Looking at stuff through the eyes of a mathematician – Gary Cornell's blog

A short post but an important one. I’ve said it before and I’ll say it again because it can’t be repeated enough: if you are told the increase in testing is why there is a rise in confirmed cases, check the test positivity ratio. If it is the same (or even worse, increasing), then the increased number of cases is because the disease has started to spread faster. Any other interpretation is just BS being tossed your way.

Let’s look at Florida for example: As I am writing this:

1. The test positivity ratio as of Tuesday June 16, 2020 is rising and is now 5.6
2. The seven day average number of cases is increasing

Undeniable conclusion? Things are getting worse in Florida quickly.

Every day when I watch the news they show graphs of which states have an increasing number of cases. Guess what? That information doesn’t tell you much if anything at all. They are to be blunt essentially useless graphs and innumeracy writ large. It makes me angry and to paraphrase a friend: “when I get angry, I do a quick blog” – so you’ll have to wait to see if you were better at understanding testing than the average Harvard medical school person for another day or so.

Yes, show the graphs of states where the number of cases is decreasing, that’s great news. Alas, showing graph of cases in states where cases are increasing, when testing is increasing constantly, not so much. Certainly a better graph to show is the daily increase in the number of hospital admissions alongside the increasing number of cases. When the number of admissions is decreasing you obviously know that the most serious cases of Covid-19 are decreasing. But we need to tell people not only whether there are more or fewer hospitalizations but whether the increase in cases is a result of increased testing.

Let me reiterate: what you (and the people making decisions about whether to reopen) need to know is if the increase in cases is only coming from an increased number of tests! And it may very well be that. Think of it this way: if you had a 100 cases a week ago and then this week you test twice as many people and you find 200 cases, do you really have a disease whose prevalence is increasing? It isn’t clear if you have really gotten an increase in cases at all, it’s more likely just an artifact of testing more.

As you might expect, to know what is really going on in less obvious situations than the one I just mentioned, requires computing a fraction expressed as a percentage. It is:

(Number of positive tests)/(Total number of tests)

This is the test positivity ratio that I talked about in an earlier blog. Recall that this is the clever idea of putting the number of tests done in the bottom of the fraction (the denominator) and the number of positive tests you found in the top (the numerator) and looking at the result. It is the easiest way to know if an increase in positive tests is just an artifact of more testing. For example, if you went from 100 cases last week to 200 cases this week but you tripled the number of tests, that’s good news. Your test positivity ratio went down. When you combine this information with just looking at the absolute number of infections found, then you have real information.

The moral of the story: when you show graphs of the test positivity ratio along with one of total hospital admissions – and both are increasing – things are getting worse. When the two graphs are straight lines, you aren’t moving the needle. When the graphs of both of these numbers are going down, that’s when you can start thinking about reopening.

While it is hard to find the test positivity ratio graphs on news programs, luckily it’s easy with google: https://coronavirus.jhu.edu/testing/individual-states

I promised you the answer to the calculation that the Harvard medical school people got so terribly wrong, but because of some news I saw in the NY Times today (https://www.nytimes.com/2020/06/01/health/masks-surgical-N95-coronavirus.html), I wanted to do a quick post updating my post:

garycornell.com/2020/04/21/multiplication-of-probabilities-or-what-to-do-when-you-have-to-go-shopping

In that post, I first guessed about what the various probabilities might be about how much safer various precautions made you. Then I used “multiplication of probabilities” to come up with a guestimate of what doing three different things together would help by.

That NY Times article I just mentioned had some actual numbers on how much your risk is lowered by various activities: “Standing more than three feet away cuts the risk of transmission to 3 percent from 13 percent, the analysis found. Use of masks reduces the odds of infection to 3 percent from 17 percent, and eye protection to 6 percent from 16 percent”
First off let’s be clear, they mean you and every other person wearing a mask! If other people don’t wear a mask, all bets are off.

Anyway, let’s assume these numbers are true and that it is reasonable to assume these are “independent events”, which I think they are. Then, if you take all three precautions and other people wear a mask, you reduce your risk of catching Covid-19 to:

.03*03*.06 = 0.000054 (!)

which, to give you a sense of perspective, is less than the chance of you dying if you drive 5000 miles. And since this number is presumably only the chance of catching Covid 19, I’m now willing to bet that if I use all three methods, and my neighbors wear a mask that I am safe. In any case, if this calculation is true, then this is a very low risk-thankfully!

Added: Tracked down the article the NY Times was referring to:

https://www.thelancet.com/pdfs/journals/lancet/PIIS0140-6736(20)31142-9.pdf

(See table 2.)

Added(2): The importance of other people wearing a mask repeatedly!

I promised in my last blog to show you how to do the calculations of whether or not you really have that disease when you tested positive. The calculations aren’t hard but they are always a bit tricky. So fair warning, if you really want to understand how to do the calculations, you should really read this blog with a pencil and paper beside you!

Anyway, to help you understand the ideas behind the calculations and also to show you how different the results can be,  I’m going to show you the calculation in two different scenarios.

DO YOU HAVE THE DISEASE?  SCENARIO 1 

For this scenario, I want you to imagine a routine physical where they do lots of screening tests. One of the screening tests is a highly accurate (99% specific and 99% sensitive) test but the disease it tests for is rare. Let’s suppose only 1/1000 (.1%) of people have this disease and so 99.9% of the people don’t have it (.999). Also, and this will turn out to be a key point, you have no other information such as some symptoms or datainformation about your genetic profile to add to the picture. 

Next, to make the arithmetic easier, and because this disease is rare, let’s suppose you test a hundred thousand people. Since the prevalence of the disease is 1/1000 (.001 of the population) and you’re looking at a hundred thousand people, you actually know you have only a hundred people  (.001*100,000) with the disease and 99,900 people don’t have the disease (.999*100,000).  

Now suppose you are one of those people who have a positive test. We need to calculate what is the probability that you have the disease!

Now I said the test is 99% sensitive and 99% specific, so because it’s 99% sensitive (very few false negatives), it’s going to find 99 out of those 100 people who have the disease. It’s only going to miss just one person who has the disease! We say you found 99 true positives.

What about false positives? Well, our test is really specific but there are an awfully lot of people who don’t have the disease in our 100,000 people sample. There are after all, 99,900 people who don’t have the disease, because only .001% of people have the disease. Since the test is 99% specific, this means you are going to find:

1% of 99,900 

false positives i.e. 999 false positives (.01*99,900). 

So now to the punchline: what is the probability that you have the disease? It’s really really low! This is so hard to believe and to understand that people actually call it the paradox of the false positive. But the math doesn’t lie, the probability really is that you don’t have the disease even though this test is incredibly accurate!

To calculate the probability, I need you to remember that, on the top of the fraction you use when calculating a probability, you put the number of times you got what you were looking for. For example, when you roll a die and ask for the probability of rolling a specific number, you use a “1” on the top of the fraction because there is only one way to roll a specific number.

In our case, you got what you were looking for (“a true positive”) 99 times. So that is the top of the fraction that we use to calculate the probability. On the bottom of the fraction you have to put all the positives you found, both true positives and false positives, because you are trying to figure out what percentage those 99 true positives were of all the “positive” results.

This means the probability of you having the disease with a positive test is:

99/(99+999) or about 9%

And so the probability of you not having the disease is about 91%!

The point is, because of the rarity of the disease, this really accurate, essentially gold standard, test still isn’t much help: you probably don’t have the disease. 

So the moral is: if you are just doing routine testing and there is no additional information, then even with a gold standard test, a positive result for a rare disease really isn’t going to tell you very much. So please don’t  immediately jump into having treatments with bad side effects- you really need to find out more.

