I thought that I would have to use the Bayes theorem but every times I try different number to plug in, I get it incorrect. Can someone explain how to do it?

Question

The Pap smear is the standard test for cervical cancer. The false-positive rate is .583; the false-negative rate is .180. Family history and age are factors that must be considered when assigning a probability of cervical cancer. Suppose that, after obtaining a medical history, a physician determines that 2% of women of his patient's age and with similar family histories have cervical cancer. Determine the effects a positive Pap smear test has on the probability that the patient has cervical cancer. (to 4 decimals).

Answer 1

Hi Carlos,

 

You're right that you should use Bayes' theorem.

To figure out the right conditional probabilities to apply,

start with an empty confusion matrix.

 

          C⁺       C^-

PS⁺   TP       FP

         -----------

PS^-    FN     TN

 

 

Here C⁺ means having cancer, C^- means not having cancer,

PS⁺ is a positive Pap smear, and PS^- is a negative Pap smear.

Also TP is the true positives, FP is the false positives,

FN is the false negatives, and TN is the true negatives.

Then what do we know? That P(C⁺) = .02.

Also P(FP) = P(PS⁺|C^-) = .583, and P(FN) = P(PS^-|C⁺) = .18.

 

What can we figure out from what we know?

P(C^-) = 1-P(C⁺) = .98.
P(PS⁺|C^-) = FP/(FP+TN), so P(TN) = P(PS^-|C^-), also called Specificity, =

TN/(FP+TN) = 1-P(PS⁺|C^-) = .417.

P(PS^-|C⁺) = FN/(TP+FN), so P(TP) = P(PS⁺|C⁺), also called Sensitivity, =

TP/(TP+FN) = 1-P(PS^-|C⁺) = .82.

 

The question is asking for P(C⁺|PS⁺), and now we have everything

we need to calculate it using Bayes' theorem.

P(C⁺|PS⁺) = TP/(TP+FP) = P(PS⁺|C⁺)*P(C⁺)/P(PS⁺) =

P(PS⁺|C⁺)*P(C⁺)/(P(PS⁺|C⁺)*P(C⁺)+P(PS⁺|C^-)*P(C^-)) =

.82*.02/(.82*.02+.583*.98) = 0.0279.