Good Thinking by David Robert Grimes

Author:David Robert Grimes
Language: eng
Format: epub
Publisher: The Experiment
Published: 2021-06-15T00:00:00+00:00

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

Returning to our example, how exactly can a test with 99.99 percent 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 percent 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 percent. 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 percent–much greater than a patient in the low-risk cohort.

The low- and high-risk scenarios can be illustrated more intuitively with a frequency tree, depicted in the figure above. This difference is extreme, and it’s worth dwelling on this finding for a moment. We might reasonably ask: Why the stratification? Why should the same test administered to one group yield an accuracy so drastically different from another group with the same test? Instinctively we may feel there is something wrong with the test, but this is not the case–the test does not discriminate, nor does its inherent precision selectively improve or disimprove given a patient’s background. The needle is not clairvoyant, remaining 99.99 percent accurate for all patients. The crux of the issue is that Bayes’s theorem shows us that this information on its own will never be enough to draw conclusions on issues that depend on other probabilities. Probabilities are often conditional, and naked numbers devoid of context need to be carefully parsed.

This serves as an illustration of the fact that, despite the ostensibly intuitive nature of probability and statistics, their seeming simplicity hides layers of complexity that are easy to misunderstand. Such misunderstandings can drive us to entirely erroneous conclusions, and dubious inference and statistical misunderstandings can all too frequently have detrimental consequences. The rationale behind these misunderstandings is no mere academic triviality, nor mathematical sleight of hand; we live in an

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