Does my friend have covid? (mixed test results, close contact with covid-positive)
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resolved Jul 21
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My friend Alice (name changed for privacy) had lunch last Thursday with a coworker who later tested positive on a PCR test on Saturday - they were sitting next to each other, indoors. Alice tested positive on a PCR test administered on Wed morning and tested negative on two Cue tests (rapid NAAT) earlier this week and today (Fri). No symptoms (so far at least). Alice is vaxxed and boosted with an mRNA vaccine.
Does Alice have Covid? Or, if it remains sufficiently uncertain, what is the probability that Alice has Covid? Resolves to our best estimate based on any additional testing and analysis here.
My initial rough estimate:
Also in spreadsheet form with some more details on the estimates and calculations at: https://docs.google.com/spreadsheets/d/1o95SAKOTFgIqrE-Y7nwZrBsqvJzyeJtMHS4QAJw0hus/edit#gid=0
Using microcovid.org/ and the stats on test accuracy from https://www.lesswrong.com/posts/cEohkb9mqbc3JwSLW/how-much-should-you-update-on-a-covid-test-result
- Baseline probability an average person in the area has covid: 0.55%
- Estimated chance of infection from close contact with Covid-positive coworker: 1.8%. Estimating this with microcovid as 1 hour, indoors, 3 ft, unmasked, normal talking volume, 3x mrna vaxxed, probably good air filtration but uncertain - I'm estimating it as probably MERV-13 with ~4 ACH but maybe less for an overall 1/2x risk multiple.
- Prior probability of infection: 0.55% + 1.8% = 2.3%. Convert to odds: .023
Bayesian evidence from test results (expressed as odds multipliers - see https://www.lesswrong.com/posts/cEohkb9mqbc3JwSLW/how-much-should-you-update-on-a-covid-test-result for how the math works):
- positive pcr: 88x
- negative cue: 0.1x
- no symptoms so far: 0.5x
Posterior odds: 0.1. Convert to probability: 9.3%
Note: PCR tests typically start returning positive at about 5 days after infection (which would be Tuesday), based on this https://twitter.com/michaelmina_lab/status/1472024457640394756 which I think is a reasonable approximation of the typical timeline of infection and test positivity.
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Update: Alice's partner Bob, who also was at lunch with the same coworker, also tested positive on a PCR test administered Wed. This seems like it should bump the probability of them having Covid up a lot - my guess is to something like 90%?
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Not relevant to the question asked for this market, but noting this here as well: We met up yesterday (Thu) with Alice, Bob, and a few others. Rough estimate for our potential exposure is: we were together indoors for about 4 hours this Thursday, if we average out with how long we were in close proximity, the exposure is probably roughly like 2 hours at 3 ft. Unmasked, 3x mrna vaxxed, HEPA air purifiers with 4-5 ACH -> that's roughly 2% chance of infection if one of Alice or Bob was infected, 4% if both.
Somewhat similar to https://manifold.markets/dreev/does-a-vaccinated-person-with-mild, you can check out the discussion in that and the linked markets for some related math/analysis.
Jun 3, 4:58pm: Update: negative rapid antigen test
Today is day 8 from the close contact, which is typically peak viral load, so a rapid test should be most likely to be positive. But given Alice is asymptomatic, that increases the chance of a false negative - the odds multiplier is 0.4x for a negative antigen test for someone asymptomatic, from https://www.lesswrong.com/posts/cEohkb9mqbc3JwSLW/how-much-should-you-update-on-a-covid-test-result#Rapid_antigen_tests__if_you_don_t_have_symptoms
Close date updated to 2022-07-12 11:59 pm
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I updated my spreadsheet with a bit more refined version of my previous analysis: https://docs.google.com/spreadsheets/d/1o95SAKOTFgIqrE-Y7nwZrBsqvJzyeJtMHS4QAJw0hus/edit#gid=0
I tried to adjust for correlation between tests a bit and figure out how much we can infer from one person's results about the other's - I think those are the biggest question marks in my stats at the moment.
My current estimate is about 15%.
