Medical False Positives and False Negatives
(Conditional Probability)

Summary:  If a test for a disease is 99% accurate, and you test positive, the probability you actually have the disease is not 99%. In fact, the more rare the disease, the lower the probability that a positive result means you actually have it, despite that 99% accuracy. The difference lies in the rules of conditional or contingent probability.

Suppose you’ve taken a test for a deadly disease D, and the doctor tells you that you’ve tested positive. How bad is the news? You need to know how accurate the test is, and specifically you need to know the probability that a positive test result actually means you have the disease.

Suppose you’re told the test for D is “99% accurate” in the following sense: If you have D, the test will be positive 99% of the time, and if you don’t have it, the test will be negative 99% of the time. (For simplicity, I’m using the same percentage for both positive and negative results. Many tests have a different accuracy for positive and negative.) Suppose further that 0.1% — one out of every thousand people — have this rare disease.

You might think that a positive result means you’re 99% likely to have the disease. But 99% is the probability that if you have the disease then you test positive, not the probability that if you test positive then you have the disease.

This kind of thing is easiest to understand if you work with numbers of people rather than percentages, and the numbers can be laid out in a chart like the one below.

Suppose 100,000 people are tested for disease D. Of these, you know that by random chance 100 actually have the disease (1 in 1000) — see column 1 total below. Since 99% of people with the disease test positive, there will be 99 positive tests and 1 negative test in column 1. There are 100,000−100 = 99,900 healthy people — see column 2 total. Of them, 99% will test negative (99%×99,900 = 98,901) and the other 999 will test positive.

Now add across and you see that there will be 1,098 positive tests and 98,902 negative tests. Out of the 1,098 tests that report positive results, 99 (9%) are correct and 999 (91%) are false positives. Therefore the probability that you actually have disease D, when you’re given a positive test result, is just 9% — for a test that is 99% accurate!

To reiterate, the conditional probability that you test positive given that you have the disease is

P(test positive | have D) = 99÷100 = 99%

but the conditional probability that you have the disease if you test positive is

P(have D | test positive) = 99÷1098 = about 9%

The exact probabilities will vary depending on the accuracy of the test and the actual incidence of the disease, but always you have to look at the conditional probability. This is one reason why, for a disease like AIDS, patients are never told they test positive until the blood has been retested with a different test, to minimize the chance of a false positive.

Doctors should be familiar with the probabilities when they give test results to patients, but if you get a positive result from a test for an uncommon disease, make sure your doctor understands.