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MAY 07, 2020 — You’re asked to measure a lamppost’s height with an inaccurate ruler. If you know the ruler’s error, say it always overestimates height by 5 cm, you could still measure the lamppost accurately. If you don’t know how inaccurate the ruler is (or even that it is inaccurate), you have two unknowns ― the height of the lamppost and the error of the ruler. In this situation, the ruler measures the lamppost, and the lamppost measures the ruler. Mathematician and philosopher Nassim Taleb names this duality ― when the measuring instrument is measured by what it tries to measure ― Wittgenstein’s Ruler.

In the COVID-19 pandemic, two lampposts need measuring: the SARS-CoV-2 community prevalence, and the infection fatality rate (IFR). The IFR isn’t the same as the (symptomatic) case fatality rate (CFR), which is the fatality rate among those diagnosed with the infection ― ie, those with symptoms severe enough to seek medical attention. The denominator for CFR omits those who had COVID-19 without symptoms or mild symptoms that were dismissed.

The CFR overestimates viral lethality but underestimates its contagiousness. Its denominator is easier to calculate than that of the IFR, whose denominator, community prevalence, eludes RT-PCR testing, which detects active infection only. For IFR, a test must uncover those who had it rather than those who have it.

Serology Testing

The latest beacon is serology, which is the diagnostic analysis of our blood. In response to coronavirus, our immune system produces antibodies, a rebellion of sorts. The rebels hang around for a while after the fight. Serum antibodies, an imprimatur of the virus, can detect undocumented infections.

Determining community prevalence of COVID-19 is straightforward, at least in theory. Say 50 people out of a random and representative sample of 1000 have antibodies to the novel coronavirus. Then the prevalence of COVID-19 is 5%. Not everyone in the community need be tested to determine prevalence. But it’s not so simple.

First, samples are seldom truly random or genuinely representative of the community, which means that raw prevalence estimates must be corrected for sampling biases. The adjusted prevalence becomes an approximation of the truth.

Continued

The more intractable problem brings us back to Wittgenstein’s Ruler ― our measuring instrument is imperfect. Any test has two attributes: sensitivity ― its ability to find disease in those with the disease; and specificity ― its ability to correctly identify those without the disease. Imperfect sensitivity leads to false negatives. Imperfect specificity leads to false positives.

To estimate community prevalence accurately, a test should have high sensitivity and specificity; specificity is more important when the prevalence is low. This is best explained with examples.

Distanceville: Social Distancing Is in Our Name

Consider Distanceville, a town of 10,000, where people practice social distancing and where the community prevalence is 1%, ie, 100 residents had COVID-19 and 9900 didn’t. Now let’s compare two antibody tests, Test A and Test B.

Test A has a sensitivity of 90% and a specificity of 95%. Of the 100 Distanceville residents with COVID-19, 90 will test positive with test A. Of their 9900 neighbors without COVID-19, 9405 will test negative (0.95 x 9900), and the remaining 495 will falsely test positive. Total positives, true and false, is 585, which is almost six times the true prevalence of 100.

Note, total positives are overwhelmed by the false positives. Test A misses 10 people with COVID-19 because of imperfect sensitivity, but this error is dwarfed by the error from imperfect specificity by a factor of 50. False positives overestimate the community prevalence.

Test B is both worse and better than Test A, with a sensitivity of 80% and a specificity of 98%. Of 100 Distancevillers wi