Statistics
Bayes' Theorem
A 90% accurate test can still be wrong most of the time it says positive. Test a crowd of a thousand people, keep only the positives, and count who is really sick: the answer depends on how rare the disease was to begin with.
sick, test caught ithealthy, false alarmsick, test missed ithealthy, cleared
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P(sick | positive)—
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Chapter
Show
P(sick | positive)—
P(sick | positive twice)—
P(sick | negative)—
Per 1,000 people
What to observe
- At 1% prevalence with a 90% / 90% test, only 9 of the 108 positive tests are sick, about 8%. The false alarms come from the 990 healthy people, so there are many more of them.
- At 20% prevalence the same test is right most of the time it says positive. Nothing about the test changed, only how many sick people there were to find.
- Back at 1%, push specificity to 99% and the answer jumps to about 48%. Sensitivity at 99% would barely move it: the false alarms are the problem.
- A second positive test starts from the 8% the first one left, not from 1%, and lands near 44%. Yesterday's answer is today's base rate.
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