How a 99% Accurate Disease Test can be 90% Wrong
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In Statistics class the other day, we were learning Baye’s Theorem and the concept of the theorem is heavily used in the medical field to find the probability of an individual having a disease after being tested positive. What was interesting to learn is the fact that a 99% “accurate” test can really be 90% wrong, which was very interesting to know. Conventional wisdom would tell you that a 99% accurate test would be 99% accurate and 1% wrong, right? Well let us explore that concept. Keep in mind that this is not limited to a disease test, but can be applied to other tests as well, such as a pregnancy test.
Let’s suppose you take a medical test with the following facts:
- If you have the disease, the probability of the test being positive is 99%
- If you do not have the disease, then the probability of the test being positive is 10%
- 1% of the population has the disease
To solve this problem, I am going to look at the statistical analysis of it first, then look at it from a non-technical perspective for those of you who do not care about the math.
Now, here is where statistics comes into play. Let us define the following random variables:
- D=1 if person has a disease, D=0 if not
- T=1 if test is positive, T=0 if negative
Thus, the probabilities of the random variables will then be (caution: heavy statistics usage)
- P(T=1|D=1) = 0.99
- P(T=0|D=1) = 1- P(T=1|D=1) = 0.01
- P(T=1|D=0) = 0.10
- P(T=0|D=0) = 1 - P(T=1|D=0) = 0.90
- P(D=1) = 0.01
- P(D=0) = 1 - P(D=1) = 0.99
Finally, the probability of you actually having the disease given the test is positive is
Thus, the probability of you actually having the disease even though the test is positive is only 10%!
The biggest issue here is the definition of “accurate” as listed by a medical test. Most people would assume that if a test is listed as 99% accurate, then the test will always be 99% accurate. However, this is not generally the case. “Accurate ” typically means that if a person has a disease, the test will be positive 99% of the time and will be negative 1% of the time. However, this does not mean that if the person does not have the disease, then the test will be negative 99% of the time. Thus, the condition of whether the person has the disease or not greatly affects the probability of whether the test will be accurate!
Therefore, the probability of an individual not have the disease and testing positive greatly outweighs the probability of the individual having the disease and being tested positive. This concept leads “accurate” medical tests to being inaccurate most of the time.
So, the next time you test positive for a disease or for a pregnancy, then caution the accuracy of the test.
Posted in Nerd Logic