## Using Bayes Theorem to Compute Posterior Probability of Disease

Biosurveillance personnel want to know how well a case detection algorithm can predict the actual diagnosis of a patient from biosurveillance data about that patient, because the real-world purpose of such an algorithm would be to screen individuals for SARS. The preferred method (and standard practice) has two parts. First, the evaluator measures and reports only sensitivity and specificity. Second, consumers of this information (system developers or biosurveillance personnel) use Bayes theorem to compute predictive probabilities from the reported sensitivity and specificity of the case detection algorithm and the current estimates of prevalence (also known as prior probability) of disease in their country or locality. We illustrate this use of Bayes theorem below.

1 ' P (Algorithm Pos I SARS) P (SARS) + P (Algorithm Pos I No SARS) P (No SARS)

(No SARS I Algorithm Neg) = —,-1 - ,-f--.-'—^--^-7

1 ' P (Algorithm Neg I SARS) P (SARS) + P (Algorithm Neg I No SARS) P (No SARS)

If the background prevalence of SARS in the population is one case per 100,000 people, the predictive probabilities of disease status given the test result are:

numeric outputs over some range of values. An evaluator selects a numerical value in this range as the cut-off point above which patients are classified as having disease and below which they are classified as not having disease.

Figure 20.1 illustrates the raw numeric results for a hypothetical classification algorithm that produces as an intermediate result a number between 0 and 15 for most cases and controls.

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