The First Positive: Computing Positive Predictive Value at the Extremes

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Abstract
Computing the positive predictive value (PPV) of a well-known test for a relatively common disease is a straightforward exercise. However, in the case of a new test for a rare disorder, the extreme numbers involved—the very low prevalence of the disorder and the lack of previous false-positive results—make it difficult to compute the PPV. As new genetic tests become available in the next decade, more and more clinicians will have to answer questions about PPVs in cases with extreme prevalence, sensitivity, and specificity. This paper presents some tools for thinking about these calculations. First, a standard PPV calculation with rough estimates of the prevalence, sensitivity, and specificity is reviewed. The “zero numerator” problem posed by not having seen any false-positive results is then discussed, and a Bayesian approach to this problem is described. The Bayesian approach requires specification of a prior distribution that describes the initial uncertainty about the false-positive rate. This prior distribution is updated as new evidence is obtained, and the updated expected false-positive rate is used to calculate PPVs. The Bayesian approach provides appropriate and defensible PPVs and can be used to estimate failure rates for other rare events as well.

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