Okay, so now we can define the positive predictive value and the negative predictive value of our screening test. It turns out, by definition, the positive predictive value is what we just calculated. It is the probability that if you tested positive, you truly have the disease. 

Positive Predictive Value = True positives / (True positives + False positives)

So again in our case it’s

99/(99+999) = 99/1098

or about 9%. 

Some people like to collect the information we used to do the calculation in a table. A table like this helps you avoid mistakes as the bottom row summarizes everything and the first two rows contain the results of all your arithmetic calculations using the sensitivity and specificity of the test.

	Have the disease	Don’t have the disease	Total
Test Positive	99	999	1,098
Test Negative	1	98,901	98,902
Totals	100	99,900	100,000

You can do the calculations by looking at the correct boxes and then, as you have seen, doing some elementary arithmetic. For example, the positive predictive value is always the value in the box in the first row, first column divided by the value in the first row, third column.

Next, what is the negative predictive value? This is how likely you don’t have the disease when you tested negative. It comes from the information we put in the second row. It’s defined as:

Negative Predictive Value = True negatives / (True negatives + False negatives)

And since we get that from the boxes in the second and third column of the second line of the table, you can see it’s very close to 100%. More precisely it’s:

98901/98902

which works out to 99.998988898%! Screening tests like this are really good at ruling out that you have a disease.

DO I HAVE THE DISEASE?  SCENARIO 2

In this case, you have some symptoms and doctors know that people with your symptoms actually have the disease in roughly 25% of the cases they look at. We are, in a sense, using the test to confirm your symptoms. Now we’re going to suppose our test isn’t anywhere near as accurate as the screening test, it’s only 95% sensitive and 95% specific. 

So let’s suppose we test ten thousand people with your symptoms. Since 25% of the people who have your symptoms have the disease, we know that 2500 people of our 10,000 person sample have the disease (.25*10,000) and 7500 (.75*10,000) people don’t.  Since our test is 95% sensitive, of the 2,500 people who have the disease, we found 2375 of them (.95*2500). Because it is 95% specific, it is only going to have 5% false positives, but this means we are going to have to have (.05*7500) = 375  false positives.  Here’s our table for this situation:

	Have the disease	Don’t have the disease	Total
Test Positive	2,375	375	2,750
Test Negative	125	7,125	7,250
Totals	2,475	7,500	10,000

So what is the probability that you have the disease when you tested positive? Just as before, on the top of the fraction, we put the number of true positives we found (that’s 2375) – again it’s the number we put in the first row, first column. The bottom of the fraction is the total number of positives we found – both true positives and false positives, and it’s the number in the first row, third column. So our fraction is:

2375/2750 = .863 or 86.3%

Thus, the positive predictive value in this case where you have some symptoms is 2375/(2375+375) or about 86%- considered a very high predictor.  

There is one more concept you will see used, it is called the “false discovery rate”. It is defined as what you get by subtracting the positive predictive value from 1. For scenario 2, our confirmatory test, the false discovery rate is about 1-.86 or 14% – which is again considered pretty good. For scenario 1, our screening test, the false discovery rate is (roughly) a horrific 91%!

*********

Finally, let me leave you with an exercise so you can do one of these calculations on your own. (I’ll give you the answer in my next blog I promise). Here is the background: in one of the most famous and (depressing) studies of how bad physicians can be at doing the calculations you just learned how to do,  20 house officers, 20 fourth-year medical students and 20 attending physicians selected at four Harvard Medical School teaching hospitals were asked the following question: 

The prevalence of the disease is 1/1000. Your test has a false positive rate of 5 per cent. It is 100% sensitive – no false negatives. What is the chance that a person found to have a positive result actually has the disease, assuming that you know nothing about the person’s symptoms or signs?

(Guess what? They failed miserably at getting the right answer. More next time.) 

