False positive paradox

The false positive paradox is a statistical result where false positive tests are more probable than true positive tests, occurring when the overall population has a low incidence of a condition and the incidence rate is lower than the false positive rate. The probability of a positive test result is determined not only by the accuracy of the test but by the characteristics of the sampled population. When the incidence, the proportion of those who have a given condition, is lower than the test's false positive rate, even tests that have a very low chance of giving a false positive in an individual case will give more false than true positives overall. So, in a society with very few infected people—fewer proportionately than the test gives false positives—there will actually be more who test positive for a disease incorrectly and don't have it than those who test positive accurately and do. The paradox has surprised many.

It is especially counter-intuitive when interpreting a positive result in a test on a low-incidence population after having dealt with positive results drawn from a high-incidence population. If the false positive rate of the test is higher than the proportion of the new population with the condition, then a test administrator whose experience has been drawn from testing in a high-incidence population may conclude from experience that a positive test result usually indicates a positive subject, when in fact a false positive is far more likely to have occurred.

Not adjusting to the scarcity of the condition in the new population, and concluding that a positive test result probably indicates a positive subject, even though population incidence is below the false positive rate, is a "base rate fallacy".

Example

High-incidence population

Number of people	Infected	Uninfected	Total
Test positive	400 (true positive)	30 (false positive)	430
Test negative	0 (false negative)	570 (true negative)	570
Total	400	600	1000

Imagine running an HIV test on population A of 1000 persons, in which 40% are infected. The test has a false positive rate of 5% (0.05) and no false negative rate. The expected outcome of the 1000 tests on population A would be:

Infected and test indicates disease (true positive)

1000 × ⁠40/100⁠ = 400 people would receive a true positive

Uninfected and test indicates disease (false positive)

1000 × ⁠100 – 40/100⁠ × 0.05 = 30 people would receive a false positive

The remaining 570 tests are correctly negative.

So, in population A, a person receiving a positive test could be over 93% confident (⁠400/30 + 400⁠) that it correctly indicates infection.

Low-incidence population

Number of people	Infected	Uninfected	Total
Test positive	20 (true positive)	49 (false positive)	69
Test negative	0 (false negative)	931 (true negative)	931
Total	20	980	1000

Now consider the same test applied to population B, in which only 2% is infected. The expected outcome of 1000 tests on population B would be:

Infected and test indicates disease (true positive)

1000 × ⁠2/100⁠ = 20 people would receive a true positive

Uninfected and test indicates disease (false positive)

1000 × ⁠100 – 2/100⁠ × 0.05 = 49 people would receive a false positive

The remaining 931 tests are correctly negative.

In population B, only 20 of the 69 total people with a positive test result are actually infected. So, the probability of actually being infected after one is told that one is infected is only 29% (⁠20/20 + 49⁠) for a test that otherwise appears to be "95% accurate".

A tester with experience of group A might find it a paradox that in group B, a result that had usually correctly indicated infection is now usually a false positive. The confusion of the posterior probability of infection with the prior probability of receiving a false positive is a natural error after receiving a life-threatening test result.

Discussion

Cory Doctorow discusses this paradox in his book Little Brother.

If you ever decide to do something as stupid as build an automatic terrorism detector, here's a math lesson you need to learn first. It's called "the paradox of the false positive," and it's a doozy.

Number (rounded)	Has Super-AIDS	Does not have Super-AIDS	Total
Test positive	1 (true positive)	10,000 (false positive)	10,001
Test negative	0 (false negative)	989,999 (true negative)	989,999
Total	1	999,999	1,000,000

Say you have a new disease, called Super-AIDS. Only one in a million people gets Super-AIDS. You develop a test for Super-AIDS that's 99 percent accurate. I mean, 99 percent of the time, it gives the correct result – true if the subject is infected, and false if the subject is healthy. You give the test to a million people.

One in a million people have Super-AIDS. One in a hundred people that you test will generate a "false positive" – the test will say he has Super-AIDS even though he doesn't. That's what "99 percent accurate" means: one percent wrong.

What's one percent of one million?

1,000,000/100 = 10,000

One in a million people has Super-AIDS. If you test a million random people, you'll probably only find one case of real Super-AIDS. But your test won't identify one person as having Super-AIDS. It will identify 10,000 people as having it. Your 99 percent accurate test will perform with 99.99 percent inaccuracy.

