So, Norway has decided not to engage in screening of asymptomatic people for COVID19.

They posted this figure to justify it.

Let's work through the math!

https://translate.google.com/translate?sl=auto&tl=en&u=https%3A%2F%2Fwww.fhi.no%2Fnyheter%2F2020%2Funodvendig-a-teste-store-grupper-av-friske-ved-lite-koronasmitte%2F

They tell us that right now they estimate the prevalence of COVID19 in Norway to be 0.01%.

This means out of every 10,000 people in Norway, 1 has COVID19.

This corresponds to the column on the right.

They give us three additional pieces of information:

#1 If you test a town with 12,250 people, 1 is a true positive (has COVID19 and is also PCR positive)

#2 For every true positive you will get 12.5 false positives(!)

#3 Probability of a positive response being true is 7.4%

Okay, let's start filling out a contingency table!

First, there is 1 person who is PCR+ and COVID+ (fact #1)

Second, we tested 12,250 total people to find that 1 true positive (also part of fact #1)

Since we know 0.01% of the population has COVID19, we can also calculate the total for the top row (all people who have COVID19). That is just 12,250*0.01%

Now the bottom right corner the SUM of the two cells above it in the right column. So we can subtract to fill in the middle.

Now, they told us for every true positive, there are 12.5 false positives (fact #2).

Since there is only 1 true positive, there are 12.5 false positives. Let's put that in the PCR+ COVID- box.

Now we can total up the left column.

Last step to fill in the table is to subtract across the rows to fill in the gap!

Okay, here is where it gets interesting!

Fact #3 was about what fraction of all positives are true positives. This number is 1/13.5=7.4%!

That is exactly what they told us it would be

Now, we can calculate sensitivity. That tells us what fraction of all COVID19 positive people we find.

We find 1 and there are 1.225 people in this town. So we find 1/1.225=81.6% of them

Sensitivity=81.6%

Lastly, let's calculate the specificity. That number tells us what fraction of all COVID19 negative people are normal on PCR testing.

There are 12,248.8 COVID19 negative people and 12,236.3 of them have a negative PCR.

12,236.3/12,248.8=99.9%

Specificity is 99.9%

So this is how a test which has specificity of 99.9% can still end up such that only 7.4% of all of those people who end up testing positive actually have the disease!

The lesson is that even really good medical tests don't perform very well in populations where the disease is quite rare.

This could have implications for testing strategies for COVID19 around the world. It doesn't mean that test & trace is hopeless.

But it does mean that tracers will track down a lot contacts of false positives along the way.

It's hard work to find needles in haystacks!

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BONUS question:

On your own, work through the same exercise for any of the other columns. See if you get the same numbers!