Why False Positives From Drug Tests Are So Common: Bayes’ Theorem
Bayes’ theorem of conditional probability provides insight into advantages of using prior knowledge in determining the likelihood of a given event. The applications of Bayes’ Theorem span almost any topic imaginable, but one that’s particularly interesting is the Bayes’ Theorem application to drug testing. With the drug overdose rates climbing, drug testing is needed now more than ever. The National Institute on Drug Abuse cites nearly 92,000 people in the U.S. who died from drug-related overdose deaths in 2020 alone. The following link presents data and visuals detailing the rise in drug related deaths from 1999-2020: https://nida.nih.gov/research-topics/trends-statistics/overdose-death-rates.
The current testing procedures, while they may seem effective on the surface level, boasting 98+ percent rates of effectiveness with detecting drug presence, have shown to have more false positives than true positives. But why is this the case? In an investopedia article, Adam Hayes presents us with a numerical example in order to explain this phenomenon, citing Bayes’ theorem. Using Bayes’ theorem, he analyzes the following situation. Given a drug test that is 98 percent accurate and a population of 0.5% of people that actually use the drug, what is the probability that a person who tested positive for the drug actually took the drug? Intuitively, it seems that a false positive is unlikely given the 98% accuracy that the drug test has. However, Bayes’ theorem proves otherwise. Hayes presents the following calculation, using Bayesian statistics, which show that the likelihood of a person who tests positive for the drug is only 19.76 percent likely to have taken the drug.
(0.98 x 0.005) / [(0.98 x 0.005) + ((1 – 0.98) x (1 – 0.005))]
= 0.0049 / (0.0049 + 0.0199) = 19.76%
Hayes states “Bayes’ Theorem shows that even if a person tested positive in this scenario, there is a roughly 80% chance the person does not take the drug.”
Using the Bayes’ Theorem, learned in Networks, CS 2850 at Cornell, the numbers presented by Hayes’ can be given in the following format which we learned in class.
(P(B|A)) * P(A)
P(A|B) = —————————————————
P(B|A)P(A) + P(B|A(c))P(A(c))
P(A|B) is the likelihood of A occurring, given the evidence, B. P(A) is the probability of A. Using this, we can construct the following definitions for the variables in the equation.
A: took the drug
A(c): didn’t take the drug
B: test positive
B(c): test negative
P(B|A) = 0.98
P(B|A(c)) = 0.02
P(A) = 0.005
P(A(c) = 0.995
When we plug these values into the equation for P(A|B), we get the following result:
(0.98) * 0.005
P(A|B) = ————————————————— = .1976 or 19.76%
(0.98) * 0.005 + 0.02 * (0.995)
Thus, P(A|B), the probability that a person took the drug given they tested positive is only 19.76%, which is significantly lower than expected. Because of this, there are a reasonable amount of false positives; this idea is illustrated in Cory Simon’s article, titled, “Why Cocaine Users Should Learn Bayes’ Theorem.” He presents a similar calculation, instead with a drug test which has 99% accuracy given the same percentage of the population taking the drug, and he finds that only 33% of positives were results from people who actually took the drug. While initially, it seems counterintuitive that a test that is 98 or 99 percent accurate produces false positives more than true positives, by examining the evidence which informs us of how small the population of drug users is in comparison to those who do not use the drug, the presence of false positives makes more sense. Bayes’ Theorem shows the importance of making informed choices by taking additional evidence into account before making assumptions.
https://www.investopedia.com/terms/b/bayes-theorem.asp
https://nida.nih.gov/research-topics/trends-statistics/overdose-death-rates
http://corysimon.github.io/articles/why-cocaine-users-should-learn-bayes-theorem/