That's the paradox of the false positive. When you try to find something really rare, your test's accuracy has to match the rarity of the thing you're looking for. If you're trying to point at a single pixel on your screen, a sharp pencil is a good pointer: the pencil-tip is a lot smaller (more accurate) than the pixels. But a pencil-tip is no good at pointing at a single atom in your screen. For that, you need a pointer – a test – that's one atom wide or less at the tip.

Number (rounded)	Is a terrorist	Is not a terrorist	Total
Test positive	10 (true positive)	200,000 (false positive)	200,010
Test negative	0 (false negative)	19,799,990 (true negative)	19,799,990
Total	10	19,999,990	20,000,000

Here is an application to terrorism:

Terrorists are really rare. In a city of twenty million like New York, there might be one or two terrorists, maybe up to ten. 10/20,000,000 = 0.00005 percent, one twenty-thousandth of a percent.

That's pretty rare. Now, say you have software that can sift through all the bank-records, or toll-pass records, or public transit records, or phone-call records in the city and catch terrorists 99 percent of the time.

In a pool of twenty million people, a 99 percent accurate test will identify two hundred thousand people as being terrorists. But only ten of them are terrorists. To catch ten bad guys, you have to investigate two hundred thousand innocent people.

The language of this subject can lead to confusion. For example, suppose the false positive rate for test A is 5%. What is the true positive rate for test A? Some might think that it’s 95%, but that will almost always be wrong. Suppose the false positive rate of a test is 12%. Does that mean that if we test a group, that group will have a false positive rate of 12%? Not necessarily. Below, this is explained in some detail.

The true positive rate of a test for some disease does not tell the true positive rate of a group of people being tested. Clearly, just because a test gives 100% true positive results doesn’t mean that 100% of the people in a group will test positive. Instead, it means that the people in the group who have the disease will all (100%) test positive. To know how many people in the group have the disease, we must know the disease’s prevalence. For example, if a disease has a prevalence of 12% in some group, and a test for that disease has a true positive rate of 100%, then 12% of the people in that group will both have the disease and test positive for the disease. In other words, the rate of true positive test results in that group will be 12%. If the prevalence of a disease is 6% in some group, and my test has a true positive rate of 80%, then 80%x6%=4.8% of that group will both have the disease and test positive. In other words, the true positive rate in that group will be 4.8%. In general, the true positive rate of a disease in a group is the true positive rate of the test multiplied by the prevalence of the disease in that group. (Notice that the true positive rate by itself doesn’t say anything about the total rate of positive test results because the test might also give false positive results.)

Similarly, the false positive rate of a test does not tell the rate of false positives in a group of people being tested. Just because a test has a false positive rate of 10% doesn’t mean that 10% of the people in a group will falsely test positive. Instead, it means that of the people in the group who don’t have the disease, 10% will test positive. How many people in the group don’t have the disease? The prevalence of the disease tells us this number. For example, if the prevalence of a disease in a group is 8%, then we know that 92% of the people in that group don’t have the disease. Any positive test result in those 92% of people without the disease is called a false positive. For example, suppose the disease prevalence in a group is 5% and a test for that disease has a false positive rate of 10%. In that group, 95% of the people don’t have the disease (because the prevalence is 5%). Of those 95%, 10% will test positive (false positive). So, the false positive rate of the group is 95%x10%, or 9.5%. Suppose the disease prevalence in another group is 30%. We test them with the same test. The false positive rate in that group is 10% of 70%, or 7%. In general, the false positive rate in any group will be: (100 minus the prevalence) times the false positive rate of the test.

Suppose a disease has a prevalence of 1/1000. A test for the disease has a true positive rate of 100% — it never misses someone who has the disease. It also has a false positive rate of 5%. A patient takes the test and the result is positive. How likely is it that he has the disease? To find this out, we need to know what percent of people who tested positive have the disease. Suppose 1000 people are tested for the disease. Since the prevalence of the disease is 1/1000, we know that, on average, 1 person in that group of 1000 will have the disease. That person will test positive (100% true positive rate). We also know that 999 people, on average, don’t have the disease, and that 5% of them will test positive (false positives). That’s 999x.05, or approximately 50 people. That means that 51 people will test positive (1 true positive and 50 false positives), and only one of those people will actually have the disease. Therefore, 1/51, or about 2% of the people who tested positive actually have the disease. So, the patient's chance of having the disease is about 2%.

See also

• Bayes' theorem
• List of paradoxes
• Prosecutor's fallacy, a mistake in reasoning that involves ignoring a low prior probability
• Simpson's paradox, another error in statistical reasoning dealing with comparing groups
• Prevention paradox

External links

• The false positive paradox explained visually (video)