A 99% accurate test is usually wrong
Ten thousand people, a test that catches 99% of cases, and 495 of them get frightening news they did not deserve.
A test detects a condition. It catches 99% of the people who have it and it returns a false positive for only 5% of the people who do not. One person in a hundred has the condition. You test positive.
What is the chance you have it?
The intuitive answer is around 95%. The correct answer is 16.7%. Most people who test positive do not have the condition, and this is not a flaw in the test: the test is doing exactly what it says on the box.
The four multiplications
Take 10,000 people.
- 100 have the condition. The test catches 99 of them.
- 9,900 do not. The test wrongly flags 5% of them: 495 people.
So 594 people test positive, and 99 of them are actually ill. 99 ÷ 594 = 16.7%.
The false positives outnumber the true positives five to one, and they do so for a reason that has nothing to do with the quality of the test: there are simply far more healthy people to be wrong about. A 5% error rate applied to 9,900 people produces more mistakes than a 99% success rate applied to 100 people produces successes.
Make the condition rarer and it gets worse fast. At a prevalence of one in a thousand, the same test gives a positive predictive value of 1.9%. Forty-nine out of fifty positives are false.
Why the intuition fails
The question people answer is not the question they were asked. "How accurate is the test" is a statement about P(positive | ill): given that you are ill, how likely is a positive result. What you want to know is P(ill | positive): given a positive result, how likely is it that you are ill. These are different numbers, and swapping them is so common it has a name: the base rate fallacy, or the prosecutor's fallacy when it happens in a courtroom.
The bridge between them is Bayes' theorem, and the term the intuition drops is the prevalence — the base rate. It is the one number the marketing material for a test never leads with, because it is not a property of the test at all. It is a property of the population you point it at.
The same test is excellent in one setting and useless in another. Applied to patients who already show symptoms, where prevalence might be 30%, our test gives a positive predictive value of 89%. Applied as a mass screen to a population where prevalence is 0.1%, it gives 1.9%. Nothing about the instrument changed.
Where this decides things
Medical screening policy. This is precisely why screening programmes are targeted by age and risk group rather than offered to everyone, and why "we should just test everybody" is a more complicated proposal than it sounds. Widening a screen to a lower-prevalence group does not merely add cost; it changes what a positive result means, and it generates anxiety, follow-up procedures and occasionally harm for people who were never ill.
Security alerting. An intrusion detector with a 1% false positive rate, watching a million events a day of which a handful are attacks, produces ten thousand alerts and a team that stops reading them. Alert fatigue is the base rate fallacy expressed as an organisational failure.
Courtrooms. "The chance of this DNA match occurring at random is one in a million" does not mean the chance the defendant is innocent is one in a million. If the match came from trawling a database of a million profiles, you would expect roughly one match by chance alone. The two statements sound identical and differ by everything.
Astrobiology. The same structure governs claims about life on other planets: a biosignature detection is only as strong as the prior probability that life is there to be found, which is exactly the quantity nobody knows. The Biosignature Bayes tool makes that dependency explicit rather than burying it.
The one habit worth forming
When you are told a test, a detector or a model is "99% accurate", ask two questions.
First: accurate in which direction? Sensitivity and specificity are different numbers and a single "accuracy" figure usually hides one of them.
Second, and more important: how common is the thing being detected? If the answer is "rare", then most of what the detector flags will be wrong, and you can work out how wrong on the back of an envelope in about thirty seconds.
The Bayes tool does the same arithmetic with three sliders. Set up the case above and run sensitivity from 50% to 100%: the answer moves from 9% to 17%. Run specificity across the same range and it moves from 2% to 100%. Prevalence and specificity decide this; sensitivity, which is the number on the box, is the one that barely matters.
The professionals get it wrong too
In 1978 Casscells, Schoenberger and Graboys put a version of this question to staff and students at Harvard Medical School: a disease affecting one person in a thousand, a test with a 5% false-positive rate, a patient who tests positive. What are the odds the patient has it?
The commonest answer was 95%. The correct answer is about 2%. Fewer than one in five of the people asked got near it, and these were people who order such tests for a living.
That result has been reproduced often enough that it is less an indictment of doctors than a description of how the human mind handles conditional probability. The instinct reaches for the accuracy of the instrument, which is printed on the box, and never reaches for the prevalence, which is not.
Watch
References (2)
- the Harvard result Casscells, Schoenberger & Graboys (1978). Interpretation by Physicians of Clinical Laboratory Results. New England Journal of Medicine 299(18).
- why counting people works Gigerenzer & Hoffrage (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review 102(4).