Let’s suppose you have an exquisitely accurate test. Hey it even meets the gold standard: it’s 99% specific and 99% sensitive. Tests don’t get much better than this. Recall that being 99% specific means you are only going to get one false positive when you test 100 people who do not have the disease and being 99% sensitive means when you test 100 people who have the disease it reports that 99 of them have it with only one false negative. (Always keep in mind: a specific test has very few false positives, a sensitive test has very few false negatives.)

So you get tested and your doctor says: Well, I’m sorry to report you tested positive for this disease. The treatment has many side effects but since you tested positive with this very accurate test, I think we have no choice but to go ahead with the treatments.”

Do you go ahead? Does this oh so accurate test have such good predictive power that you should start a series of treatments that have many many side effects or do you get a second opinion and more tests before starting the treatments?

The answer to this question depends on calculating the odds of you having the disease correctly. Yes it means someone actually doing the mathematics and you and your doctor not jumping to conclusions until you see the result of the calculation. (And as you will see it’s really just arithmetic, no higher math needed!). But to give you the key idea:

The odds of you having the disease depend not only on how good the test is but also on how prevalent the disease is given your symptoms.

This is probably the single most important fact you need to know about testing for a disease!

Still, it’s natural when you get tested for a disease to ask your doctor: Hey I got a positive test, what are the odds I really have the disease?” This is called the positive predictive value of the test. Similarly, you could ask: “Hey I tested negative, what are the odds the test screwed up and I have the disease”. This is called the negative predictive value. But I can’t stress enough these are among the most difficult things to understand about testing because the correct answer always require somebody doing some arithmetic using additional information that is outside the test itself.

Why? Because yes it can’t be repeated enough: predicative value doesn’t only depend on the sensitivity and specificity of the test, it also depends on the prevalence of the disease given your symptoms. In particular if you have the positive result because of routine testing and you have no symptoms, you really really want to do (or have someone do) the calculation for you!

Let me reiterate it once more a different way: a positive test for a rare disease may still mean the odds are very much against you having the disease. This is why it is sometimes called the paradox of the false positive. And I also can’t stress enough, surveys show doctors get this wrong a lot.

Of course, as you might expect, there is a number that measures these odds: it is called the false discovery rate. You really really care about the false discovery rate! (And, by the way, the same interaction between the prevalence of the disease and the accuracy of the test that determines the false discovery rate also comes up when using antibody testing to determine who had the disease. It’s why the CDC issued their warning about antibody testing yesterday.)

But since this blog entry is already fairly long, I will explain the (elementary) math behind all these concepts as well as explain what is going on with the CDC’s warning/calculation, in my next blog!

Recall that a test’s sensitivity is the percentage of false negatives you get. If you test 100 people known to have the disease and your test says 99 of them have the disease that is a 99% sensitive test. Or, in other words, you take the number of really positive people your test found (99) and divide it by the number of people you tested (100). Similarly, specificity is the percentage of false positives you get. Here you should think about testing people who don’t have the disease. You test 100 of them and 5 test positive. Or, if you prefer, you can think of it as taking the number of really negative people you found (95 in our case) and divide it by all the people you tested. This is still 100 because you reported 5 people as having the disease when they didn’t. We say the test is 95% specific. Specific tests are the mirror of sensitive tests, so in this case a negative result on a specific test isn’t necessarily informative. (This time think of a broken test which reports everyone as not having the disease, no false positives. But, as before, not much information.)

O.K. in a previous blog I promised you the definition of what it means for a test to be accurate. I keep my promises. Still, before I do that, I want to stress, this can and usually is a dangerously wrong number to be interested in. The “accuracy” number for a test may make for good advertising by the manufacturer, but it is almost always the wrong number to look at. (But of course I am writing this blog to help you make sense of all that statistical excrement that is thrown out at you, which is why I will give you the definition shortly.). Let me reiterate:

Unless they can tell you a test is very close to 100% accurate i.e. the so called gold standard, they are using a single number to describe a situation that needs to be described by two numbers.  Having an easy to administer test that meets the gold standard is awfully rare. Gold standard tests are usually expensive  and not widely available.

