Bayes Theorem Lecture
Bayes’s Theorem (pages 291–297)
It is a sad state when even physicians in the United States lack the tools to reason well. Consider this (from G. Gigerenzer’s Calculated Risks): “The probability that a woman of age 40 has breast cancer is about 1 percent. If she has breast cancer, the probability that she tests positive on a screening mammogram is 90 percent. If she does not have breast cancer, the probability that she nevertheless tests positive is 9 percent. What are the
chances that a woman who tests positive actually has breast cancer?” Do you know the answer? Most doctors who were presented with this common medical situation got the answer wildly wrong (off by 80 percent!). Providing poor information leads to poor choices. Consider the possibility that a woman would take unnecessary invasive action on the basis of her or her physician not knowing how to perform this simple probability calculation. The fact is many people have made poor medical choices. So, we could even say that what you will learn in this course might save your life!
A personal story: when my spouse was pregnant with our first of two daughters, she took a standard screening test for Down syndrome and got a positive result. The doctor told us that the next step was to perform a more invasive examination. The doctors were very clear why the invasive examination was required—it carried a lower “false positive” rate. But what, we wanted to know in our nervous state, was the chance that our baby potentially had Down syndrome? The nurse had no answer for us. We were given a pamphlet that told us not to worry because the first test had a high “false positive” rate. There’s that term again! We knew what it meant—it meant that the test sometimes indicates that there is a condition when there really isn’t. The question was: how does that rate affect the calculation for determining whether the fetus has Down? No answer. We consulted a “genetic
specialist” who pulled out charts, threw some numbers at us, etc. We were confused and very frustrated. Why couldn’t someone tell us a straight answer? It was then and there, nearly ten years ago, that I decided to learn
more about probability theory. When I found out that the calculation is relatively easy, I was determined to teach it to all my critical thinking classes. Our daughter, now nine, does not have Down, and the chances that she did,
even with that first positive screening, were very low. I don’t recall the specific conditions that would allow us to calculate the numbers, but it is the procedure that is important, and that’s what I’ll teach you, following the method laid out on textbook pages 294–296, which involves simple charts.
As your authors indicate on page 292, the reason why even seasoned doctors fail to correctly answer questions like the breast cancer or Down syndrome case is that people tend to focus too much on the rate of true positives and ignore the rarity of the condition in the first place. In both the breast cancer and Down syndrome cases there were no other signs or symptoms that the patient had the condition in the first place. So, they were no more likely than anyone in the general public to have the condition in the first place. Now, had there been other signs—for example, had the woman been given a mammogram because she felt a lump—then the value of true positives would have been higher. Nevertheless, the important point is that to determine one’s chance of having the condition given the result of the screening test requires us to use all the information available and not focus exclusively on one particular value (like the rate of true positives). Another important point is that the calculation involves a conditional probability: it is the chance that, say, our daughter would have Down’s given that she tested positive in the screening test.
On pages 292 and 293 the authors give us a little history—the theorem that allows us to calculate conditional probabilities where information changes our initial rates is due to some old guy born in the early eighteenth century. They also reveal to us the ugly formula associated with the old guy’s name. Never mind all that (unless you are a history buff). Let’s just get to the simple system that allows us to make accurate assessments (starting on page 294).
The authors do an excellent job of showing us how the simple system works. They use an example to guide us. The example concerns Wendy, who has received a positive result from colon cancer screening. The initial numbers are given on page 292. Notice that the probability that a person in the general population has colon cancer is very low, 0.003 or 0.3% or (better for quick calculations) 300/100,000. Wendy is assumed to have no other symptoms of having colon cancer—so she falls within the “general population” parameters. Had she had one of the symptoms, alas, the initial probability would have been higher.
Notice the labeling of the charts on pages 294–295. The trick is to consider all the relevant probabilities, not just one. At the very bottom of page 294 you will already get a sense for why Wendy’s chances are low. The total number of people getting colon cancer is only 300 out of 100,000. Because this is a total value for the column “Colon Cancer,” the two numbers we will plug into the cells above on the same column will add up to 300. Right away you should know that because colon cancer is rare, the chance of Wendy having colon cancer is low regardless of whether she tests positive or not. This is what my spouse and I didn’t really understand when we took the screen test for Down’s. Use the methods described in the book to calculate the probability. The answer should come out as roughly 9.2%, or .092-ish.