Easy math?

A concept which is closely related to Bayes' theorem, and may be a somewhat easier way of understanding the relevant qualitative issues, is absolute risk vs relative risk. As you know if you are a faithful reader, the culture in the United Kingdom is much more favorable than ours to both the rational allocation of medical resources, and to promoting understanding of cost-effective care among the general public. So, fly across the pond with me for a good explanation.

In a nutshell, if I tell you that taking this test or swallowing a pill every day for the rest of your life will cut your risk of going tone deaf from Oppernockity's disease* in half, it might sound like a good idea to go for it. But if I told you that your chance right now is 1 in a million, and doing the test or the pill will make it 1 in 2 million, you might not be so inclined.

In fact it doesn't have to be some weird disease I made up that you never heard of. Your chance of getting some specific form of cancer, for example, is quite low. You may have heard otherwise -- that the chance of a woman developing breast cancer, for example, is 1 in 9. The risk of cancer rises with age, so if you live to 80 there may be a 1 in 9 chance of having a detectable lesion, but that doesn't mean it's going to kill you or make you seriously ill before something else gets you. Almost 100% of men develop prostate cancer but if we weren't aggressively looking for it the vast majority of them would never know it.

So public health specialists generally like to put risks inside of time frames -- what is the chance of this happening to you within 10 years, might be a typical question. The reason is that a lot can happen in that time. You could die of something else, or have bigger problems to worry about; or a miracle cure could be developed, or a better and cheaper and safer preventive.

As the linked article explains, a useful metric is the Number Needed to Treat. If the prior risk is 4/100, and a treatment reduces the risk to 3/100, that's a 25% risk reduction, which sounds pretty good, but you'd have to treat 100 people to prevent one instance of disease. That means there is a 1% chance it would benefit you. That's what you have to weigh against side effects, cost, and hassle. Put it that way, and lots of interventions and tests that are commonly done don't really seem worth it after all.


* He only tunes once.