# 6.3 - Another Example

6.3 - Another Example

## Example 6-2

A common blood test indicates the presence of a disease 95% of the time when the disease is actually present in an individual. Joe's doctor draws some of Joe's blood, and performs the test on his drawn blood. The results indicate that the disease is present in Joe.

Here's the information that Joe's doctor knows about the disease and the diagnostic blood test:

• One-percent (that is, 1 in 100) people have the disease. That is, if $$D$$ is the event that a randomly selected individual has the disease, then $$P(D)=0.01$$.
• If $$H$$ is the event that a randomly selected individual is disease-free, that is, healthy, then $$P(H)=1-P(D)=0.99$$.
• The sensitivity of the test is 0.95. That is, if a person has the disease, then the probability that the diagnostic blood test comes back positive is 0.95. That is, \P(T+|D)=0.95\).
• The specificity of the test is 0.95. That is, if a person is free of the disease, then the probability that the diagnostic test comes back negative is 0.95. That is, $$P(T-|H)=0.95$$.
• If a person is free of the disease, then the probability that the diagnostic test comes back positive is $$1-P(T-|H)=0.05$$. That is, $$P(T+|H)=0.05$$.

What is the positive predictive value of the test? That is, given that the blood test is positive for the disease, what is the probability that Joe actually has the disease?

The test is seemingly not all that accurate! Even though Joe tested positive for the disease, our calculation indicates that he has only a 16% chance of actually having the disease. Is the test bogus? Should the test be discarded? Not at all! This kind of result is quite typical of screening tests in which the disease is fairly unusual. It is informative after all to know that, to begin with, not many people have the disease. Knowing that Joe has tested positive increases his chances of actually having the disease (from 1% to 16%), but the fact still remains that not many people have the disease. Therefore, it should still be fairly unlikely that Joe has the disease.

One strategy doctors often employ with inexpensive, not-too-invasive screening tests, such as Joe's blood test, is to perform the test again if the first test comes back positive. In that case, the population of interest is not all people, but instead those people who got a positive result on a first test. If a second blood test on Joe comes back positive for the disease, what is the probability that Joe actually has the disease now?

Incidentally, there is an alternative way of finding "reverse conditional probabilities," such as finding $$PD|T+)$$, when you know the the "forward conditional probability" $$P(T+|D)$$. Let's take a look: