# 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.