Antibodies Test Statistical Analysis Reconsidered

The other day, concerning my antibodies test, I wrote that “If the test shows a false positive only 1% of the time {as my test was claimed to}, that doesn’t mean that, if you test positive, there’s only a 1% chance it’s a false positive. Your chance of a false positive is affected by how many people who take the test are positive, and how many are not. For example, if out of 1,000 people who take the test, only 10% are positive, out of the 900 people who aren’t positive, 1% –– 9 people –– will test false positive. So, if you are test taker #1,001, you’re roughly nine times more likely to be a false positive than a true positive.” Then I said, “I think I have that right.” Reflecting upon this in the middle of the night, I began to doubt it. Suppose 10,000 people, instead of 1,000, have taken the test. Then of the 9,000 people who aren’t positive, 1% –– 90 people –– will test positive. So, by my logic, if you’re test taker #10,001, you’re 90 times more likely to be a false positive than a true positive.

That can’t be right. Why would the likelihood of your being a false positive depend on how many people have taken the test, and vary so widely? Something’s wrong here. I’ll investigate further.