I also found this study about PCR accuracy for asymptomatic individuals: https://bmcmedicine.biomedcentral.com/articles/10.1186/s12916-021-01982-x. "We estimated that the probability that the PCR test detected infection peaked at 77% (54–88%) 4 days after infection, decreasing to 50% (38–65%) by 10 days after infection."
So that's a lot lower than the usual figures we see based on symptomatic infections. Reading from fig 3, looks like ballpark 70% 6 days after infection, 60% at 8 days.
This lower sensitivity estimate mainly means we should update less on the negative test results, so the chances of Covid are higher.
Guesstimating from the study in the previous comment:
For Omicron in vaccinated people, it appears typical for symptoms to start about 2-3 days after exposure. I'll just simplify and use 2 days, and interpret the charts as if they had experienced symptoms starting at 2 days after exposure (even though they were actually asymptomatic).
The positive tests were at 6 days from the close contact, so reading the chart 4 days from symptom onset is about 90%. The negative tests were at 8 days from the close contact, so reading the chart 6 days from symptom onset is about 80%.
Rapid tests are much lower sensitivity in asymptomatic individuals (~60% vs ~80%), and it seems PCR tests are also based in the study on PCR for asymptomatic testing. So we should adjust downwards for asymptomatic, which puts us pretty similar to the above.
Some other notes:
- the fact that the two tests were from different labs I think makes it more accurate to treat them as closer to independent events when looking at the false positive/negative rates. It's still correlated though because e.g. low viral load makes one positive and one negative more likely I guess.
- this particular close contact was responsible for roughly about 3/4 of their chance of infection that week. So it's also possible but less likely that they could have gotten infected earlier or later, which would impact the timeline. I'll ignore that for now and focus on guesstimating the most likely infection scenario.
Update: Bob's PCR test from Friday also came back negative. So Alice and Bob both got a positive PCR on Wednesday from one lab (bioiq), and a negative PCR on Friday from another lab (county testing site, not sure what lab it was).
Trying to find out what proportion of people test positive vs negative on PCR tests after recovery was kind of difficult because most of the reporting on the topic just says that you _may_ continue testing positive for a month, not how likely it is. But eventually I found this meta-review https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7609379/ - see figure 4, which shows percentage of positive PCR tests vs days since symptoms start. Look at the upper respiratory tract curve (that includes nose and throat, so this should be most of the tests we're used to). The figure shows that it drops to 50% positive at about 15 days from symptoms start. (Most of the studies here did not look at asymptomatic infections.)
Also, note that it's not the case that tests consistently go from negative to positive some time after infection to negative some time after recovery. Tests can give intermittent positive/negative results. Another study https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7956402/ that analyzed a population of 20k Covid patients in Wuhan who had a positive PCR test followed by 2 consecutive negatives at least 24 hours apart (which was the criteria for hospital discharge) found that 12% had a positive PCR test afterwards. So getting a negative and then staying negative is most common but re-positives are not rare.
@fortenforge My impression was that PCRs generally kept returning positive for about a month. But I'm not sure if that's still true with very low viral loads - although my impression was also that they were tuned to be very sensitive.
I plan to resolve to the best estimate I can come up with based on discussion here. We're also waiting for Bob's second PCR test which should be helpful.
Update: negative PCR test from yesterday. This is quite surprising and causes me to update down substantially.
So that makes 1 positive PCR (on day 6 from the close contact), 1 negative PCR (day 8), 2 negative rapid NAATs, and 1 negative rapid antigen.
If we just looked at those tests I think it would be very likely a false positive. If we also look at Bob's positive PCR test result, it seems less clear.
I'm not familiar with the research on interpreting inconsistent test results, but I think PCR tests are estimated around 95% sensitivity and 99% specificity and I updated my rough estimates above assuming the test results are mostly independent to get to a overall estimate of 15% (see spreadsheet)
Anecdotally, the rapid tests take a few days to show results. Not sure about Cue, they might be more accurate, but I had been symptomatic and testing daily for about 3 days before my first positive: https://manifold.markets/Austin/do-i-have-covid
@Austin Yeah I agree, the rough timeline I've seen is that PCR starts returning positive at ~5 days and rapid antigen at ~6 days. Here the positive PCR was at 6 days and the latest negative Cue was at 7 days. So I don't think the Cue negative is a matter of it being too early.