Example — Epidemiological test

Example — Epidemiological test

A test for a rare illness has $P(A) = 0,003$ (the illness strikes $3$ people in $1000$), $P(A\mid B) = 0,7$ (precision) and $P(A\mid \overline{B}) = 0,0001$. Let us fill in the contingency table using $10\,000$ people as the reference population.

	$A$ (ill)	$\overline{A}$ (healthy)	Total
$B$ (test $+$)	$21$	$9$	$30$
$\overline{B}$ (test $-$)	$9$	$9\,961$	$9\,970$
Total	$30$	$9\,970$	$10\,000$

From here the three characteristic quantities of the test can be read directly:

• Sensitivity $= P(B\mid A) = \dfrac{21}{30} = 70\%$
• Specificity $= P(\overline{B}\mid\overline{A}) = \dfrac{9961}{9970} \approx 99,9\%$
• Precision $= P(A\mid B) = \dfrac{21}{30} = 70\%$

A well-used test: few false positives and a very high specificity.

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Topics: Probability
Concepts: Precision · Sensitivity · Specificity · Contingency table
Methods: Bayes · Total probability
Skills: Probability calculation