This document presents some computations regarding the following question:
How much information does a negative test result convey?
A bit more precisely: Everyone can quickly look up the official numbers for their given region. I am based in Vienna, so I check the Austrian authorities’ numbers; today, there are 49113 active cases. It is said that the true number of active cases is 5-10 times larger than the measured one, so let us say 300 thousand in Austria. Austria has a population of about 9 million, so if I walk around the streets and think that everyone is COVID-free then I am wrong about \(\frac{300000}{9000000} = 3.33\%\) of cases.
Now, let us say that I meet with a friend. Naturally, I assume that the friend is COVID-free. I am interested in comparing two scenarios:
In the second case, there is a 3.33% chance that I am wrong. Put differently, every 30th friend without a test result is expected to have COVID today.
How much more do I know about my friends’ health thanks to the negative test result?
Source for this document can be found here.
I am a statistician doing some dirty work here. In other words:
Source: Wikipedia
I use input from different sources. The sources were not cherry-picked; however, I did not search very thoroughly for various studies. The first study is taken that looks trustworthy.
This study cited by the CDC. There vastly different values for people with and without COVID symptoms. If you are symptomatic, then
If you are asymptomatic, then
These numbers are estimates and they have their own error. I ignore that.
As discussed in the intro, I assume that the true number of active cases in Austria is about 300000 today. That is, in the following I will use \(P(\text{COVID})=0.0333\); that means that a random person is infected with 3.33% probability.
I will look into pooled salive-based PCR tests because that is the method used by “Alles gurgelt”—an initiative of the city of Vienna.
I take the numbers from this article and the cited study therein.
The figures for the pooled nasopharyngeal swab (not saliva-based) are similar.
The interesting question is the probability if being negative given a negative result. The figures above give us the reverted probability: e.g. getting a negative result given that you are negative (that is specificity).
One uses Bayes’ theorem to revert the probability: \[P(A\mid B)=\frac{P(B\cap A)}{P(B)}\]
In our case, we need a more detailed variant of that formula: \[P(\text{no COVID}\mid \text{negative antigen}) = \frac{P(\text{negative antigen}\mid\text{no COVID})P(\text{no COVID})}{P(\text{negative antigen}\mid\text{no COVID})P(\text{no COVID}) + P(\text{negative antigen}\mid\text{COVID})P(\text{COVID})}\] where \(P(\text{negative antigen}\mid\text{COVID})=1-P(\text{positive antigen}\mid\text{COVID})\).
That gives for asymptomatic people \[P(\text{no COVID}\mid \text{negative antigen}) \approx \frac{0.984 \cdot 0.9667}{0.984 \cdot 0.9667 + (1-0.412)\cdot (1-0.9667)}=97.98\%\]
So every 50th negative COVID antigen test gives false inference for asymptomatic people.
And that gives for symptomatic people \[P(\text{no COVID}\mid \text{negative antigen}) \approx \frac{0.989 \cdot 0.9667}{0.989 \cdot 0.9667 + (1-0.8)\cdot (1-0.9667)}=99.31\%\] Only 7 out of 1000 negative test results give false inference for symptomatic people.
Very similar computation leads to the following result: \[P(\text{no COVID}\mid \text{negative PCR}) \approx \frac{0.992 \cdot 0.9667}{0.992 \cdot 0.9667 + (1-0.832)\cdot (1-0.9667)}=99.42\%\]
Only 6 out of 1000 test results give false inference in case of a setup similar to Alles gurgelt. That is ca. every 166th negative PCR test results in wrong inference.
If it is antigen testing, then it is important to combine the test result with observing symptoms. A negative antigen test result for an asymptomatic person makes you only a bit more certain (less than twice as certain) about that person being healthy.
“Still, that 97.98% is not soooo bad.”
It is not but without the test you already know 96.67%. The error is ~2% with the antigen test which is not much better than the ~3.3% error of the base rate.
That would be the main message of the document: the interesting comparison is between the base rate (96.67%) and the rate given a negative test (97.98% for asymptomatic). The 41.2% from above is not the final answer you are interested in because what you know is whether your friend has a negative test result; you do not know whether your friend has COVID. It is the 97.98% that tells you the probability for your friend having COVID given that you know that she/he has a negative antigen test result.
Trust PCR tests more; they are more than three times as good as antigen tests and more than five times as good as the base rate guess. Do Alles gurgelt.
Again, Bayes’ theorem can be applied to answer the interesting question: how much more do I know about my friend if they have two negative tests compared to the base rate/one negative test result?
\[P(\text{no COVID}\mid \text{2 negative antigens}) \approx \frac{0.984^2 \cdot 0.9667}{0.984^2 \cdot 0.9667 + (1-0.412)^2\cdot (1-0.9667)}=98.78\%\] That is every 82nd asymptomatic person is expected to have COVID when they have two negative antigen test results. Here, I assume that test results can be faulty independently; I consider it to be a realistic assumption.
The same idea, Bayes’ rule: \[P(\text{no COVID}\mid \text{2 negative PCRs}) \approx \frac{0.992^2 \cdot 0.9667}{0.992^2 \cdot 0.9667 + (1-0.832)^2\cdot (1-0.9667)}=99.901\%\] Every 1000th person with two negative PCR test results is expected to have COVID.