Mammogram Math

Ok, so longtime readers here know I don't get mammograms, the simplest explanation being that the death rates from breast cancer between people who get mammos and those who don't are roughly similar, so I decided to skip the bad energy and spend my time sniffing flowers instead.

Here's an EXCELLENT article by mathematician Jonathan Allan Paulos about why these numbers bear out what the studies say, and having women do less mammograms is actually BETTER, not anti-woman. He brings up the other good point that if mammograms are only beneficial, then why not start screening at twenty? In the teens? There have now been reported CHILDREN having breast cancer. Why not start at six?

Much of our discomfort with the panel’s findings stems from a basic intuition: since earlier and more frequent screening increases the likelihood of detecting a possibly fatal cancer, it is always desirable. But is this really so? Consider the technique mathematicians call a reductio ad absurdum, taking a statement to an extreme in order to refute it. Applying it to the contention that more screening is always better leads us to note that if screening catches the breast cancers of some asymptomatic women in their 40s, then it would also catch those of some asymptomatic women in their 30s. But why stop there? Why not monthly mammograms beginning at age 15?

The answer, of course, is that they would cause more harm than good. Alas, it’s not easy to weigh the dangers of breast cancer against the cumulative effects of radiation from dozens of mammograms, the invasiveness of biopsies (some of them minor operations) and the aggressive and debilitating treatment of slow-growing tumors that would never prove fatal...

Assume there is a screening test for a certain cancer that is 95 percent accurate; that is, if someone has the cancer, the test will be positive 95 percent of the time. Let’s also assume that if someone doesn’t have the cancer, the test will be positive just 1 percent of the time. Assume further that 0.5 percent — one out of 200 people — actually have this type of cancer. Now imagine that you’ve taken the test and that your doctor somberly intones that you’ve tested positive. Does this mean you’re likely to have the cancer? Surprisingly, the answer is no.

To see why, let’s suppose 100,000 screenings for this cancer are conducted. Of these, how many are positive? On average, 500 of these 100,000 people (0.5 percent of 100,000) will have cancer, and so, since 95 percent of these 500 people will test positive, we will have, on average, 475 positive tests (.95 x 500). Of the 99,500 people without cancer, 1 percent will test positive for a total of 995 false-positive tests (.01 x 99,500 = 995). Thus of the total of 1,470 positive tests (995 + 475 = 1,470), most of them (995) will be false positives, and so the probability of having this cancer given that you tested positive for it is only 475/1,470, or about 32 percent! This is to be contrasted with the probability that you will test positive given that you have the cancer, which by assumption is 95 percent.

The arithmetic may be trivial, but the answer is decidedly counterintuitive and hence easy to reject or ignore.

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