# 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:

**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.

**Statistical Analysis**

Now, here is where statistics comes into play. Let us define the following random variables:

**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%!

**Afterthought**

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

KrunkNovember 11th, 2007 at 6:55 pm

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in/home/nathanng/nathanng.com/blog/wp-includes/formatting.phpon line74As Nathan and I were talking about this earlier, I mentioned that there were a few assumptions made in this calculation. The #s work out if the assumptions are true, but one of the assumptions made was that the numbers only apply if a person was randomly selected out of the general population to take this test.

I mentioned that generally people who do take these tests (be it disease or pregnancy), have at least some reason for doing so, be that it was they’re experiencing some symptoms, been exposed to something known to cause it, or that they missed their period, etc. A random woman who hasn’t had sex for awhile wouldn’t just go and take a pregnancy test.

Therefore, I argued that the probability of the person having the disease is not just 1%, but something higher. We randomly chose 20% just to see what it would come out to and by replace P(D=1) with .2 and P(D=0) with .8, both of us were shocked that that the probability of you actually having the disease increased from 10% to over 70%.

The 20% figure was just something we made up and we really have no idea what percentage of woman who do take the pregnancy test are indeed pregnant, but the fact that the number can change so drastically depending on how we determine who is actually taking the test is mind bending.

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