Sunday, March 21, 2010

About False Positives. Do you have XYZ disease?

We spend a good portion of the Research Statistics classes discussing Type I and Type II errors. Type I error is also known as a False Positive. If you go to the doctor and are tested for a rare disease and the results are positive, does this necessarily mean you have the disease. Sometimes the tests are in error and will give a positive answer when they should give a negative result.

Here is a brain teaser I offer to my classes:

"Let's discuss a situation that can occur in real life in the medical field. Let's assume you go in for a test for a certain disease and the test returns positive. It could be cancer, HIV or other things. Suppose the lab test has a false positive (type I) rate probability of 0.05 or 5% of the time it returns a false positive. Now considering this the incidence of the disease in the general population is 0.01 or 1% or about 1 in 100. Can you calculate the probability that you actually have the disease? Incidentally, most Doctors are not good with probabilities and are likely to give you the wrong information in this situation."

Most people will see the false positive rate as 5% and assume there is only a 5% chance the test was wrong. Doctors will often advise you there is 95% certainty you have the disease. They will get it wrong also.

The answer is best seen though the use of a Venn Diagram:

Let's assume 100 people are tested for a certain disease - XYZ.
  • Knowing we have a 5% false positive rate we will have we will have 5 people test positive but these will not have the disease.
  • Knowing the incidence rate is 1 in 100 or 0.1% we will have 1 person that tests positive that actually has the disease.
  • 94 people will test negative for XYZ


To answer the question about the probability of actually having the disease it is realtively easy to calculate. 6 people tested positive and 1 person had the disease. The probability is 1/6 or about 16.7% chance of having the disease.

The conclusion is that whenever we test positive for a medical issue it is important to know three things.

  1. What is the false positive rate for the test.
  2. What is the incidence of the disease in the general population.
  3. If I am not in the high risk group what is the incidence of the disease outside of the high risk group.
It's an interesting problem for students to think about especially when studying probabilities and statistics.
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