Figure the Odds
We ended yesterday's post by promising to show you how to more easily understand the debate over breast cancer screening. Here's a handy way to calculate odds like the ones being discussed in the breast cancer debate.
First, let's start with another test. Assume the following:
a) The accuracy rate of mammography is 95%
b) The false positive rate for mammography is only 3%
c) Only 1% of women over 50 have breast cancer
d) A woman over 50 has a positive mammogram
Question: What are the odds that she actually has breast cancer?
Before we give you the answer let's talk a little bit about percentages. First most people think they understand them, second they don't, and third even when they do most people tend to have a very difficult time reaching the right answer to a question like the one above. On the other hand, people tend to do better when dealing with rates or frequencies. So before we introduce you to Bayes' theorem (not today) let's try solving the question using rates.
If 1% of women over 50 have breast cancer that means that out of 10,000 women 100 of them will have breast cancer. If all 10,000 women are screened by mammography and the accuracy rate is 95% then the test will detect 95 of the 100 cases.
However, if all 10,000 women are screened and the false positive rate is 3% then, of the 9,900 who don't have breast cancer, 297 (3% x 9,900) of them will have a mammogram indicating that they do have breast cancer.
The total number of positive mammograms thus equals the 95 who actually have breast cancer and whose cancers were detected plus the 297 who don't have breast cancer but who had a positive mammogram for a total of 392 possible cases of breast cancer. So, if only 95 of the 392 women with positive mammograms actually have breast cancer what are the odds that your hypothetical patient is one of them?
Well, 95 is only 24% of 392 (95 / 392) - slightly less than a one in four chance that she actually has breast cancer. So how did you do?
Risk is hard because it's counter-intuitive. Comparing percentages is inevitably an apples to oranges trap. Instead of thinking that a 95% accuracy rate is really high and 3% false positive rate is really low, maybe try asking yourself whether you'd rather have 95% of $100 or 3% of $10,000.
