Question

Your patient’s mammogram is positive for breast cancer which has a fairly low rate in your state of 1 case per 1,000 women annually. You know from the literature that the mammogram test has a sensitivity of around 92% and a specificity of 95% depending on the study. What do you tell your patient when she asks do I have breast cancer.

Show your work in both a contingency table and using the shortcut Bayes Theorem.

Answer 1

Contingency table:

	Disease (D)	No disease (N)	Total
Positive (+)	TP =0.00092	FP =0.99805	0.99897
Negative (-)	FN =0.00008	TN =0.00095	0.00103
Total	0.001	0.999	1

(TP =True Positive; FP =False Positive; FN =False Negative; TN =True Negative).

Let the probability =P

P(D) =1 in 1000 =0.001

P(N) =1 - P(D) =1 - 0.001 =0.999

Sensitivity:

Sensitivity =P(+|D) =0.92 P(-|D) =1 - 0.92 =0.08

P(+|D) =P(+D)/P(D) =0.92 P(+D)/0.001 =0.92 P(+D) =0.92*0.001 =0.00092

Specificity:

Specificity =P(-|N) =0.95

P(-|N) =P(-N)/P(N) =0.95 P(-N)/0.001 =0.95 P(-N) =0.95*0.001 =0.00095

Now,

We have P(+) =P(+D)+P(+N) =0.00092+0.99805 =0.99897

The probability that the woman has disease (breast cancer) given that the test result is positive =P(D|+) =P(D+)/P(+) =0.00092/0.99897 =0.000921. It means 921 in a million or 0.921 in a thousand.

So, I would tell her that she had a very little chance being less than 1 in 1,000 chance of having a breast cancer and not to worry until further tests were done to confirm.