Uh, Oh…

by Dean Wortmier

Because the panacea bonfire of the “New Abbott Labs Test” is being stoked, I have been forced to revive some brain cells that have been comfortably soaking up rum for the past 8 years.

How much help is testing for CoV2 going to be in deciding to ‘Open Up’ Virginia?

Let’s use Abbott’s Emergency Use Authorization (EUA) application[1] for our Gold Standard.

From the tests that Abbott submitted to get the EUA, the 95% confidence interval for the Probability of Detection, Pd, of the virus is (94.0, 100), greater than 94 but less than 100.  Moreover, the document tells us that the 95% confidence interval for the Probability of a False Alarm, Pfa, is (0, 11).

Pretty impressive, well, impressive enough to secure a EUA to soak up $Billions of taxpayer money, but is really going to help with decisions to end the quarantine measures for Virginia? (Did you see how I kept it relevant to Bacon’s Rebllion’s blog?) Remember, we’re making an executive decision that could cost granny her life.

We need to know that, if I select someone from the population and his test returns a positive result, then the person is really sick (immune).

We need what is known as the conditional probability, P(A|B), read Probability of event A given event B has occurred. In our case, P(sick person | +test), or P(s|+).  Mathematically, this written as

P(s|+) = P(+|s)P(s) / P(+)

where P(+|s) is probability of getting a + from a sick person (that’s just Pd), P(s) is the probability the guy REALLY is sick/immune from CoV2, and P(+) is the probability of getting a positive result at all, either through a good detection or a false alarm, or P(+|s)P(s) +P(+|not s)P(not s).

We’re all set to go except we don’t know P(s), and P(not s) = 1-P(s). But we can estimate. The U.S. has 600,000 reported cases, and estimates for total sickened is 10x that, 6M, out of a population of 360M so,

P(s) =approx 0.1666… Meh, let’s call it 2%

Let’s grab some values, Pd = 97%, Pfa = 5%, P(not s) = 98% and this

P(s|+) = 0.97×0.02/(0.97×0.02 + 0.05×0.98) = 0.0194/(0.0194 + 0.049) = 0.0194/0.0684 = 0.2836

P(not sick|+) = 1.0-0.2836

Do you see the problem? The test sucks for any decisions to ‘Open Up’ since nearly ¾ of the time a non-immune person will be declared to be immune.

Dean Wortmier is the pen name of a retiree living in Hampton Roads. A former employee of government-contracting businesses, he spent 23 years as an adjunct professor in mathematics, physics and computer science, and published more than a dozen degrees in information theory.

[1] https://www.molecular.abbott/sal/9N77-095_SARS-CoV-2_US_EUA_Amp_PI.pdf