The Irrational Ape by David Robert Grimes

Author:David Robert Grimes [Grimes, David Robert]
Language: eng
Format: epub
ISBN: 9781471178276
Publisher: Simon & Schuster UK

Frequency trees depicting reliability of HIV tests for (a) low-risk cohort and (b) high-risk cohort

Imagine you’re given an HIV test, which you’re informed is 99.99 per cent accurate. The test comes back positive; then what are the odds you have HIV? For most of us, our instinct quite reasonably tells us it is almost certain we have the disease, yet this is generally wrong. The actual answer is instead closer to 50 per cent for most of us. If you’re left somewhat perplexed by that result, you’d be in good company; most people, including medical professionals, tend to be equally flummoxed by this seemingly bizarre assertion.

This curious result is explained by Bayes’ theorem, a mathematical framework for combining conditional probabilities, mapping how probability branches out. Bayes’ theorem tells us that the probability of having HIV in the event of a positive test is dependent not only on the test but on how likely one truly is to have the illness. While the test itself is almost perfect, its accuracy is dependent upon another condition, namely the a priori chance that a person has the virus in the first place. We’ll avoid a formal statement of Bayes’ theorem as it is beyond our scope and is needlessly intimidating to those unfamiliar with mathematical notation. However, the logic behind it is easy to follow and vital to illustrate, as it lurks behind countless seemingly paradoxical statistics.

Returning to our example, how exactly can a test with 99.99 per cent accuracy only be half-sure a typical patient has HIV? For a typical low-risk subject, baseline infection rate is about 1 in 10,000. Now, imagine 10,000 such people walk in for a HIV test; one of them has the virus and will almost certainly test positive. But in the remaining 9,999, another will test positive due to the accuracy limits of the test, leaving two positive tests, only one of which is a true positive, meaning that with a positive test, a person is 50 per cent likely to have the illness.

Crucially, this jarring result does not indicate that the test is inadequate; the HIV test in our example is incredibly accurate. Rather, due to the limited prevalence of the illness, the conditional probability is much lower than what we may intuitively expect. In truth, the a priori likelihood of a particular subject being infected is inextricably entangled with the precision of the result. Consider the same test administered to a high-risk population, such as intravenous drug users. The infection rate in this cohort is roughly 1.5 per cent. Let’s again envision 10,000 such patients getting tested. In this cohort, roughly 150 will have the virus and flag positive. Of the remaining 9,850 patients, there should be approximately one false positive. In this instance, the odds of HIV infection given a positive test are not 50/50 any more. The likelihood of a high-risk patient having HIV given a positive test is 150/151 or 99.34 per cent – much greater than a patient in the low-risk cohort.

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