Are you sure about that Doc?
During a routine check-up, your doctor runs a test that reveals that you have chicken pox. You know that the test might be wrong, so how do you determine the chance that you are actually sick?
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The doctor tells you that the test will report that you have chicken pox when you do not, one out of every 1,000 times. In other words, the test is 99.9% accurate, suggesting that there is a very good chance that you have chicken pox. The accuracy of the test, however, is not the only thing that we need to consider. We also need to consider the chance that you would be sick, regardless of the test. For example, a young boy tests positive on a pregnancy test that is wrong 1 out of 1,000 times. Surely we wouldn't say that the chance that he is pregnant is 99.9%. We need to consider the fact that it is impossible for him to be pregnant (test or no test). |
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The probability that you are sick given a positive test result is the probability that the test result is positive given that you are sick, times the probability that you are sick (this is known as Bayes' rule, after the mathematician Thomas Bayes, 1702-1761). We can express this more formally as:
P(S|T) = P(T|S) P(S)where P() denotes the probability of an event, S=you are sick, T=positive test result and "|" should be read as "given" (e.g., P(S|T) is the probability that you are sick given a positive test result).
For example, the probability that the chicken pox test is positive given that you have chicken pox is:
P(T|S)= 0.999this is the known error of the test. If your younger brother came down with the chicken pox, and you have never before had chicken pox, then the chance that you have chicken pox is pretty high, let's say 90%:
P(S)=0.90The overall probability that you have chicken pox is then:
0.999 x 0.90 = 0.8991or just under 90%. If, however, you had chicken pox two years ago, and you have not come into contact with anyone with chicken pox, then the probability that you have chicken pox is low, let's say 20%:
P(S)=0.20The overall probability that you have chicken pox is then:
0.999 x 0.20 = 0.1998or just under 20%, much smaller than the 99.9% based only on the test.
When evaluating the results of a medical test, you should take into consideration your inherent risk of having contracted whatever it is that you may have tested positive for. A surprising number of doctors don't understand this. So the next time the doctor returns to you a test result, ask them if their diagnosis has taken into consideration the full Bayesian likelihood.