December 9, 2022

# Inferring Causes from Effects in False Positives – 10/18/2022 – Marcelo Viana

Officials in Oz are concerned about a disease afflicting the population. The disease is serious but treatable if caught early. It may be normal to test everyone immediately, but tests are not guaranteed: the chance of a sick person testing negative is 2%, and the chance that a healthy person tests positive is 3%.

False positives are particularly problematic: in addition to causing a person to suffer from the belief that their life is in danger, they also indicate expensive, inconvenient and in this case unnecessary treatment. But the chances of error seem so small, isn’t it worth the risk anyway?

The central question is the following: when the result is positive, what is the chance that the person will be healthy and therefore treatment is unjustified? The answer has already been given Inverse probability theorywhich was created in the eighteenth century by the work of Briton Thomas Bayes and Frenchman Pierre-Simon de Laplace (the expression “inverse probability” was first used in 1837 by Augustus de Morgan, another Briton).

Estimated 1% of the Oz population is infected: This is the probability that a randomly selected person will be sick. We write this in simplified form: P (sick) = 1%, so P (healthy) = 99%. We want to know what is the probability of a P (healthy if positive) that the person is fine, given that they test positive. Bayes’ theorem explains how to calculate, and the result may surprise you.

The first step is to calculate P (healthy) times P (positive if healthy). Since P (healthy) is 99% and the probability of false positives is 3%, this gives 99% multiplied by 3%, which is 2.97%. The second step is to do the same calculation for patients, i.e. P (sick) times P (positive if sick). Since P (patient) is 1% and the probability of false-negative results is 2%, this calculation gives 1% multiplied by 98%, or 0.98%.

The last step is to divide the first two numbers by the sum of both, that is, P (healthy if positive) equals 2.97% divided by 2.97 + 0.98%. The result of this split is 75.2%. So, in this case, the vast majority of positive results – more than 3/4 of them – are false positives!

This does not mean that the government should necessarily abandon the idea of ​​testing the population. It just goes to show that test results need to be careful.

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