Diagnostic test with a 10 percent prevalence

Text

Let us take up again the diagnostic test example from the chapter (sensitivity $99\%$, false-positive rate $2\%$), but with prevalence $P(M) = 10\%$. Compute $P(M\mid T^+)$ and comment on how it changes compared with a lower prevalence.

Solution

With $P(M)=0,1$ the probability of a positive test is obtained from the total probability: $P(T^+) = 0,99\cdot 0,1 + 0,02\cdot 0,9 = 0,099 + 0,018 = 0,108.$ By Bayes’ theorem the precision is $P(M\mid T^+) = \frac{0,99\cdot 0,1}{0,108} = \frac{0,099}{0,108} \approx \boxed{91,7\%}$ Compared with the low-prevalence cases the predictive value improves drastically: when the disease is more widespread, a positive test is far more reliable.

Links

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