Why 15% is more likely 5%

We have been hearing news reports for a week or more now that say that it is likely that COVID-19 actual infections are multiples of reported cases. THis is likely true. But probably not by as much as some reports.

That is because of the expected level of false positives from the antibody tests that are being used. Let’s break this down…

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Let’s focus on the New York state figures reported above. They represent that the antibody testing of 7500 people in NY State showed antibodies for 14.9% of the sample, while reported cases run about 1.5% of the population of NY State. But, as is often the case, the most important information is in the footnote. It said that these figures are not corrected for test accuracy.

Now, accuracy for the antibody tests is reported to be about 90%. So many people would read that to mean that the result is good +/- 10%.

But that is not the case at all. In fact, a news report that I heard on the local news program yesterday said that it is quite possible that more than 2/3 of the people who got an indication of antibodies present do not in fact have any antibodies! How can that be you ask?

This result falls out of a little Baysian thinking. Like this…

1. Let’s assume that we have a population of 1 million people where 5% of those people or 50,000 have been infected and 950,000 have never been infected.
2. And that we have tested them all with an antibody test that is 90% accurate.
3. So when we test the 50,000 who were infected, the test will tell us that 90% or 45,000 have antibodies (which is correct) and that 5,000 do not (which is incorrect – the 10% error rate). So far so good.
4. But when we come to testing the 950,000 people who have not been infected, the test will find that 85.5% or 855,000 people are antibody free (correct for 90% of tests) and that 9.5% or 95,000 people have antibodies (the 10% error).
5. So in total, the testing told us that 45,000 + 95,000 = 140,000 people (14%) have antibodies. And that 5000 + 855,000 = 860,000 people do not. So the error rate for Positives is 95,000 / 140,000 = 67% of the positives are WRONG. Error rate for negatives is 5000/860,000 = 0.6% wrong. Not bad on the negatives.
6. So in this example, a test that told us that the rate of positives is 14%, when it is actually 5%.

So that 14.9% reported for New York State is likely to be closer to 5% if the test was 90% accurate. This is the adjustment for Test Accuracy that the footnote says was not made.

If you make a correction based on this example (which seems to almost fit the data), you get a corrected result of 5% (33% of the 14.9% reported). The 5% is still more than 3 times the reported 1.5% infection rate.

The Smell Test

This result is more consistent with reported statistics from China, where they report that about 50% of the cases are asymptotic or common cold like symptoms. People like that are unlikely to have been tested in New York when tests were in short supply. Of the other 50% who showed symptoms, China reported that about a third (15%) required treatment and a tenth (5% Included in the 15%) required treatment in an ICU. So if NY State was capturing all of the 15% who needed hospital care and three quarters of the other people with symptoms with testing, then 30% of the cases would be reported, as the 5%/1.5% ratio would indicate.

The 15% rate indicated by the test results without correction would suggest that either the disease is much milder in the US than in China or that NY State was not capturing more than 2/3 of 15% of the infections needing hospitalizations and none of the people with clear symptoms who did not go to the hospital. That just doesn’t seem likely to me.

So, to me the idea that 15% is more likely 5% passes my smell test.

What do you think?

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