In the most recent issue of JAMA is an article by Sirovich and Welch, Cervical Cancer Screening Among Women Without a Cervix [Abstract] that estimates almost 10 million women in the United States have received a pap smear unnecessarily because they have had a full hysterectomy and no longer have a cervix. For women who have had only a partial hysterectomy or where the hysterectomy was done for cervical neoplasia, regular pap smears are recommended. For the other women, though, this is an unnecessary test, because the pap smear is trying to detect cancer in an organ that the woman no longer has.
It makes for an interesting thought experiment. What would the sensitivity, specificity, positive predictive value, and negative predictive value be for such a test? It's a useful way to review the fundamental definitions for these terms. Try to work this out in your head before reading any further.
Let's tackle the list in reverse order. The negative predictive value is the probability that you do not have the disease, given that you tested negative. The denominator is the number of women who test negative and the numerator is the number of women among these who do not have the disease. The answer has to be one since all of the women who test negative do not have cervical cancer.
The positive predictive value is the probability that you do have the disease given that you tested positive. The denominator is the number of women who test positive and the numerator is the number of women among these who actually have the disease. The answer here is zero, because the numerator is zero.
Specificity is the probability that you test negative given that you don't have the disease. The denominator in this case is the number of women without the disease (which in this case is everyone). You can't actually estimate this value without more information, but specificity would end up being the same as the proportion of women without a cervix who end up testing negative.
Sensitivity is the probability that you test positive given that you have the disease. The denominator is the number of women with the disease. Here the denominator is zero, which produces a fraction that most mathematicians will sneer at. You cannot estimate sensitivity.
You can also view sensitivity as one minus the false negative rate. You could argue that it is impossible to have a false negative, so the sensitivity is one. Mathematicians would quibble with this answer, since both the numerator and denominator for the false negative rate have to be zero. Zero divided by zero makes for a very unhappy fraction.
Realistically, no one would ever estimate or report any of these quantities in such an extreme case. But I think it's a good excuse to review these definitions. This example is, perhaps, a bit of an oversimplification. A pap smear performed on a woman without a cervix does have a limited ability to detect vaginal cancer, but with an unacceptably high false positive rate.
You can find an earlier version of this page on my original website.