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Seven months ago I wrote a column about the paradoxical arithmetic of mass testing for a rare disease, showing that even very accurate tests are wrong more often than we think.

It was a fun intellectual exercise thrown together as a last-minute replacement for a column that fell through. It got some reaction from intrigued readers but that’s about it, and we all moved on.

Then SARS-CoV2 moved in. Suddenly the use of widespread medical testing is one of the central issues of life.

So I thought I’d revisit the column and take it slightly further because the original included a simplification. A warning: You’re going to have to keep the terms “sensitivity” and “specificity” straight, which I never can.

First, a quick reminder of the paradox.

Let’s say we test 1 million New   Hampshire residents for COVID-19 – roughly everybody over age 18. And let’s say 4% of people have the disease, which is the percentage of tests which have been turning up positive so far in New Hampshire. We use a test that is 95% accurate, which is a very good test.

In this scenario 40,000 people have COVID-19 (that’s 4% of 1 million). The test accurately finds 38,000 of them (95% of 40,000) and misses 2,000 (5% of 40,000).

On the other hand, 960,000 people (96% of 1 million) don’t have the disease. The test accurately tells 912,000 (95% of 960,000) that they are free of COVID-19 – but, and this is the point, it inaccurately tells 5% of those people that they do have the disease. In other words, it gives 48,000 false positives.

So a total of 86,000 people will get positive results in our statewide test (38,000 plus 48,000) but more than half of those results – 48,000 of them – are false.

It’s very counter-intuitive. Even though each individual test is 95% likely to be right, overall there’s a 56% chance that a positive result is wrong. Mind-blowing, I know, but you can’t argue with arithmetic.

Even if we have a test that is 99% accurate it would give false positive one out of every 12 times in this scenario. And if COVID-19 is less widespread than in 4% of the population, the problem gets worse.

Now we get to the new part: Specificity and sensitivity. These are confusingly similar terms that refer to how accurate a test is when giving a positive result, and how accurate when giving a negative result.

My original column assumed that all tests have the same specificity and sensitivity, but that isn’t true. It’s common for a test to be more accurate when saying “yea” than when saying “nay,” or vice-versa. It comes into play because society needs two types of tests to move on post-COVID.

One, currently performed by the tens of thousands daily, can spot the disease. It needs high sensitivity (meaning its negative results must be very accurate) but its specificity is less important.

Why? If a COVID-19 test gives me a false positive, telling me I have the disease when I don’t, then I will get worried but my life won’t be in danger. On the other hand, if a COVID-19 test gives a false negative, saying I’m free of virus when I actually have it, then I could walk around spreading it to other people, and then die. Which would be bad.

So when developing these tests – deciding how to collect mucus so you’re sure it contains virus when it’s present, how to transport and store it, how to expand samples with the process known as PCR without tainting the result, how to quickly and accurately interpret what you see – what really matters is that the negative results can be counted on. Being super-accurate with positive results is less important.

The opposite is the case for antibody tests.

As you know, these tests don’t try to determine if you have COVID-19 right now but whether you have had COVID-19 in the past. That involves looking in your blood to find antibodies, the virus-fighting mechanism created by your immune system during the earlier infection. The hope, although we don’t know this yet, is that once you have antibodies against COVID-19 you will be immune and can resume a fairly normal life.

Antibody tests must be highly specific. When they give a positive result, you must be very sure that the result is accurate.

Why? If it gives a false positive, I’ll think I’m protected by antibodies when I’m not. I’ll go out into the world unprotected and possibly get COVID-19, then maybe even die. Once again, that would be bad.

If an antibody test gives me a false negative, on the other hand, I will be inconvenienced but not endangered because I am protected even though I think I’m not. I’ll stay at home and wear a mask unnecessarily, but nothing more.

This difference between the need for specificity and sensitivity is one reason why developing tests is so hard. It’s not just that they have to be accurate, they have to be accurate in certain ways.

And then we decide how and when to use them, to decide whether the harm from inevitable false positives and false negatives outweighs the benefit of the true results.

COVID-19 is so contagious and so damaging that widespread testing for the disease is not only legitimate but we should have been doing it months ago as part of national preparation. The administration has been almost criminally negligent in failing to do that.

As for antibody tests, when good ones are released – some commercial products are out there but they’re very suspect; don’t count on them – the issue of false results will probably solve itself. Eventually, large swaths of the population will have antibodies so the arithmetic paradox that I’ve described above will go away.

If you don’t believe that, crunch the numbers from my example above but assume that 20% of the population has the disease. Then do it again with 40% or 60%. You’ll see that the ratio of false positives declines to almost zero.

So, yes, we should test for the disease and for antibodies, but we also need keep in mind what I said when I ended the first column last October: Public health is complicated. In some ways, I think it’s the most complicated thing that society does, since it has to factor in biology, economics, individual preferences, group behavior and politics.

And statistics. Whatever you do, don’t forget statistics.