COLUMNIST: Of false positive mammograms and blue and green taxicabs
Pretend that you are judging a case involving a nighttime hit-and-run by a taxi cab. An eyewitness testifies that the cab was blue. In the city where the accident occurred there are 100 cabs: 15 are blue, 85 are green. When tested for accuracy under the same nighttime conditions as occurred in the accident, the eyewitness was accurate 80 percent of the time. Should you convict based upon this testimony?
If you said “yes,” you are probably making the common mistake of not taking the rate that an event occurs, known as base rate, into account. In this case — called the Cab Problem — it is not only the eyewitness’s accuracy that needs to be taken into account, but also the likelihood of encountering a blue or green cab. While the eyewitness would correctly identify 12 out of 15 blue cabs as “blue,” he would also incorrectly identify 17 out of 85 green cabs as “blue.” Thus, only 41 percent of his identifications of “blue” would actually be blue cabs.
This is a basic problem of screening tests. The base rate of a disease becomes important when tests are not 100 percent accurate. And no test is 100 percent accurate: either it will miss a diagnosis it should have caught, or it will falsely identify the disease when it isn’t present — called a false positive. In the case of rare diseases, because most people don’t have the disease, more people can show up as false positives than actually have the disease.
This is why the proposal back in the 1990s that everyone receive an HIV test would have been counterproductive. There would have been far more false positives from those not infected with HIV than correct identifications — known as hits — of those who did.
Many women intuitively know this to be true of mammography; they have been called back for further testing after a positive mammogram, only to discover that their anxiety was for naught — they did not have breast cancer. This happens because even though the hit rate for mammograms is high, about 75 percent — meaning that it catches three out of four women with breast cancer — when combined with a low incidence of occurrence and a modest false positive rate, 8 percent, you end up with a lot more women receiving a positive mammogram than those who actually have cancer. The problem is even more acute for women in their 40s. Only about 1.5 percent of women in their 40s develop breast cancer, but mammography’s hit rate for this group slips to about 68 percent. (Sensitivity slips because denser breasts make it harder to interpret mammograms.) But the rate of false positives increases slightly. Thus, for women in their 40s only 1 in 53 receiving a positive mammogram screening have cancer.
Compare that to 1 in 17 for women in their 60s. Thus, if a woman has a positive mammogram in her 40s, in all likelihood she doesn’t have cancer. However, she might undergo further testing or even treatment that can have other complications.
For women, the unfortunate truth is that breast cancer can occur at any age. Women in their 30s have about half a percent chance of developing breast cancer.
Should we then begin routine mammography screening at age 30, with the potential for even lower hit rates and higher false positive rates for this age group?
For this reason as well as others, the benefit of mammography screening for younger women has been difficult to demonstrate. The task force developing new recommendations on mammography screening, about half of whom are women, has tried to take these realities into account when recommending that for women under 50, only those in higher-risk populations, such as those with a family history of breast cancer, seek routine screening.
They are trying to follow the Hippocratic Oath, to help where possible without doing harm. These recommendations are always matters of judgment; we can all hope they are wise ones.
But if we are truly interested in preventing breast cancer deaths, rather than arguing about whether these new recommendations represent rationing, we might be more concerned about giving all women access to recommended mammography screenings and appropriate treatment if they do have breast cancer.
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John Horner is a psychology professor at Colorado College. Horners email address is: JHorner@ColoradoCollege.edu
The data presented in this article comes from two primary sources. The study that details mammography accuracy was done by Patricia Carney and her colleagues in the Annals of Internal Medicine, 2003, Vol. 138(3): 168-175. From this study I used specificity to calculate the false positive rates as well as their estimates of true-positives to true cancer rates to access hit rates for women of varying age groups.
The percentage of women who develop cancer in different age cohorts comes from MJ Horner (no relation to the author) and colleagues in SEER Cancer Statistics Review, 1975–2006, National Cancer Institute. Bethesda, MD.
