Imagine you and a friend go fishing, and after a long day with no luck your friend says, “I guess there aren’t any fish here.” This would obviously be unsound reasoning, yet it’s present in many of the current discussions about COVID-19 testing. Policies being crafted and implemented now will require people to show negative tests before traveling, attending sporting events, seeing a doctor, going to college, and returning to work. A positive test would mean they had the virus, so a negative one must mean they don’t. If you don’t catch a fish, there must not be any.
The problem is that virus tests, particularly the standard PCR tests using nasal swabs, have a substantial false negative rate. The test looks for the virus’s genetic material, which, if discovered, means the person is almost certainly infected. But the virus could still be present in low concentrations even if the test doesn’t find it. A study in a hospital in Wuhan found that even at the height of sickness, confirmed COVID-19 patients tested negative about 30 percent of the time. For asymptomatic carriers or people in the early stages of infection, the rate could be much higher. Such people can certainly be contagious, though.
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Because of this possibility, negative test results contain little truly usable information. As a rule of thumb, a 30 percent false negative rate means a person’s odds of having the virus, after testing negative, go down by a factor of about three. Consider a very sick patient with fever, shortness of breath, a chest X-ray with the telltale signs of lung damage, etc. A doctor might say this person’s odds of having the virus are 100-to-1 in favor, so a negative PCR test might only lower those odds to 33-to-1 in favor. That is still a 97 percent probability. At the other extreme, an asymptomatic person from a population where the virus isn’t widespread might reasonably have odds of 1,000-to-1 against being infected, so a negative test might steepen the odds to 3,000-to-1. In neither case does the difference matter much. Would you treat someone with a 99.97 percent chance of being virus-free differently from someone whose chance was only 99.90 percent?
Widespread testing is essential because a positive test can trigger someone to go into quarantine and begin contact tracing. But the tests’ usefulness does not swing both ways, because the number of contagious people without symptoms at any given time is relatively small. Anyone you encounter already has a low chance of having the virus. The stakes are high, so caution dictates that we protect ourselves by keeping our distance and wearing masks. But that would remain true even if everyone were to test negative. Any activity that was risky before the tests should be considered practically just as risky after them.
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False negatives are common, so a negative test can’t reduce the risks to zero. When you fail to catch a fish, assume the fish are probably there and keep fishing.
Aubrey Clayton is a mathematician living in Boston and the author of the forthcoming book “Bernoulli’s Fallacy.” Follow him on Twitter @aubreyclayton.