Reading a Medical Test

A test that is 99% accurate can still be wrong most of the time when a disease is rare. Bayes' theorem combines the test's accuracy with how common the disease is.

How it works

Bayes' theorem

In one sentence

The probability of disease given a positive test is $P(D\mid +) = \dfrac{P(+\mid D)P(D)}{P(+\mid D)P(D) + P(+\mid\neg D)P(\neg D)}$ .

General formula

{\htmlData{sym=0}{P(D\mid +)}} = \frac{{\htmlData{sym=1}{P(+\mid D)}}\,{\htmlData{sym=4}{P(D)}}}{{\htmlData{sym=1}{P(+\mid D)}}\,{\htmlData{sym=4}{P(D)}} + {\htmlData{sym=2}{P(+\mid\neg D)}}\,{\htmlData{sym=3}{P(\neg D)}}}

Of all positive tests, what share come from people who are sick? True positives divided by all positives.

What the symbols mean

$P(D\mid +)$
Chance of disease given a positive test
What the patient actually wants to know.
$P(+\mid D)$
Sensitivity
The chance the test comes back positive when the disease is there.
$P(+\mid\neg D)$
False positive rate
The chance a healthy person tests positive anyway.
$P(\neg D)$
Healthy fraction
The share of people without the disease: 1 − prevalence.
$P(D)$
Prevalence
How common the disease is before any test.

Sensitivity $s = P(+\mid D)$ : how often sick people test positive.
False positive rate $f = P(+\mid\neg D)$ : how often healthy people test positive anyway.
With a rare disease, false positives from the huge healthy group outnumber the true positives.

Learn the math

Bayes' TheoremStats · Lesson 12, Probability