a mini post

There are many references in the news to how good viruses tests are and what conclusions you can draw from them. Unfortunately some are wrong and I haven't seen any easy to understand explanations of how to figure these things out. It's all conditional probability. The standard approach is to use Bayes' Theorem. Unfortunately most people have never heard of it or vaguely recall a formula with P's and |'s they once memorized without knowing what it really did

Here's an easy way to think about it in the form of an example. Imagine a population, say a city, where 1% of the people have a viral disease (I'll call the disease 'virus'). There's a test with a 90% chance of returning a correct result. So if you have the disease there's a 90% chance the test returns a positive, and if you don't have the test there's a 90% chance the test returns a negative. So what's the probability that someone who has just been tested has the disease?

For simplicity imagine an unbiased group of 1000 people in the city. Let's screen them and see what happens. Ten people will have the disease, 990 will be disease-free.Of the ten people with the disease the test will correctly identify nine of them and incorrectly identify one. Looking at the people who do not have the disease 90% of 990 or 891 will correctly test negative, but 10% of the 990 or 99 will incorrectly test positive.

So the probability a person who tests positive actually has the disease is the number who test positive and have the disease divided by the total number who test positive: 9/108 or about about an 8% chance.

The example has nothing to do with the specifics of CV-19, but maybe will give you a way to think about how to sort out what the real tests are saying when an accuracy is published. Going deeper really requires tools like Bayes' theorem, but hopefully this is clear.

Armed with this technique perhaps this otherwise well-written piece in the* NY Times* will make sense. Also note this is for an unbiased initial sample. If you're screening on existing conditions before testing (which is common outside of Iceland), additional complications appear. But at least this should give you an idea.

## Comments

You can follow this conversation by subscribing to the comment feed for this post.