Stats Chat

There’s a new COVID case from the Marist College cluster today.  The person previously tested negative, but had been in isolation (perhaps, though the Herald doesn’t say, because of the combination of symptoms and being a contact).  Now that we have plenty of testing capacity there has been follow-up testing of some clusters as well as testing of some apparently healthy people in high-risk jobs.

From what I’ve seen on social media, this has led some more people to find out about the false negative rates of the current tests.  It’s not a secret that the swab+PCR test we use in NZ misses maybe a third of infections (because there isn’t enough virus on the swab), though it hasn’t exactly been emphasised.  So, how is this acceptable? Well, “acceptable” depends on the alternative. It’s the best test we have. Researchers (and companies) are working on better ones, and things are likely to improve over time, as they did with HIV testing.  If you’ve been in contact with a case and have COVID-like symptoms and test negative, you’re still going to need to isolate until you recover.  That, plus the fact that the testing does pick up the majority of cases, means a test/trace/isolate strategy, done right, should be nearly enough to control an outbreak that’s caught early.

The current tests give basically no false positives. That’s really helpful for a test/trace/isolate strategy — we’ve done 200,000 tests, and if, say, 5% of them were false positives, that would be another 10,000 cases. Before counting all the contacts of those 10,000 people.  The low false positive rate also means the health system can say, “yes, you need to get tested”,  and then after a positive results, “yes, you absolutely must stay home”, “yes, you need to tell us about all the places you’ve been, even if some of them are embarrassing or illegal”.

There’s another testing-accuracy story in the New York Times, unhelpfully headlined Coronavirus Testing Used by the White House Could Miss Infections. It turns out that they don’t mean that it could miss infections the way all the other tests do; they mean it could miss infections that other tests detect.  The test in question is a portable testing machine from Abbott that takes only 5 minutes to process a sample, quite a bit faster than the standard testing systems.  Researchers from a testing lab at New York University’s Langone Medical Center (who liked the idea of a faster test, given the number of tests they perform) did comparisons to the machines they are currently using and published a PDF about it– some tests on the same swabs and some on different swabs taken at the same time from the patient.  They say the Abbott machine missed 1/3 (out of 15) or 1/2 (out of 30) samples where the current machine found the virus.

Abbott, on the other hand, said their evaluations showed 0.02% false negative rate.

You might wonder how two evaluations could be so different: one false negative in 50,000 samples vs 5 in 15 samples?  As usual, when two numbers don’t fit, it’s probably because they don’t mean the same. Abbott will have been referring to an estimated false negative rate in viral samples with a known virus concentration — an evaluation of the assay itself.  The NYU researchers are talking about live clinical use in samples where they know the virus is present in the swab.  And when we talk about false negatives in the NZ sampling system, we’re talking about samples where the virus probably isn’t present in the swab.

Abbott argue that the NYU researchers were using the machine incorrectly. That could be true, but it’s only reassuring to the extent that you think the White House will be better at it than a pretty highly regarded New York hospital and research centre.

From the Twitter account of the White House Council of Economic Advisors

From the Washington Post

• Models based on understanding the actual process that’s causing the trends.  The SIR models and their extensions, which we’ve seen a lot of in NZ, are based on a simplified but useful representation of how epidemics work.  Weather forecasting works this way. So do predictions of populations for each NZ region into the future.
• Models based on simplifying and matching previous inputs.  When Google can distinguish cat pictures from dog pictures, it’s because it has seen a bazillion of each and has worked out a summary of what cat pictures and dog pictures look like. It will compare your picture to  those two summaries and see what matches best.  Risk models for heart disease are like this: does your data look like the data of people who had heart attacks. Fraud risk models for banks, insurance companies, and the IRD work this way. It still helps a lot to understand about the process you’re modelling, so you know what sort of data to put in or leave out, and what sort of summaries to try to match.
• Models based on extrapolating previous inputs.  In business and economics you often need predictions of the near future.  These can be constructed by summarising existing recent trends and the variation around them, then assuming the trends and variation will stay roughly the same in the short term.  Expertise in both statistics and in the process you’re predicting is useful, so that you know what sorts of trends there are and what information is available to model them. A key part of these time-series models is getting the uncertainty right, but even when you do a good job the predictions won’t work when the underlying trends change.

