19/10/2022

Column by Marcelo Viana: False positives in medicine

Crédito: Marcelo Camargo/Agência Brasil

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Authorities in Oz are concerned about a disease affecting the population. The illness is serious, but treatable if detected early. It would be natural to test everyone immediately, but tests are not infallible: the chance of a sick person testing negative is 2%, and the chance of a healthy person testing positive is 3%.

False positives are especially problematic: besides causing the person distress at the thought of risking their life, they also point to expensive, uncomfortable, and, in this case, unnecessary treatment. But the chances of error seem quite small, so isn't it worth taking the risk anyway?

A central question is this: when the result is positive, what is the chance that the person is healthy and therefore treatment is not justified? The answer is given by the theory of inverse probability , created in the 18th century by the work of the British Thomas Bayes and the Frenchman Pierre-Simon de Laplace (the expression "inverse probability" was first used in 1837 by Augustus de Morgan, another Briton).

It is estimated that 1% of the population of Oz is infected: this is the probability that a randomly chosen person is sick. We write this in a simplified form: P(sick) = 1%, therefore P(healthy) = 99%. We want to know the probability P(healthy if positive) that the person is well, knowing that they tested positive. Bayes' theorem explains how to calculate it, 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 chance of false positives is 3%, this gives 99% times 3%, which is 2.97%. The second step is to do the same calculation for sick people, that is, P(sick) times P(positive if sick). Since P(sick) is 1% and the chance of false negatives is 2%, this calculation gives 1% times 98%, or 0.98%.

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