In a previous blog I said that we say that a test is very sensitive if it has very few false negatives. I then said that if all you know about the test is that it is a very sensitive test, only it’s negative results are truly informative! Getting a positive result i.e. you are told you have the disease by this oh so very sensitive test, may not give you all that much information.
To repeat: just because you tested positive on a very sensitive test, you may or may not have the disease. In fact, if you continue to read this blog, you will learn why, even with an exquisitely sensitive test, when you test positive for a disease, the odds may still be very much against you having the disease! (I will cover this topic, which has the very fancy name of Bayesian Updating very soon, please stay tuned.)
But back to the idea that a positive result on a very sensitive test isn’t necessarily very informative. Some readers told me they found this a little strange and had trouble wrapping their head around it. At first glance it seems that both positive and negative results should be informative. But what I said is certainly true and understanding why it is true is actually a good use of one of the first tools a mathematician uses to think about a problem – looking at the boundary conditions!
To a mathematician, looking at the boundary conditions first is automatic. Boundary conditions are the most extreme possibilities, all in or all out so to speak. Once you look at the boundary conditions for a test, it becomes clear why a positive result on a very sensitive test may mean nothing!
O.K. here’s how a mathematician thinks about this: We say to ourselves “Hmmm by definition a very sensitive test is one that has very few false negatives. O.K. suppose the test never gives a false negative because it is broken and reports every one as positive.” (That’s a pretty good boundary condition to a mathematician!) In this case being told you are positive from this test means absolutely nothing. This shows a mathematician that without more information about the test, only negative results from very sensitive tests are truly informative.
What we obviously need is a good definition of a test being accurate – and that will be the subject of a blog very soon.