What you want is that every test should report the test’s sensitivity and specificity – I can’t stress enough that you want and need those two numbers- not a blended number, which is what accuracy is defined as!  (This is a good example of how too often the news tries to reduce situations that need to be described by multiple pieces of information into one.)

Anyway, as you might expect, a test’s accuracy is a fraction expressed as a percentage.  On the top of the fraction you put the sum of the correctly identified positive patients and the correctly identified negative patients. So let’s suppose we tested 100 people, and we know that 70 are truly positive and 30 are truly negative. We found 65 of the 70 truly positive people to be positive and found 28 of the truly negative people to be negative. The top of our fraction is then:

65+28

The bottom of the fraction is the number of people we tested – so 100 in our case.

The accuracy of this test is then (65+28)/100  = 93/100 = 93%

Let me end by stressing again: sensitive tests are good for telling you you don’t have the disease. A negative result on a sensitive test is a good thing to get.   Specific tests are the opposite: they are good for telling you do have the disease. A positive result on a very specific test is depressing to get but at least you are likely to be treated for a good reason.  These two numbers are what you are really interested in, not a blended number! Always ask for them when you are given a diagnostic test. Don’t be distracted by having them tell you about the “accuracy” of the test.

For those who are interested, here is a list of common tests with their sensitivity and specificity:

• AIDS testing meets the gold standard: is both highly specific (99.5%) and highly sensitive (99.5%).
• Mamograms are highly sensitive (97%) but not highly specific (64.5%)
• A PSA test of 4.0 as the cutoff had a sensitivity of 21%(!) and a specificity of 91%. It is a pretty bad test-which is why it isn’t routinely used anymore..

So what about COVID 19 nasopharyngeal swab tests? Still the most common test, it involves sticking a really long swab up a patient’s nose. Its specificty and sensitivity is actually really hard information to dig out. If done by really experienced people in the lab under ideal conditions, they aren’t bad, but in the real world, not so much. Thankfully we are moving to saliva tests. I have not been able to find a definitive answer to how sensitive and specific saliva tests are, although all the papers I have looked at say that they should be at least as good as the nasopharyngeal swab tests done under perfect lab conditions-without needing those lab conditions!. One paper out of Australia I found seems to indicate that the saliva test they developed was  98% specific and 84.6% sensitive.

There was some good (if very preliminary) news from Moderna on their vaccine candidate:

https://investors.modernatx.com/news-releases/news-release-details/moderna-announces-positive-interim-phase-1-data-its-mrna-vaccine

And yes it does seem to be good news. But leaving aside that they so far only report results for 8 people from their 45 person Phase 1 trial, some 4th grade arithmetic shows that even if (as we all hope) they give us a usable vaccine, it’s not going to be a total game changer – and certainly not in 2021.

Why?  First off, from reading the press release it is pretty clear that a minimum of two doses will be needed to get the kind of immune response we need.  Optimistically, they are talking of ramping up to 1 billion doses a year:

https://www.biopharmadive.com/news/coronavirus-moderna-lonza-mrna-vaccine-manufacturing/577169/

That will cover 1/2 a billion people at most. Again using some fourth grade arithmetic, that’s enough to treat  about 7% of the population of the world, roughly  enough to treat the people living in the United States, Canada and Mexico.  Or, more hopefully, health care workers, first responders and essential workers world wide. If we end up with vaccine nationalism, does that gives us Smoots-Hawley effects on the world’s economy (https://en.wikipedia.org/wiki/Smoot%E2%80%93Hawley_Tariff_Act)? 

