Monday, April 04, 2016

Blood tests and illness: do the arithmetic.

(Thanks to John Paulos for inspiration)

Suppose a nasty but relatively rare disease that shows up every spring. Suppose that in any given year, 5% of the population will get it, and many of them will die. Suppose that researchers have discovered that if you catch the virus early enough, a short but expensive treatment will cure you. Would be nice to have a simple and cheap blood test, wouldn’t it?

Suppose now you read a news story that a lab has developed a blood test that will find evidence of the disease before you have any serious symptoms. It’s cheap enough to use as a screening test every spring. Suppose it is 95% accurate. That means, it will catch 95% of the people who have it.

Sounds pretty good, right? 

Think again.

Test 1,000 people for the disease. 5%, that is 50 people, will have it. You will find 45 of them.

What about the 950 that don’t have it? At a 95% accuracy rate, 95% of those will test negative and 5% will test positive. 5% of 950 is 47.5. So 47 or 48 people will test positive that don’t have it. Let’s go with 47.

So after 1,000 people are tested, we have:

5 false negatives
903 true negatives
Ratio of false to true negatives: 5 to 903, or 1 to 180.6, or 0.006%, or very low.
If you test negative, the odds are close to 200 to one that you don’t have it. Pretty good.

45 true positives
47 false positives
Ratio of true to false positives: 45 to 47, or 0.95 to 1, or 96%, or almost even.
If you test positive, the odds are almost even that you do not have it. Leaves you pretty much where you were before the test.

So if you take the test, and it comes up positive, you have a roughly 50% chance that you don’t have it. If it comes up negative, you have a roughly 99% chance that you don’t have it.

You realise that a vaccine would be better.

Read the news with a numerically critical eye.

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