How accurate must the test be?

Question

A disease affects 1\% of a population, and you have a test with 99\% sensitivity. You get to choose the specificity. What specificity do you need so that a positive result means at least a 50\% chance of disease? And for a 90\% chance?

Accepted Answer

Work in odds. The prior odds of disease are 1{:}99, and a positive multiplies them by the likelihood ratio 	ext{LR} = \dfrac{	ext{sensitivity}}{1 - 	ext{specificity}} = \dfrac{0.99}{1 - s}, where s is specificity. For a 50\% posterior, you need posterior odds \ge 1: \frac{1}{99} 	imes \frac{0.99}{1 - s} \ge 1 \;\Longrightarrow\; \frac{0.01}{1 - s} \ge 1 \;\Longrightarrow\; 1 - s \le 0.01. So you need specificity \ge 99\% (a false-positive rate of at most 1\%). For a 90\% posterior, you need posterior odds \ge 9 (since 	frac{9}{9+1} = 0.9): \frac{0.01}{1 - s} \ge 9 \;\Longrightarrow\; 1 -…

How accurate must the test be?

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