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ABSTRACT
In a wide variety of biomedical and clinical research studies, sample statistics from diagnostic marker measurements are presented as a means of distinguishing between two populations, such as with and without disease. Intuitively, a larger difference between the mean values of a marker for the two populations, and a smaller spread of values within each population, should lead to more reliable classification rules based on this marker. We formalize this intuitive notion by deriving practical, new, closed-form expressions for the sensitivity and specificity of three different discriminant tests defined in terms of the sample means and standard deviations of diagnostic marker measurements. The three discriminant tests evaluated are based, respectively, on the Euclidean distance and the Mahalanobis distance between means, and a likelihood ratio analysis. Expressions for the effects of measurement error are also presented. Our final expressions assume that the diagnostic markers follow independent normal distributions for the two populations, although it will be clear that other known distributions may be similarly analyzed. We then discuss applications drawn from the medical literature, although the formalism is clearly not restricted to that application.
Acknowledgments
The authors thank Dr. Christopher Morrell and Dr. Eleanor Simonsick for their interest and discussion. The authors thank an anonymous reviewer for helpful comments and for bringing to our attention the work of Fluss, Faraggi, and Reiser (Citation2005), and an anonymous TAS associate editor and editor Dr. Nicole Lazar for important input and insight.
Funding
This work was supported by the Intramural Research Program of the NIH, National Institute on Aging.
