An application of Bayes’ theorem on Drug Testing.

1. Drug Testing
An application of Bayes’ theorem by Rafael Aguiar

2. Context
❖ Let's say you're from a drug
company;
❖ And you are interested in
measure the presence of a drug
that you produced, in a population;
❖ To measure, you need to TEST. So
we get to an interesting question:
how often your test gonna fail?
2

3. Context
❖ If a randomly selected individual
tests positive, what is the
probability that he or she is a
user of your drug?
❖ To answer that, we gonna make
use of some statistical
concepts(Sensitivity, Specificity)
and Bayes’ Theorem(“posteriori
probability”).
3

4. Context
❖ Sensitivity measures the
proportion of actual positives which
are correctly identified as such (e.g.
the percentage of drug users who
are correctly identified);
❖ Specificity measures the proportion
of negatives which are correctly
identified (e.g. the percentage of
non-drug users who are correctly
identified).
4

5. Context
❖ A perfect predictor would be
described as 100%
sensitivity (i.e. predict all
people from the drug user’s
group as drug users) and
100% specificity (i.e. not
predict anyone from the
non-drug group as drug
user).
5

6. Example
❖ Suppose a drug test is 99% sensitive and 99%
specific. That is, the test will produce 99% true
positive results for drug users and 99% true negative
results for non-drug users. Suppose that 0.5% of
people are users of the drug.
6

7. Diagram
7

8. Resolution
8

9. Conclusion
❖ Despite the apparent accuracy of the test, if an individual tests
positive, it is more likely that they do not use the drug than that
they do;
❖ This surprising result arises because the number of non-users is
very large compared to the number of users, such that the number
of false positives (0.995%) outweighs the number of true positives
(0.495%). To use concrete numbers, if 1000 individuals are tested,
there are expected to be 995 non-users and 5 users. From the 995
non-users, false positives are expected. From the 5 users, true
positives are expected. Out of 15 positive results, only 5, about 33%,
are genuine.
9

10. Rafael Aguiar[rfna]
@rafadaguiar
about.me/rafaelaguiar
10