6.3 - Another Example6.3 - Another Example
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:
Some Miscellaneous Comments
- It is quite common, even for people in the seeming know, to confuse forward and reverse conditional probabilities. A 1978 article in the New England Journal of Medicine reports how a problem similar to the one above was presented to 60 doctors at four Harvard Medical School teaching hospitals. Only eleven doctors gave the correct answer, and almost half gave the answer 95%.
- A person can be easily misled if he or she doesn't pay close attention to the difference between probabilities and conditional probabilities. As an example, consider that some people buy sport utility vehicles (SUV's) so that they will be safer on the road. In one way, they are actually correct. If they are in a crash, they would be safer in an SUV. (What kind of probability is this? A conditional probability!) Conditioned on an accident, the probability that a driver or passenger will be safe is better when in an SUV. But you might not necessarily care about this conditional probability. You might instead care more about the probability that you are in an accident. The probability that you are in an accident is actually higher when in an SUV! (What kind of probability is this? Just a probability, not conditioned on anything.) The moral of the story is that, when you draw conclusions, you need to make sure that you are using the right kind of probability to support your claim.
- The Reverend Thomas Bayes (1702-1761), a Nonconformist clergyman who rejected most of the rituals of the Church of England, did not publish his own theorem. It was only published posthumously after a friend had found it among Bayes' papers after his death. The theorem has since had an enormous influence on scientific and statistical thinking.