And remember, to be able to go back to life something like it once was,  we need to vaccinate enough people to get herd immunity. For that we would need about 10 billion doses worldwide (500 million doses more or less for the United States alone). So please keep in  mind this quote from the CEO of Moderna:

“The odds that every program works are really low, obviously, but I really hope we have three, four, five vaccines, because no manufacturer can make enough doses for the planet,”

So, to reiterate, yes potentially very good news but please people temper your enthusiasm. Keep in mind not only will there be the difficulty of producing enough doses, you can’t ever forget that most vaccines candidates don’t get through phase 3 trials even if they have a successful phase 1. Even if we assume the Moderna results are similar to what a Phase 2 might give (they aren’t, although they have some elements of a Phase 2), then for a random vaccine in development, the historical odds are roughly ⅕ that they can go from Phase 2 to Phase 3 and even lower that they can go on from Phase 3 to a successful vaccine.

“The foremost challenge in vaccine development is reducing the average transition rate from clinical phase II to III. Between these value chain phases the risk profile incorporating all data has an estimated transition probability of 0,21. It represents the highest attrition rate when compared to the productivity of the other phases.” (Taken from:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3603987/

In a previous blog I said that we say that a test is very sensitive if it has very few false negatives. I then said that if all you know about the test is that it is a very sensitive test, only it’s negative results are truly informative! Getting a positive result i.e. you are told you have the disease by this oh so very sensitive test, may not give you all that much information.

To repeat: just because you tested positive on a very sensitive test, you may or may not have the disease. In fact, if you continue to read this blog, you will learn why, even with an exquisitely sensitive test, when you test positive for a disease, the odds may still be very much against you having the disease! (I will cover this topic, which has the very fancy name of Bayesian Updating very soon, please stay tuned.)

But back to the idea that a positive result on a very sensitive test isn’t necessarily very informative. Some readers told me they found this a little strange and had trouble wrapping their head around it. At first glance it seems that both positive and negative results should be informative. But what I said is certainly true and understanding why it is true is actually a good use of one of the first tools a mathematician uses to think about a problem – looking at the boundary conditions!

To a mathematician, looking at the boundary conditions first is automatic. Boundary conditions are the most extreme possibilities, all in or all out so to speak. Once you look at the boundary conditions for a test, it becomes clear why a positive result on a very sensitive test may mean nothing!

O.K. here’s how a mathematician thinks about this: We say to ourselves “Hmmm by definition a very sensitive test is one that has very few false negatives. O.K. suppose the test never gives a false negative because it is broken and reports every one as positive.” (That’s a pretty good boundary condition to a mathematician!) In this case being told you are positive from this test means absolutely nothing. This shows a mathematician that without more information about the test, only negative results from very sensitive tests are truly informative.

What we obviously need is a good definition of a test being accurate – and that will be the subject of a blog very soon.

I don’t know if what is mentioned in this article is true:

https://www.thedailybeast.com/team-trump-pushes-cdc-to-dial-down-covid-death-counts

but if it is, I have completely lost respect for Dr. Deborah Brix (not that I had much before considering her appearances on Fox news defending the indefensible – like Trump’s “musings” about bleach and UV light).

The question of how many deaths we have from Covid 19 is something that can be addressed by a mathematician’s “first cousin”, the actuary using their concept of “excess deaths”. (Excess deaths are typically defined as “the difference between observed numbers of deaths and the expected number of deaths” – simple subtraction.) Actuaries have spent a long time understanding “expected” death rates because life insurance companies would go out of business without this information and so excess deaths are something they think long and hard about. There is absolutely no question the United States has had an unbelievable number of “excess deaths” since the virus hit us. All I had to do was read this: https://www.cdc.gov/nchs/nvss/vsrr/covid19/excess_deaths.htm.

While it isn’t for a mathematician to say that these excess deaths are only caused by Covid 19, common sense says, what other explanation is there? Car accidents are down for example…. I conclude the (mathematical) evidence strongly suggests the number of reported deaths is lower than the actual number of deaths caused by this horrid virus.

