Today we learned something interesting in class!

We learned about Bayesian statistics and why they're important to the way you read the news, or live your life. This is how it goes: say you're a woman in your 40s and your doctor says to you, "In order to save yourself from breast cancer, you should have a mammogram every year." You know, knowing things as you do, that a mammogram isn't the most unobtrusive or benign tests there are to be taken. For one, you're exposed to radiation, which can cause cancer. It takes time out of the work day, it is uncomfortable, and it is costly. If your test is a false positive, the consequences of taking such a test can become even more serious. Some people upon hearing that they have a dangerous cancer can become depressed, angry, or suicidal. It is a shock wave into their lives... the couple of weeks that it takes to prove that it was a false alarm can be agony, and they have to suffer for no good reason. Sometimes they have to take biopsies to further test for cancer, which can be invasive and sometimes disfiguring.

But come ON! What are the chances of a false positive, especially when the test is 95% accurate!

Actually, no. It is not 95% accurate. If you have cancer, the test will come back positive 95% of the time. That means that 5% of the time, you will get a false negative. This is the worst result because the patient who has cancer might not know about it until her next mammogram, a year or two away. 2% of the time you get a false positive. So 98% of the time, if you don't have cancer, the test comes back negative. But let's turn that statistic on its head. Let's say that you get a positive test result. How likely is it that you really have cancer? Actually it turns out to be about 4.5%!! Whaaaaa, you say, but the test is right 98% of the time, how can your positive test result have 95.5% chance of being false???
Well think about it... let's say we have a hundred thousand women who are between the ages of 40 and 50. In this age range, the chance that you have cancer is about .001%. Thus 100 of these women have cancer and 99900 of them don't. Of the 100 women who do, 95 of them will test positive. Of the 99900 who don't, 2% will test positive. That's still 1998 people!! That means 1998 + 95 = 2093 people will test positive all together. So if 95 out of those 2093 people have cancer, that means that if you test positive, you have a 95/2093 chance of having cancer... which is 4.5%.

Of course, when you get between 50 and 60, the chance that you have cancer is more than .001%, so much so that if you test positive, you have about a 30% chance of actually having cancer. So the discussion goes, "Should we test women between the ages of 40 and 50 as often as we do, knowing that more 99.9% are going to test negative, and knowing that out of those who test positive, 95.5% of those are going to be false positives, causing grievous emotional harm and stress not only to the woman herself but for her entire family?

My opinion of the matter is yes, we should still test them, because despite all of these statistics, you'll either have cancer or you won't. If you have cancer, your chance of having cancer is 1. BUT, your chance of beating said cancer can be much higher if they catch it early. So I think the "emotional trauma" done to the 1998 women who get false positives is worth possibly saving the lives of the 95 who actually have cancer. Wouldn't even the 1998 false positives still say that, even after going through their ordeals? If you asked 19 women to go through three weeks of emotional hell in order to increase the chances of saving the life of one woman whom they don't know, would they agree to do it?
In order to do this mathematically, you'd have to see the statistics on how many people actually fall into depressions or commit suicide after hearing such a diagnosis, how many biopsies are botched, how much radiation one gets from each mammogram, etc., and see how that balances against risking not catching the breast cancer right away in the 95 women out of 100,000 who will have it and be diagnosed correctly. Then you could decide if you could minimize potential harm by having the mammograms a bit less often through your 40s.

Anyway, Bayesian statistics are really important to think about if you're reading the newspaper and they talk about the reliability of a test or something like it, because it isn't always the reliability of the test that's important (if you have cancer, how good are they at catching it?) but also the probability that another condition is true (how probable is it that you actually have cancer?) As you can see, figuring that in can really change your answer!

So I guess the moral of the story is to know that if you are diagnosed with cancer when you are young you really shouldn't freak out, because there's a large chance that it was just a mistake!

Learned about that in Bioinformatics class. Those informaticists (informaticians? Fucked if I know!) really like that kind of shit.
» ranor on 2007-04-13 12:02:03

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