Bayes, and the Horse He Rode In On

#1
A professor presented this question: "If a cancer screen comes back positive in 68 of 75 cases of actual cancer and in 5 of 191 cases of no cancer, when the actual cancer rate in the total sample is 1/3, what is the probability of a false negative test result (patient has cancer but test comes back negative)?"

I was thinking the simplest calculation is as follows: the cancer screen identifies 68/75 = 91% of cases where the patient has cancer. It fails to identify 7/75 = 9% of cases that it should identify. The probability of a false negative is 9%.

One person tells me that it is necessary to use Bayes' Theorem to arrive at a correct answer. I don't know how to do that. He says the correct answer is 4.57%. But when I use the Bayesian calculator at http://www.vassarstats.net/clin1.html, I don't get that number.

Possibly I am using the calculator wrong. Even so, I don't know why the 9% figure wouldn't be right. I am wondering whether this is a situation where Bayesian and non-Bayesian statisticians diverge.

In case anyone is wondering, this is not a homework question. It used to be a quiz question, but right now it's just spilt milk.

Thanks for any insights.
 

Dason

Ambassador to the humans
#2
I am wondering whether this is a situation where Bayesian and non-Bayesian statisticians diverge.
No because this is just an example of using Bayes theorem. It's straight probability and frequentists and bayesians both have no problem with this.
 

Mean Joe

TS Contributor
#3
I was thinking the simplest calculation is as follows: the cancer screen identifies 68/75 = 91% of cases where the patient has cancer. It fails to identify 7/75 = 9% of cases that it should identify. The probability of a false negative is 9%.
Write it out, you made a common mistake.
P{positive test | cancer} = 68/75,
therefore
P{negative test | cancer} = 7/75.
This is what you actually calculated. But the question of course is asking P{cancer | negative test}.

It is a very common mistake; see the example(s) in this article -- I myself just read the first half: http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/