Imani Brown
Chapter 5 Diagnostic Testing
Consider the mammogram for detecting breast cancer in women. One recent study estimated
sensitivity=0.86 and specificity=0.88. Of the women who undergo mammograms at any given
time, about 1% are typically estimated to actually have breast cancer; that is, P(S)=0.01. Of the
women who receive a positive mammogram result, what proportion actually have breast cancer?
Use the following diagram to help you solve this problem:
Sensitivity is the probability that the
test detects cancer given that it is
present.
Specificity is the probability that the
test does not detect cancer given that
it is not present.
Both refer to correct results.
Show your work:
The question asks us to find the proportion of women who actually have cancer given that they received
a positive test result. That is, we’re asked to find P(S|POS) where S represents the presence of cancer.
The formula for conditional probability gives us P(S|POS) =
Now to find P(S and POS) we cannot just multiply P(S) x P(POS) because these events are not
independent. So let’s start with P(POS|S) because this is the sensitivity and it is given to us. Don’t forget
to use the sensitivity for younger women, which is .001 rather than .01. So using the conditional
formula on this quantity, we have
P(POS|S) =
.86=(P(S and POS))/(.001) = .001*.86 = .00086
Since we are given P(S) = .01, we can solve for P(S and POS)
P(S and POS) = .00086
To complete the answer, we still need to find P(POS). From the chart, we can see that there are two
ways to get a positive result – a true positive (which is the sensitivity) and a false positive. In the chart,
these are represented by the two branches: YES and positive (sensitivity) = (.01)(.86) and the other
branch NO and positive = (.99)(.12). Since a positive result can happen either way, we add these two
branches to get .0086 + .1188 = .1274 and now we are ready to finish the problem:
P(S|POS) =
= .999 = (P(S and POS))/(.12) = .999*.12 = .11988
When you return this assignment, you will receive the solution to the above question so that you can
check your work.
Now answer the question, “Why is the proportion in error likely to be larger for a young population than
for an older population?” Because there are more young women in the negative category opposed to
the positive so there will be a larger number of false positives than correct positives.