The SIR epidemiology models that you might have seen in the Herald are based on knowledge of how epidemics work.  The IHME/UW models are at least based on knowledge of what epidemics look like. The cubic model isn’t.

The cubic fit is a model of the second type, based just on simplifying and matching the available data.  It could be useful for smoothing the data — as the tweet says, “with irregular data, curve fitting can improve data visualization”.  In particular, the weekly up-and-down pattern comes from limitations in the death reporting process, so filtering it out will give more insight into current trends.

The particular model that produces the red curve is extremely simple (Lucy D’Agostino McGowan duplicated it).  If you write t for day of the year, so t starts at 1 for January 1, the model is

log(number of deaths + 0.5) = -0.0000393× t³ +0.00804× t² -0.329×t – 0.865

What you can’t do with a smoothing/matching model like this is to extrapolate outside the data you have.  If you have a model trained to distinguish cat and dog pictures and you give it a picture of a turkey, it is likely to be certain that the picture is a cat, or certain that the picture is a dog, but wrong either way.  If you have a simple matching model where the predicted number of deaths depends only on the date, and the model matches data from dates in March and April, you can’t use it to predict deaths in June. The model has never heard of June. If it gets good predictions in June, that’s entirely an accident.

When you extrapolate the model forward in time, the right-hand side becomes very large and negative, so the predicted number of deaths is zero with extreme certainty.  If you were to extrapolate backwards in time, the predicted number of deaths would explode to millions and billions during December. There’s obviously no rational basis for using the model to extrapolate backwards into December, but there isn’t much more for using it to extrapolate forward — nothing in the model fitting process cares about the direction of time.

The Chairman of the Council of Economic Advisers until the middle of last year was Dr Kevin Hassett (there’s no Chairman at the moment). He’s now a White House advisor, and the Washington Post attributes the cubic model to him.  Hassett is famous for having written a book in 1999 predicting that the Dow Jones index would reach 36,000 in the next few years.  It didn’t — though he was a bit unlucky in having his book appear just before the dot-com crash.  Various unkind people on the internet have suggested a connection between these two predictive efforts.  That’s actually completely unfair.  Dow 36,000 was based on a model for how the stock and bond markets worked, in two parts: a theory that stocks were undervalued because of their relative riskiness, and a theory that the markets would realise this underpricing in the very near future.  The predictions were wrong because the theory was wrong, but that’s at least the right way to try to make predictions. Extrapolating a polynomial isn’t.

We often hear groups of people described as ‘high risk’ in the context of COVID.  The problem is that this means three different things, and they often aren’t distinguished clearly

• People who have a relatively high exposure to coronavirus, so they are more likely to catch it: nurses, doctors, supermarket staff, police (high probability)
• People who are more likely to get seriously sick if they do become infected: elderly, immunocompromised, people with chronic lung disease (high consequence)
• People who are more likely to spread the infection if they get it. Some overlap with the first group, but also migrant workers, prisoners, and at least in the US, meat processing workers. (high transfer)

The second group are very different from the others. Suppose you were doing intensive testing to try to see if there was undetected community transmission of COVID.  You’d definitely want to test the first group, because that’s where you’re most likely to find the virus, and you might want to test the last group, because missing it there would be serious (as it was in Singapore).  You might well not go after the second group, because the safest thing for them is isolation — having a bunch of health workers barge in and stick swabs up their noses is unpleasant and possibly risky.  You’d absolutely want to test the second group if there was any indication of symptoms or exposure, but not just in the ordinary course of business.  The three groups are different.

Even Cory Doctorow confused the first and second groups a bit, in his rant about the risks of contact-notification apps

The proximity sensing they do is going to miss out on people who don’t have smartphones and/or don’t have the technological savvy to install them. That overlaps broadly with the most at-risk groups: elderly people and poor people.

