Now that you learned Bayes Theorem here, time to put it to use.
Here's a situation that doctors might encounter:
1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?
What do you think the answer is?
WARNING: Most doctors get the same wrong answer on this problem - usually, only around 15% of doctors get it right. See Casscells, Schoenberger, and Grayboys 1978; Eddy 1982; Gigerenzer and Hoffrage 1995; and many other studies.
If you get it right, you are smarter than 85 percent of the doctors in the United States!
Maybe I'm oversimplifying this, but I don't see how Bayes applies here. If 9.6% of positive mammographies are false, that implies that 90.4% of them are true; so there ought to be a 90.4% chance that she actually has cancer. The only thing that potentially complicates this is if the 80% and 9.6% numbers apply to all women, as opposed to women in the age group.
Now if I run the numbers through Bayes, and I'm almost positive that I'm running them incorrectly, I get only about a 7.76% chance that the woman in question has breast cancer. It seems that the way I plugged the numbers, Bayes overvalues the 1% chance if we have no additional information.
I still say the 90.4% value follows from The 9.6% figure. If 9.6% of positives are false, then the remaining 90.4% of positives must be true, meaning there's a 90.4% chance that the woman in question has cancer.
Posted by: tgirsch | June 28, 2006 at 00:23
I don't think that this quite fits the Bayes you talked about earlier. We know that the women has a positive scan, whihc, I think, invalidates the 1% probabality, becasue the mere presence of a scan immediately takes her out of the set of all women and puts her into the set of women who have had a mammogram. We aren't talking about the two bags of marbles anymore, but one bag of marbles and what the odds are of pulling a white one.
Or, at least, thats the way it seems to me. I lvoe stats. I need to get a decent refresher book ...
Posted by: kevin | June 28, 2006 at 10:03
As I point out in the next thread, nevermind. :)
Posted by: tgirsch | June 28, 2006 at 19:26