According to today’s New York Times,

The OraQuick test is imperfect. It is nearly 100 percent accurate when it indicates that someone is not infected and, in fact, is not. But it is only about 93 percent accurate when it says that someone is not infected and the person actually does have the virus, though the body is not yet producing the antibodies that the test detects.

You’ve got to hand it to the makers of this test: it can’t have been easy to devise a test that remains 93% accurate even in situations where it gives the wrong result. On the other hand, it’s only “nearly 100% accurate” in situations where it gives the right result, so there’s room for improvement.

Since this is the textbook example demonstrating the role of prior probability in Bayesian analysis, I was certain this post would be a demonstration of the proper way to interpret such numbers.

In other words (assuming 99.9% for “almost 100″): Of course, we have 4 cases: someone is infected and the test say he is or isn’t and the same for someone not infected. Suppose someone takes the test, not knowing his actual status. In each case (test indicates infection or not), what is the probability that the result is correct? (Note: there is one piece of information missing.)

A couple of comments on the article demonstrate what I was talking about: a misperception and its Bayesian correction:

So in other words, if you both test negative, it’s ok to forget the condoms…but oops …oh I’m sorry, there’s was a 7 percent chance that this wasn’t true..and ooops now I have infected you. * My bad, dude.*No. The infection rate in the US is 0.6%. From above, perhaps 10% will lie. Let’s say 10% don’t know, so 20% of the infected are “dangerous”. 0.006 * 0.2 * 0.07 = 0.000084. So if you use this test and have no skill at detecting liars, then you’re taking a 0.0084% chance of, not catching AIDS, but being exposed to AIDS.Of course, if you want to test someone, their prior probability of being infected is probably higher than 0.6%, so adjust accordingly. However, the impression that 7% of the time when there is a negative result, someone is actually infected is still wrong.

I think he is poking fun at the wording used in the article. The wording points at an individual. If Joe is not infected and the test indicates that Joe is not infected, I’d say the test (i.e. Joe’s test) is 100% accurate. On the other hand, if Susie does have the virus and the test indicates Susie does not have the virus, well, that test was 0% accurate.