In seeking to reassure the public about rising covid-19 case numbers in several states, President Trump continues to attribute the new cases to the result of improved testing in the United States. At his rally Saturday, he called testing a “double-edged sword.” When you test 25 million people, he said — as the United States has done — “you’re going to find more people, you’re going to find more cases.” Because that leads to bad news, he said he’d urged officials to slow down testing.

In the past, the president has gone even further, saying: “If we stop testing right now, we’d have very few cases, actually.” Trump’s sentiments have been echoed by Vice President Pence, who recently urged governors to cite increased testing as the reason for any growth in cases as they “encourage people with the news that we are safely reopening the country.”

The extreme version of the argument obviously makes no sense: If we stopped testing for cancer, cancer diagnoses would also go away, though cancer’s prevalence in the population would remain the same. But at least in theory, the milder version could be right: As we test more people, we may see more positive tests (that is, cases). The increase in cases might not necessarily mean that the virus is spreading aggressively: Rather, it could be an artifact of increased surveillance.

But a closer look at the state of testing in the United States shows that this theory is dangerously wrong. To understand why, we have to look not just at the number of tests each state is doing, but at the positivity rate of those tests. In combination, those two metrics give a sense of whether infections are rising, declining or holding steady. And in too many states, the dispiriting answer is that the virus is racing ahead of public health measures to contain it.

It is true that the United States has dramatically expanded testing for the virus that causes covid-19. Currently, more than 400,000 tests are administered per day in this country, compared with fewer than 20,000 tests per day in early March. As President Trump has boasted, more tests for the coronavirus have been conducted in the United States than in any other country.

Of course, the main reason we have to conduct more tests than other countries is because we have the largest epidemic in the world. And we don’t test out of curiosity: The key reason to do so is so we know whom to isolate, to prevent further spread.

Tests can also help tell us whether such measures are working. The way to tell whether a rise in cases is indicative of increased spread in the population — rather than a byproduct of conducting more tests — is by seeing how many tests are identifying infections.

To get a sense for this dynamic, consider New York. At the beginning of April, when testing was constrained at around 20,000 tests per day, positivity reached a peak of 50 percent. No one thinks that anywhere near half the state’s population had covid-19 at that time, so those results would imply that testing was overly focused on a subset of the population highly likely to be infected. Indeed, health officials were limiting testing to health-care workers and patients requiring hospitalization. These days, upward of 50,000 tests per day are conducted in New York and positivity is less than 2 percent. Case numbers in New York have fallen considerably since April — from a peak of more than 10,000 cases per day to around 600 to 700 per day — although the state is now casting a wider net to look for infections. It is testing people with milder symptoms that may be due to covid-19, but could also be caused by other conditions.

That’s the pattern health officials would like to see in testing, but it is not the prevailing pattern across the United States. What is an appropriate test positivity level? The World Health Organization recommends that countries seeking to lift social distancing restrictions should be doing enough testing to maintain a test positivity of less than 5 percent. This target is based on the experience of countries that have driven their numbers of cases down and largely stopped viral spread. Many places, such as South Korea, Australia and New Zealand have been keeping their positivity much lower — at 2 percent or less. Such experiences show that very low positivity rates are an important benchmark to gauge progress against the pandemic.

Overall test positivity in the United States has fallen dramatically (partly because we tested so few people early on). Across the entire country, positivity is currently running at just under 5 percent, down from a peak of 21 percent in early April. That’s a good sign, in general. However, more than 20 states — most of them in the South and Midwest — still have positivity rates higher than 5 percent. These include Arizona, at 17 percent; Alabama, at 12 percent; Florida, at 10 percent; Texas, at 9 percent; and Georgia, at 8 percent.

These states have also made news for their rising covid-19 case numbers. Arizona had 14,107 cases last week, when it tested 67,235 people — up from 8,802 cases and 56,424 tests in the previous week At a glance, it may look as if Arizona found more cases last week because it tested more people, but the reality is that new cases outpace the spread of testing.

In the last week, the incidence of new cases has increased in nearly half of the states. (Nationally, an average of about 20,000 cases have occurred each day since the beginning of June). For some of these states, such as West Virginia and Oregon, these increases may be due to week-to-week fluctuation, not increasing transmission. Recently, West Virginia and Oregon have both increased the number of people tested and seen a decline in test positivity. But for some states with rising case growth, it is almost certainly because more people are becoming infected.

Disconcertingly, in the past two weeks, more than 10 states have reported increases in both new cases and positivity rates. These trends are most pronounced in Alabama, Arizona, Arkansas, Florida, Georgia, North Carolina and Texas. These states are all doing more testing than in previous weeks, but if the infection rate were holding steady, we’d expect the positivity rate to at least stay level.

That states are finding more cases relative to the amount of tests they are conducting provides the strongest rebuttal to the administration’s assertion that case numbers are rising because we’re getting better at finding cases through increased testing. They tell us the opposite — that each of these states needs to do even more testing to find infections — followed by more rigorous contact tracing and isolation. And if states are failing to control their case numbers, they should reopen more slowly and be sure that people are adhering to social-distancing recommendations.

Testing, in short, is not an obstacle to Trump’s goal of reopening the economy. It is one of the tools that will allow us to control disease spread, to keep people safe and confident as they begin reengaging outside of their houses. Ignoring the data is not a serious strategy: It is a threat to health and the economy.