statistic result. In the particular case of the AUC variance estimate we compare our result to DeLong's AUC variance estimate.We extend the result to obtain an unbiased estimate of variance for a linear combination of possibly correlated AUCs. An example of this extension is the 25 estimate of variance of the difference of the areas under two correlated AUCs.
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