It forgets to consider false negatives, which will not really change much for this example as the accuracy at 99% is pretty high and the incidence is pretty low, and false negatives thus would not really change much.

However, when it comes to mammograms, where the false negative rate is up to 12.5% and the false positive rate is up to 50%, and the incidence of breast cancer in women (12.5% over a lifetime) is rather high, false negatives would play more of a role.

It doesn't need to consider false-negatives. The premise is that you received a positive test.

The math is, given the test shows a positive, what is the probability that you actually have the disease?

Well, if we assume there is a 0% chance of false negatives (which seems implicit in the post, even though the definition of "test accuracy" doesn't make that assumption), then we don't need to consider it.

If we are not making that assumption, then the 50% chance claim at the end is incorrect, since there's a chance that both of the two positive results are false positives and there is a false negative somewhere else in the remaining 99 (out of 101) people.

The 50% chance of it being a false positive (conditional on two positive results being observed) is actually a lower bound, and will increase as a function of the false negative rate.

What? No. That's not how this works.

First, I'm assuming that top comment meant the false-positive rate was 1%, even though it was poorly stated as "accuracy." If you're commenting on this original ambiguity, then the whole thing is ambiguous.

But if we accept a 1% false-positive rate, then it makes no difference at all what the false-negative rate is, once we have a positive test.

Out of a 100,000 tests, on average 1000 will be falsely positive. That's what a 1% false-positive means. It makes no difference how many are truly positive, how many are truly negative, and how many are falsely negative. The definition of 1% false-positive means that, out of 100,000 tests, on average 1000 will show a positive test when the condition isn't actually there.

So if 100,000 people each took two tests, on average 10 will have two false-positive tests. Again, this is independent of the number of false-negatives, and is simply the definition of false-positive. (Some tests may have correlated false-positives, i.e. you're more likely to get two in a row, but that's not standard and not part of the definition.)

So if you have a disease whose incidence is 10 people out of 100,000, and you took two tests which are both positive, you are equally-likely to be in the group of 10 people who really have the disease, and the group of 10 people sitting there with two positive tests who don't have the disease.

You seem to be saying that the percentages will be affected by whether any of the 10 out of 100,000 who really do have the disease got a negative test. But it doesn't. 1% false-positive means that, on average, 10 people who don't have the disease will get two false positive tests. It doesn't matter what the people who really have it get.

I believe this is wrong.

Positive predictive value is the likelihood that, if you have gotten a positive test result, you actually have the disease. It’s calculated as TP/TP+FP.

TP, ie true positives, is a function of false negatives.

Here's a common sense explanation. Assume the false negative rate is 100% and so you have zero true positives. Then every positive test result will be a false positive, and therefore a positive test result shows that you don't have the disease.

On the other hand, assume the false negative rate is 0. Then a positive test result will mean that you probably do have the disease as long as the false positive rate isn't too high.

Remember your post I was replying to was all about the probability that you actually have the disease conditional on a positive test result. That's the question I am addressing. For this question, you must consider the false negative rate.

I'm reasonably sure the false negative rate is 1%, because the least misleading definition of "99% accurate" means it fails 1% of the time at random.

A 1% false negative rate has a negligible impact on the math. 50% isn't perfect down to the decimal point, but it's very close.

After the first round you have 99.99 false positives and .99 true positives.

After the second round, if independent, you have .9999 false positives and .9801 true positives.

And in fact the lower bound, if you had no false negatives, is slightly below 50%. .9999 vs 1.

Their definition of accuracy is 1-false_positive_rate. They were quite explicit about that.

No, they didn't say that was the definition in their one post.

Their math shows that those are the same number, and they didn't elaborate on definitions.

The least misleading definition of accuracy is that accuracy = 1-false_positive_rate = 1-false_negative_rate. So unless told otherwise, I will use that definition.

Yes they did. They say it twice. I have italicised both places where they say it.

"Let's say there is a terrible disease that 0.01% (1 in 10,000) of the population is infected with, and we develop a test that is 99% accurate. You go to the doctor and get a test. Oh no, it's positive. What are the chances that it's a false positive?

Intuition would tell you 1% since the test is 99% accurate. Intuition, as it often is, is wrong. Take a sample size of 'x', we'll use 10,000. How many people will be infected? 1. How many false positives will there be? 1% * 10,000 = 100"

Then they conclude by saying:

  "So there will be 100 false positives, and 1 real positive."

The fact there is one real positive necessarily implies they are assuming the false negative rate is exactly 0, given that the disease prevalence is in 0.01 percent of the population.

> The fact there is one real positive necessarily implies they are assuming the false negative rate is exactly 0, given that the disease prevalence is in 0.01 percent of the population.

Or that they rounded .99 people to 1 person.

And given that they did round the false positives, I don't see why that's a strange idea.

If the right, show the proof (or at least the real math).

Specifically, why would the 10,001st test carry a 50% chance of being a false positive?

You misunderstood the scenario: 10,000 people would receive a test. Out of those, ~100 would be expected to test positive. Only those ~100 people would receive a second test.

Thus, the population for tests 1..10,000 is different than the population for tests 10,001..10,100.

Ok... and why would that second population carry a 50% risk of the second test being a false positive?

Edit: I misunderstood the scenario.

In this example? Because we know that one is a true positive. We tested 100 people with a false positive, and 1 with a true positive.

If we test again, and the test is "fair", then of these 101 people, we will retest 99% (or 99) true negative, 1% (or 1) false positive, and 1 true positive. So two positives, one false positive and one true positive. 1 out of 2 is 50%.

(Tho, there is a chance also that the true positive might come back as a false negative, then 1 out of 1 or 100% of positive results would be false positive)

There is a 1/10000 (the OP said 0.01%) chance of having the disease. That is, in 10000 people, one person will have the disease. But also, if you test everyone with a 99% accurate test, 100 people will test positive. If you test them again, 1 person will get another positive test. So, at the end of the day, your test will identify the one person with the disease with a 99% certainty (or 98.01 if you require two tests to be positive). But also identify another individual as being doubly positive. So, after two rounds of tests, 50% of the marked individuals truly have the disease.

The silly thing with this demonstration is that accuracy is poorly defined. Tests are normally conditional on the actual presence of the disease or not.

The second population contains 100 people without the disease. Of them we expect one person to have a false positive. In addition the person who actually has the disease will test positive. So on average 2 people will test positive, one of whom has the disease and one of whom doesn't. If you are one of those two people you have a 50% chance of your (positive) diagnosis being false.

Yes, it's not a 50% chance of false positive. It's 50% chance of false, given positive test.