# Post #1298: William and Mary, five new student COVID cases this week.

Posted on October 15, 2021

Source:  Calculated from William and Mary COVID-19 dashboard.

Is there anything else worth noting?  I don’t think so.

With numbers this low, I’d love to start talking about the potential for false positives.  False positives in the sense of tests results showing the presence of COVID-19 in individuals who never actually had COVID.  If that were common enough, then maybe the trickle of cases currently being observed is actually zero true new cases, and we’re just seeing these results of some small false positive rate.

But I think that’s wishful thinking.  It’s far more likely that , in reality, there is still some low rate of infections circulating among the student body.

Let me work through the arithmetic on that as best I can.

### False positive COVID-19 DNA (PCR) tests

It’s tough to talk about false positive DNA (PCR) tests for COVID-19, for several reasons.

First, all the pandemic-denier nuts come out in full force as soon as you raise the topic.  That tends to poison rational discussion when people start claiming that (e.g.) it’s all a hoax, it’s all false testing, and so on.

So I need to start this by averring that I am not a nut.  I’m just trying to run down the numbers.

Second, that aside, it’s difficult to get an estimate of the false-positive rate for DNA (PCR) COVID-19 tests, because, hey, how else can you be sure somebody had COVID-19?  In practice, a positive PRC test is taken as the gold standard for somebody actually having COVID-19.  How can you test the gold standard?

That said, the American College of Pathology (ACP) says that, in practice, COVID-19 DNA (PCR) tests have 98% to 99% “specificity”.  That is, there’s a roughly 1 to 2 percent false positive rate.  (Per this reference).  Other seemingly legitimate studies put the false-positive rate between 0.8% and 4.0% (per this reference, in The Lancet).

I’d call that the same ballpark.  How they know that, I haven’t a clue.  But two seemingly authoritative source more-or-less agree, that’s good enough for me.

Third, there’s a separate confounding issue of PCR tests flagging individuals who actually had COVID-19, have recovered, and yet retain fragments of (dead) viral DNA on and in their nasal membranes.  I have yet to see any quantitative estimate of that, but that is, as I understand it, part of the reason that they don’t want you to get re-tested once you’ve tested positive and have recovered from all symptoms.  Not sure if that’s contained within the false-positive rates cited above, or not.

Fourth, there may be a re-testing protocol for positives.  I don’t think that’s done for COVID-19, but if so — if you double-checked every positive with a second test, and required two positives in a row — that would eliminate false positives due to (e.g.) mishandling of specimens in the lab.  (It would also likely eliminate a lot of true positives, as the DNA test has a pretty substantial false negative rate.)

Finally, and relevant here, the false-positive rate depends on the pre-testing probability that disease is present.  It’s much more of an issue for screening testing — where you test everybody on a population, regardless of symptoms — than testing-for-cause (for those with symptoms or known exposure).

For that last point, I would love to have a single-sentence explanation that anybody could understand, but I don’t.  Will it suffice to say that if somebody has every symptom of COVID-19 and tests positive, there’s little reason to question the validity of the test.  By contrast, an out-of-the-blue positive result, in somebody with no symptoms and no know exposure, should be subject to a higher degree of skepticism.

That pretest-probability effect contributes to the decision not to do screening testing on populations thought to have low probability of infection (such as vaccinated students, in our case).  Aside from the cost of the test, if the actual infection rate is low enough, you reach the point where much of what you are doing is incorrectly forcing healthy people to quarantine, due to false-positive test results.

So, crudely put, false positives are mostly a problem for screening testing, in a population with a low true positive rate.

As I understand it, the only screening testing at William and Mary is weekly testing of the un-vaccinated.  Those now account for just 2% of the student population, or maybe 130 students of the roughly 6600 student residents in and around the Williamsburg campus.

For the vaccinated 98%, testing is only being done for cause.  William and Mary only tests for symptoms or known close contact with an infected individual.  That should be true even for high-risk populations such as student athletes, as the NCAA says that screening testing is not necessary for vaccinated athletes (per the “Resocialization” document on the NCAA website).  The dashboard also includes positives that reported by students who had themselves tested, without having the testing done by William and Mary.

Consistent with that, last week William and Mary performed 162 new tests, but only tested 20 new individuals.  I’m guessing that the bulk of those tests were weekly screenings for the un-vaccinated, and so did not include new (not-previously-tested) students.

If the false-positive rate really is about what the ACP said it was — maybe 1 to 2 percent — then you’d expect maybe a false positive or two per week to arise out of the screening testing.  It’s tough to say.  A lot would depend on the particulars of which test, which lab, which procedures, and so on.  Out of testing-for-cause, I’d guess the expected false positive rate would be a tiny fraction of one test per week.

In other words, no matter how I slice it, using a realistic estimate of the in-the-field false positive rate from a reputable source, I can’t come up with five false positives.

If we knew more we could probably rule it out definitively.  But given the information that is public, my conclusion is that we have to assume that COVID-19 is still, in fact, circulating at low levels within the William and Mary student population.