Sheesh

We know testing is all important to get through this. People need to know if they have active virus so they can adjust their behavior accordingly. Anyone who tests positive should self quarantine for 14 days and we also need to do contact tracing to find the people that they have been near to. (I feel that forced quarantine in one of those many nice empty hotel rooms at government expense is the way to go for people who won’t (or can’t) self quarantine. My understanding of quarantine law is that this is a permitted use of governmental power. I know Singapore is doing this. Heck it would even help the hotel industry-which is slowly dying.)

Next, if having (enough) antibodies to Covid 19 means you can’t infect people any more, that would be a game changer. This means accurate testing for antibodies could become really important. As I write this we don’t actually know if having enough antibodies means you no longer can spread the disease. So at this point, antibody testing  just tells us what percentage of the population had Covid 19 but were never diagnosed. This is certainly useful information but nowhere as important as knowing if you can’t spread the disease would be. And as I write this, there are way too many crappy antibody tests out there because the FDA sure screwed up at first. Thankfully, they are now trying to fix that (https://www.fda.gov/news-events/fda-voices/insight-fdas-revised-policy-antibody-tests-prioritizing-access-and-accuracy).

But testing can’t be perfect. As the old engineering joke has it: you always tell the customer that they can have it fast, cheap or good. They need to pick the two they really want. In medical testing you also get to choose from three options: how fast you want the result is one but more importantly you almost always have to have to choose to optimize your testing for false positives or false negatives. A false positive means, naturally enough, the test says you are infected, but you are really not. A false negative is the opposite: it says you are not infected but you are. Note: while almost no medical test can be optimized for both-at least not at the start of the disease, it is not impossible (eventually). For example, current HIV tests have almost no false positives or false negatives.

So what are you really interested in? Well the percentages of false positives and negatives of course. For a diagnostic test, in a situation like Covid 19, you want a very small percentage of false negatives. You don’t want to tell someone they don’t have the disease when they really do. In testing speak, you want a very sensitive test. Negative results from highly sensitive tests are very useful. (Interestingly enough, a positive result from a very sensitive test is often less useful. More on that in my next blog entry.)

So, suppose for example, you have a test which, when you give it to a 100 people who are infected, it only reports 85% of them as having the disease. We say that the test is 85% sensitive and has a 15% false negative rate. I think we can agree that is not a great diagnostic test. By way of contrast. a single modern AIDs test is 99.5% sensitive and one can do multiple independent tests if you want to increase accuracy. The best home pregnancy test is at least 95% sensitive and probably a bit higher than that when used correctly. Highly sensitive tests are quite common.

Let’s go back to our not so great 85% sensitive (15% false negative) test. Guess what? The Abbott ID NOW test, which has created lots of excitement because it can get results in less than 15 minutes (and is used by the White House to do daily testing of white house employees now ) is about 85% sensitive-and that is actually pretty good for a Covid 19 test. (Well 85.2% https://www.npr.org/sections/health-shots/2020/04/21/838794281/study-raises-questions-about-false-negatives-from-quick-covid-19-test). More generally, the most used tests for Covid 19 are basically crap, they all have lousy sensitivity. They are plagued by false negatives.

(Interestingly enough, many of the tests are quite sensitive when done under laboratory conditions, it’s how they are done in the field that makes them insensitive. The main problem may simply be that any test that relies on sticking a swab way up a person’s nose may not be done right in the field. An MD friend of mine says that having a red line near the end of the swab would be a cheap low tech solution to improve accuracy! Turns out nasal swabs are not easy to use correctly.)

A saliva test works much better in field conditions, people rarely screw up when asked to spit into a tube. And they are beginning to come out, in particular the Rutgers test (https://www.rutgers.edu/news/new-rutgers-saliva-test-coronavirus-gets-fda-approval) looks to be a breakthrough because it is both sensitive and easy to administer. Alas (see the rules for optimization), it seems to take a minimum of 24 hours to get a result.