It seems kind of important that a doctor can correctly interpret the following scenario:
- A 40-year old woman has a positive mammography in a routine screening.
The doctor is told the following about scanning for breast cancer:
- 1% of women at age forty who participate in routine screening have breast cancer.
- 80% of women with breast cancer will get positive mammographies (which means there are 20% false negative).
- 9.6% of women without breast cancer will also get positive mammographies (known as a 9.6% false positive)
What is the probability that she actually has breast cancer?
Only 15% of the doctors surveyed estimated the correct probability; most doctors estimated the probability to be between 70% and 80%, which is wildly incorrect.
That’s kind of scary.
This is a classical example of where Bayes’ Theorem should be used.
The problem, according to an editor at Read the Sequences, is that Bayesian reasoning is very counterintuitive. People do not employ Bayesian reasoning intuitively, find it very difficult to learn Bayesian reasoning when tutored, and rapidly forget Bayesian methods once the tutoring is over. This holds equally true for novice students and highly trained professionals in a field.
Here is Bayes Theorem:
Clear as mud, right?
No wonder people claim that it is not intuitive.
But let’s try to apply it to the mammography screening.
We are trying to solve for the probability of having cancer, given a positive screening. This is P(A|B).
Here is what the other terms mean:
- P(A) = the overall probability of cancer; in this example, this probability is one percent, or .01
- P(B|A) = probability of positive screening, given that you have cancer; in this example, this probability is 80 percent, or .80
- P(B) = the overall probability of getting a positive result; this is the most challenging number to calculate.
To calculate the P(B), let’s reword the original scenario.
Let’s assume that 1,000 women of age 40 take the mammography screening. Of these 1,000 women, 1% have cancer, which is 10 women. Of these 10 women, the mammography test will give the correct result for 80% of them, or 8 women. Of the 990 women in this group who do not have cancer, the mammography test will falsely say that 9.6% of the women do have cancer, which is approximately 95 women.
Thus, in total, out of the 1,000 women who take the mammography test, 103 (8 + 95) will test positive, or about a 10.3 percent chance, or .103 (103/1000). This is P(B).
Let’s put it all together now, using the first formula in the blue box above:
P(B|A) * P(A) = =.8*.01 = .008 (in English, these are the odds of testing positive, given that you have cancer times the chance of having cancer)
P(B) = .103 (from above) (in English, this is the probability of testing positive)
Dividing one by the other, we get P(A|B) = .008/.103 = .078, or 7.8%
We could have also calculated the denominator using the alternative formula shown above. We already know that
P(B|A) * P(A) = =.8*.01 = .008 these are the odds of testing positive given that you have cancer times the chance of having cancer
P(B|not A) * P(not A) = .096 * .99 = .095 these are the odds of testing positive given that you do not have cancer times the chance you do not have cancer.
P(B|A) * P(A) plus P(B|not A) * P(not A) = .008 + .095 = .103 (in English, this is the probability of testing positive)
So to answer the original question: if a woman tests positive for cancer, there is only a 7.8% chance that she has cancer; dramatically different than the 70 to 80% probability that doctors had estimated.
Perhaps an easier way to think of this would be to go back to the example of 1,000 women.
We have shown 103 of these women will test positive; however, only 8 of them will have cancer, 95 will not have cancer. Thus the probability of having cancer, given that you tested positive is 8/103, or 7.8%.
As noted above, this may not be an intuitive result.
When you are told that 80% of women with cancer will test positive, that is not the same question as asking that if a woman tests positive, what are her chances of having cancer.
Since there are so many women more women who do not have breast cancer, and some of those women will test positive, that creates a much larger population of women who test positive and don’t have cancer (95) compared to the number who test positive and have cancer (8).
In conclusion, if you made it this far, then:
- you are a stat nerd like me (less than one percent chance)
- you need to find better reading material (greater than 99 percent chance)
I also wonder, given that a person has read this far, what the odds are that they will hit the like button…
- top image from LinkedIn