INVESTED: In Markets, in the Future, and in Society.

You’ve been experiencing headaches and dizziness for the past few weeks, and eventually, you decide it’s time to pay the doctor a visit. Your doctor is stumped, and after a few unsuccessful remedies, they decide to test you for a rare blood disease that affects .01 percent of the population. The test that they recommend is 99.9 percent accurate. You agree and take the test.

A few days later you’re back at the doctor’s office with your spouse, and the doctor tells you that the test came back positive. Your spouse begins to tear up, shaking their head and asking how this could happen. You tell them not to worry — there is actually only about a 10 percent chance you have the disease. How? We need to take into account the base rate of the disease. According to Robert Matthews of Aston University, “people typically ignore base-rate effects and overlook the need to know both false positive and false negative rates when assessing predictive or diagnostic tests.”

These results are surprising to us because life as we experience it typically flows from cause to effect. The baseball flies off to center field because the bat struck it. The flowers bloom because spring has brought warmer weather and longer days. The hiker is running because he startled the mama grizzly bear. Our brains have evolved to be highly specialized in causal reasoning. You have a rare blood disease, and therefore your diagnostic test comes back positive. What we are much worse at, however, is reasoning in reverse. If we start with the effect and try to reason about the cause, our brains seem to short circuit. Your diagnostic test comes back positive – what does that tell you about your likelihood of a correct diagnosis? The tools of probability, such as Bayes method, can help us put our brain in reverse and reason from effect to cause. While it seems paradoxical to many that a positive result on a test with 99.9 percent accuracy would still mean it’s unlikely you have the disease, if we work through a concrete example it becomes clear. 

In a city of 1 million, only 100 people will have the disease. If we give our 99.9 percent accurate test to everyone in the city, 0.1 percent of the population will receive a false positive. That means there will be on average 1,000 false positives and 100 true positives. If your test comes back positive, you only have about a 10 percent chance of actually being positive, because the condition itself is 10 times rarer than a false positive.
Our brains are wonderfully powerful tools in so many ways but that doesn’t mean they come without fault. You can read more about how our brains process probabilities and Bayesian reasoning in this article from the Royal Statistical Society.