Bayes Theorem

Drug TestingAn application of Bayes’ theorem by Rafael Aguiar

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?

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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”).

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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).

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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).

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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.

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Diagram

Resolution8

Conclusion

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❖ 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.

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Rafael Aguiar[rfna]

@rafadaguiar

about.me/rafaelaguiar