Very accurate, very hidden, covid test

So my friend’s news got me thinking once more about testing. In particular, what does it really mean, at least statistically, to test positive for coronavirus?

Much of the early testing for the virus used the RT-PCR method. It was uncomfortable to endure, because it required the tester to insert a swab deep into your nostrils, and another into your throat. Still, it was always considered to be very accurate, especially when performed properly by a trained tester.

Here’s a question worth asking: what do we mean by a “very accurate” test?

Let me try explaining that here with some hypothetical numbers. We can divide the population of the country into two camps: those who are actually infected with the virus, and those who are not. Of course, some who have the virus, like my friend before he took the test, don’t know they are infected. They have suspicious symptoms, that’s all. They want to be sure one way or another, and that’s why they get tested.

Now, let’s say that an RT-PCR test administered in India produces a positive result for 99% of those who already have the virus. Let’s say the converse is even better: the test produces a negative result for 99.5% of those who are not infected. (Remember, just hypothetical numbers.)

No doubt you’d agree that this is a very accurate test. In fact, with the numbers above, you’d most likely call it better than 99% accurate and that’s a good way to describe it.

So, given the accuracy of this test, what is the chance that my friend, who tested positive, actually does have the virus? At first glance, we might think the answer is 99% or more, simple. Isn’t that what the numbers tell us? And if so, and even if covid is on the wane, that is still a pretty unsettling number. After all, it was a pretty deadly pandemic.

But suppose we take a closer, maybe deeper, look at my friend’s case? Consider these numbers:

* India has about 1.4 billion people. Let’s say all of us Indians are getting ourselves tested. This is of course not true, but we are still being hypothetical.

* The Ministry of Health and Family Welfare says today (April 5) there are 23,091 active cases of covid in the country. Those are the diagnosed and recorded cases, of course. It’s possible there are many more who are infected but don’t know it. So, let’s assume—we can be hypothetical, remember?—that for every known infection, there are almost nine unknown ones. That is, let’s assume that there are about 250,000 people in India right now who are infected with coronavirus. This means there are 1,399,750,000 (1.4 billion-250,000) Indians who are not infected.

* If the test produces a negative result for 99.5% of uninfected people, that means it produces a (false) positive result in 0.5% (that is, 100% – 99.5%) of them.

So: Among the 250,000 infected Indians, the RT-PCR test will produce a positive result 99% of the time, meaning in 247,500 people. Among the uninfected 1,399,750,000, 0.5%, or 6,998,750 people, will test falsely positive.

So, if everyone is getting tested, the total number of Indians who will get a positive result is:

247,500 + 6,998,750 = 7,246,250

It always takes me a moment to fully grasp the implication here. Of this total of 7,246,250 positives, only 247,500 actually do have covid. That is, if you do test positive, and if everyone around is getting tested, the chance that you actually have the virus is: 247,500 / 7,246,250 = 3.42%

Think of that. You have tested positive on a test that is “better than 99% accurate”, remember. Naturally, that result worries you. But on the other hand, the probability that you are really infected is 3.42%, which is tiny.

Should you be worried at all? Well, that depends partly on what you make of some of the hypotheticals here: that everyone is getting tested, that there are many undetected infections, and more. My feeling is, be worried, but hold on to some perspective as well.

But that apart, we still need to reconcile that 99% test accuracy with this 3.42% figure. It looks like a logical progression from one to the other, so why this seeming and yawning contradiction?

To answer that, let me suggest that you’ve actually taken two tests here, though we call only one of them – the RT-PCR – a test.

\The other one is hiding in plain view, implicit in the figure above of 250,000 infections. Again, what that means is that 1,399,750,000 Indians are not infected. So, here’s the hidden test: pick a random Indian. What’s the chance that she is not infected? Easy: 1,399,750,000/1,400,000,000, or about 99.98%.

But what kind of a test is that? Whatever it is, wouldn’t you say it is pretty seriously accurate? After all, if you pulled out a random Indian and told her “You are not infected with the coronavirus”, you would be right 99.98% of the time. Whereas in coming to that same conclusion, our “accurate” RT-PCR test is actually accurate only 99.5% of the time. So in fact, the hidden test is much superior to the RT-PCR.

You have taken two tests. Because it is superior, meaning far more accurate, the result of one of those matters more than the result of the other. It’s the gap between the accuracies of these two tests that explains the tiny chance that you do have the virus, just 3.42%.

Put it another way: In effect, this means that if you have symptoms that start you thinking “corona!”, better than taking a RT-PCR test might be to look around, realize that the overwhelming majority of people is not infected, and conclude that it’s very likely you too are not.

I’m not suggesting complacency, just perspective. Besides, the hidden test costs substantially less than a RT-PCR.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.