# Positives

No, this isn’t a cheery post. I want to discuss three aspects of the news that three Trump-Pence staff members have tested positive for COVID-19. (That’s Trump’s military valet, Pence’s press secretary, and now a “personal assistant” to Ivanka Trump.)

The first point is that these results strongly suggest the folly of having staff go around without masks for fear someone might take their picture.  It’s really dumb.

The second point is that these results show the complete hypocrisy of the Trump administration’s failure to go all-out for a national max testing policy: the White House will respond to this news by doing more testing — something it says the rest of the country doesn’t need.

As the links above show, both these points are getting some airing.

But there’s a third point I haven’t seen in the news yet: no test is totally reliable. I read that some people in the White House are being tested every day.  Let’s assume that the White House has the best test.  What’s the rate of false positives for asymptomatic people? I can’t figure it out. I read that, at least under lab conditions,  Abbot’s new test for people who have had symptoms for a couple of weeks is very very accurate: 99.9% specificity, or about one false positive out of a thousand healthy patients, and 100% sensitivity, or a complete lack of false negative results in patients confirmed to have had COVID-19.  But that’s for people with full-blown disease, and also it’s not clear if anyone is using it yet.

What the false positive rate might be for asymptomatic people will vary with the test, and the quality of the implementation.  If it’s 99.9% then ignore all of what follows. But suppose the accuracy rate is ‘only’ 99%.  In other words, suppose that 1 out 100 flagged as positive are in fact not carrying the virus.  What are the odds of a false result if someone is tested every day?

The way you work that out, if I remember Freshman math, is to take the odds of the thing not happening (.99), and multiply it by itself for the number of events.  So (.99) to the 30th power gives you the odds that all that month’s tests will be accurate, which google tells me is about .74.  So there’s about a 1 in 4  chance of an erroneous result if we use a test that is 99% accurate on one person for 30 days.  That’s pretty high. Increase the number of people being tested daily, and the odds of a false positive on someone go up quickly.

So maybe they don’t all have it.  But it’s still very likely that at least some of them do, and given the no-mask rule, there’s a quite decent chance they will have exposed someone else.

That said, in the grand scheme of things, a 1% false positive rate is not much to worry about — the victim quarantines unnecessarily, but no one else is harmed. It’s the false negatives that are the worrying problem, because they allow the unknowing to go out and spread the disease.  And we also don’t know what the false negative rate for asymptomatic persons is for whatever test the White House is using.  Want to bet it’s not below 1%?

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### 3 Responses to Positives

1. le says:

One thing to consider is that there might be a mechanism for the false positive rate. If that is the case, the false positive rate of 99.9% might mean that 1 in a thousand people have a characteristic that breaks the test. In that case, the false positive rate among a particular group might be different than that for the larger population.

2. Vic says:

That’s NOT how odds work. If there is a 99 percent chance of something being true, then EACH time the even happens, the odds are 99 percent. It’s not an additive (so to speak) thing.

If you flip a coin, there is a 50 percent chance you’ll get heads. You get tails. If you flip it again, there is STILL a 50 percent chance you get heads, not a 75 percent chance. You get tails again. If you flip at yet again, there is NOT a 87.5 percent chance now of getting heads.

• You’ve confused two things. What you describe is the fact that the probability of each even is independent of those before it. So each time you flip the coin, take the test, whatever, the odds are the same as every other time. That is assumed.

But what I’m talking about is totally different: the cumulative odds. If we were talking about fair coins, the question would be, if we flip it, say, ten times, what are the odds that I will get zero heads in that process. And the answer is NOT 50%. Rather it is .5 ^ 10 which equals a giant 0.0009765625 aka circa 0.098%, or rounding a bit more, just under 1/100th of a percent.

When we read a report that N staffers have (or don’t have) the disease, we need to think not just of the odds that the tests were wrong today, but the odds that the tests will be wrong eventually. Or, if you prefer, we can totally forget the past, and just think if we are testing 100 people today, what are the odds of a false result appearing? Hint: it’s not the same as the odds of an individual test being wrong — it’s much greater.

All this is very basic statistics, and one of the many reasons I tell students that statistics is an essential course in law school if you didn’t do it in college.