Precision with a 2 percent prevalence

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A test for a certain infection has sensitivity $95\%$ and specificity $90\%$. The prevalence of the infection is $2\%$. Compute the precision of the test, i.e. $P(A\mid B)$ with $A=$ «infected» and $B=$ «positive test».

Solution

Let us reason on a sample population of $1000$ people. With prevalence $2\%$ there are $20$ sick and $980$ healthy.

• True positives (sick people who test positive): $0,95\cdot 20 = 19$.
• False positives (healthy people who test positive, specificity $90\%$ hence $10\%$ error): $0,10\cdot 980 = 98$.

The precision is the proportion of sick people among all the positives: $P(A\mid B) = \frac{19}{19+98} = \frac{19}{117} \approx \boxed{16,2\%}$ Despite the fine figures for sensitivity and specificity, the test is unreliable on a low-prevalence population: most of the positives are false positives.

Links

Topics: Probabilita
Concepts: Precisione · Teorema di bayes
Methods: Bayes
Skills: Calcolo probabilita
Exercise type: Problema probabilita