Take the National Institutes of Health. In a two-year study about to begin, it will perform an antibody test on 10,000 volunteers from around the country. The number is large. But the N.I.H. decided to apply it on an “opt in” basis, to anybody who asked for it, so it will not amount to a representative sample of the population.

It will represent only those who want it most, capturing mostly Americans who fear they are sick. It will miss many asymptomatic carriers of the virus, who see no point in getting tested. The exercise will then fail to provide a faithful picture of how the new coronavirus has spread across the country.

There is other valuable testing to be done. Some health experts argue that, early in an epidemic, it is more useful to test specifically those suspected of carrying the virus, to trace their contacts and introduce more nuanced separation and quarantining policies.

“I don’t care about the prevalence of the infection in the population,” said Stefano Bertozzi, a professor of health policy and management at the University of California, Berkeley, who was the last director of the World Health Organization Global Program on AIDS. “I care about when the infection rate will overwhelm the capacity of the health system.”

There are more urgent questions at this stage than the overall infection rate in the population, he noted. For instance, what is the share of infected people who get sick, and how fast is the infection rate growing across thousands of cities? There is hope that epidemiological modeling based on testing of infected people can approximate this answer effectively, he said.

And yet, Mr. Mogstad argued, tracking the infection rate in the population is essential to track the evolution of the virus, including asymptomatic cases, and how it affects people of different races and income levels. That is critical to deciding when to relax restrictions, evaluating the effectiveness of measures taken and calibrating epidemiological models.

Figuring out this rate of infection is not that costly. “It’s not about how many you test — it is about who you test,” Mr. Mogstad said, adding that 5,000 tests could do the trick.

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