Let’s talk about why it’s critical NOT to assume you are immune to covid-19 when you have a positive antibody test.
I will argue that this problem can perhaps be overcome by requiring three positive tests in order to prove immunity.
Let’s make Levi’s assumption that 1 percent of the population actually has antibodies. Having antibodies is a good thing, because it means you already have had the virus and we can hope that this makes you immune.
For rounder numbers, assume that the test gives a false positive reading in 5 percent of cases where people don’t really have antibodies, and also a false negative reading in 5 percent of the cases when people do have antibodies.
Out of 10,000 people, 1 percent will have antibodies. That is 100 people, with the other 9900 not having antibodies.
Of the 100 people who have antibodies, 5 will falsely be reported as not having them. 95 will correctly be reported as having them.
Of the 9900 who do not have antibodies, 495 will falsely be reported as having them.
Altogether, 495 + 95 = 580 people will be reported as having antibodies, but most of these people will not have them! So you would not want to tell all 580 people that they don’t have to worry about getting infected if they go out and play. Most of them in fact can get infected.
My recommended solution to this problem would be to require a second test for those who test positive the first time. The second test also has to be positive in order to say that the person has antibodies. Assuming that the results are independent, the chances of two tests incorrectly reporting positive result is .0025, or 25 out of 10000.
That still might be too high. But we can take the people who test positive on two tests and make them take a third test. Only if this last test is also positive would you give the person freedom to roam.
Note that if the probability of a false positive depends more on the person being tested than on pure chance, this proposed solution will not work.
When a half the population has anti bodies. Better numbers.
” false positive depends more on the person ”
They are antibodies, you are looking for a concentration in a Known fluid, so your method should work.
Hello — in order for this to work, we also need to assume that test errors are uncorrelated, i.e. just random inaccuracies in the execution of the test.
If, on the other side, test errors reflect underlying conditions that make the test less accurate (say certain characteristics of the tested subject, or local environmental conditions), then repeating the test will not reduce false positives/negatives fast enough.
However, in the same spirit of the original post, supplementing medical tests with other independent risk assessments, such a historical proximity to other “at risk” individuals (as provided, for instance, by the Blue Trace program developed by the Singapore government), might enable to further reduce false positive and negative rates. Of course, this only works if tech-driven tracing is widespread.
Reading elsewhere on the web, my impression is that it’s more about the person, and in particular what other similar-to-but-not-the-same-as-covid-19 antibodies they have.
(I believe the issue is “selectivity” – the test will report positive for an antibodies that are super similar but not functionally the same.)
That of course applies to the (first?) test under discussion, test with better selectivity will hopefully soon appear.
Consider the two groups are equal in size, the immune and the infected, and the one changes into the other at a fixed period, equal for both. The tester collects a third group, those with inconsistent tests. And those who have switched and thus show tncoinsistent test.
We impose a rule. When that third group gets too big, to a certain smalll amount the tester retests. No matter how often he est, the members who legitimately switched will be balanced by members in each group who tested consistent by error. That is the member whoe switched in the error group must have men zero. I am using a half and half ratio on switch times, they are equal.
The This system is adaptive, variance is conserved. The variances in he two groups are huge, this does not converge. So the tester begins to look at the error group and sees 0,1; 1,0; 1,0′ 0,1 and only retests when the two parts are unevenly matched, and will retest the group needing to fill in. No it is evening the queue of tests. The tester reduces the group that is out of variance by giving it more tests, more samples. The effect is to rive variances of the two large groups down.
All of the sample are of 11,00,01,10; the later two only appearing in the error group. The tester keeping the two other groups balanced in count. The testers best strategy is to test the under sampled until it is over sampled then test the other until it is oversampled.
The tester is quantizing, separating the groups into partitions. They will be groups if about the same size that get sampled at once. And the shape of the distribution in each group is altered, as necessary. His tester queue queue is:
00,00,01,11,10,01,01,01,01. and so on. He is testing until the exception is found. The tester is segmenting the group members so each group looks like a gaussian arrival h that creates these two alternating sequences of varying length. And the tester has a market point between alternations.
This is whack a mole, it is how the docs drill don o the proper neighborhood so that hospital use can be a bound queue.. elf sampling systems.
In real life, let us assume the immune period forty times the infected period Further the docs accept either error as significant, and boumd. Now, when the docs segment the groups and neighborhoods with the appropriate arrival times, the infected group will be way over sampled. That is, once the docs find an infected group, they attack it, essentially. They are looking for a testing inconsistency, where an infected appear immune, that is a victory, a patient out the door with some error.. Then they attack the immunes, to fill in the empty hospital bed, all the while keeping error ratio below minimum. Value added chains do this to bound the inventory space.
What about market serving value chains?
The same. This more or less is the automated savings and loan, a standard nonprofit machine does this. One difference is the market maker has the entire queue in the computer and can just go ahead and organize the two group by structured queue. This is need not work with a fixed ratio. The other difference is flow is positive from loans to deposits.
I am going to write a book on Arnold’s blog.
Go back to the immune/effected balance. We have converted this to a self sampled system, and we see us creating two Poisson process, adapted along the way. The Poisson queues ar symmetrical, by construction, so the mean of their difference is zero and variance the sum of the two means. Half the samples show a consistency equal to the other half, and they balance over the complete groups. What is the variance of the 01 and 10 pattern? The variance of that pattern is the variance of the word size, the number of samples needed to flip the pattern. Both patterns produce message at the neighborhood rate. The both fill equal spaces, the alternating 01 10 pattern will occupy one bit of a fixed channel. The immune count neighborhoods in four bits, so do the invfcted, the separator gets one bit. It become a channel packing problem/ Serf sampling minimizes samples to meet the error criteria. So they always drive the separator bandwidth to a minimum bound variance, mean zero.
The last thing.
The tester only test as much as is needed. But each new bunch of consistencies looks like a Gaussian arrive, the larger bunches appear less often. But the sequence of consistencies can be identified by the nuber of samples needed, it is Log2((Number samples)). If a message takes 18 samples, then it needs about four bits of bandwidth to be identified. Under these circumstance, our error rate is the accumulation of losses from making a histogram instead of a smooth distribution, the round off error and it is selected by the neighborhood size, which it the typical message size a hospital can handle.. Remember the hospital has an entry and exit at each inconsistency, one in and one out. They are stuck with mantaining that balance flow and finding the neighborhood size, which is equivalent to the message rate.
Another excellent post! In addition, doesn’t the logic of inverse probabilities apply to the pandemic writ large? That is, if the base rate of infection is low, then most “positive” test results might well be false positives! See: https://priorprobability.com/2020/04/10/bayes-papineau-and-the-pandemic/
Working against this is the fact that testing is not random. Most tests are conducted on people who either are showing symptoms or believe they have been in contact with people who have the disease.