Epidemiology is a team sport and the most vulnerable people are the MVPs on the team. “Our app will tell you if you came in contact with an infected person (but not if that person is from the most likely group of infected people)” is a fundamentally broken premise.

Elderly people are a ‘high consequence’ group — infection is serious for them. They aren’t a ‘high probability’ group — there’s no special reason why elderly people in the community would be more likely to get infected (or in residential facilities, if good care is taken)

Nice story by Farah Hancock at Newsroom, on NZ mortality data.

In places with less successful control of COViD-19, there has been a spike in deaths confirmed as due to coronavirus, and also a spike in other deaths.  In New Zealand, there hasn’t been — there isn’t any clear excess over the average for the time of year.  There undoubtedly have been deaths due to coronavirus, but there have also been deaths prevented by the lockdown (on the roads, for example), and there may well have been deaths caused by the lockdown (eg, people not getting heart attacks treated promptly), but the overall trends are spectacularly unlike those in New York City, which has not quite twice the population of New Zealand and over 20,000 net excess deaths.

There’s still plenty of time for NZ to catch up to outbreaks in other parts of the world. Let’s not do that.

There are over 100 potential vaccines being developed, and several are already in preliminary testing in humans.  There are three steps to testing a vaccine: showing that it doesn’t have any common, nasty side effects; showing that it raises antibodies; showing that vaccinated people don’t get COVID-19.

The last step is the big one, especially if you want it fast. I knew that in principle, but I was prompted to run the numbers by hearing (from Hilda Bastian) of a Danish trial in 6000 people looking at whether wearing masks reduces infection risk.  With 3000 people in each group, and with the no-mask people having a 2% infection rate over two months, and with masks halving the infection rate, the trial would still have more than a 1 in 10 chance of missing the effect.  Reality is less favorable:  2% infections is more than 10 times the population percentage of confirmed cases so far in Denmark (more than 50 times the NZ rate), and halving the infection rate seems unreasonably optimistic.

That’s what we’re looking at for a vaccine. We don’t expect perfection, and if a vaccine truly reduces the infection rate by 50% it would be a serious mistake to discard it as useless. But if the control-group infection rate over a couple of months is a high-but-maybe-plausible 0.2%  that means 600,000 people in the trial — one of the largest clinical trials in history.

How can that be reduced?  If the trial was done somewhere with out-of-control disease transmission, the rate of infection in controls might be 5% and a moderately large trial would be sufficient. But doing a randomised trial in setting like that is hard — and ethically dubious if it’s a developing-world population that won’t be getting a successful vaccine any time soon.  If the trial took a couple of years, rather than a couple of months, the infection rate could be 3-4 times lower — but we can’t afford to wait a couple of years.

The other possibility is deliberate infection. If you deliberately exposed trial participants to the coronavirus, you could run a trial with only hundreds of participants, and no more COVID deaths, in total, than a larger trial. But signing people up for deliberate exposure to a potentially deadly infection when half of them are getting placebo is something you don’t want to do without very careful consideration and widespread consultation.  I’m fairly far out on the ‘individual consent over paternalism’ end of the bioethics spectrum, and even I’d be a bit worried that consenting to coronavirus infection could be a sign that you weren’t giving free, informed, consent.

This turned up via Twitter, with the headline Pitt researchers developing a nasal spray that could prevent covid-19

“The nice thing about Q-griffithsin is that it has a number of activities against other viruses and pathogens,” said Lisa Rohan, an associate professor in Pitt’s School of Pharmacy and one of the lead researchers in the collaboration, in a statement. “It’s been shown to be effective against Ebola, herpes and hepatitis, as well as a broad spectrum of coronaviruses, including SARS and MERS.”

The active ingredient is the synthetic form of protein extracted from a seaweed found in Australia and NZ. Guess how many human studies there have been of this treatment?

Clinicaltrials.gov reports one completed safety study of a vaginal gel, and one ongoing safety study of rectal administration, both aimed at HIV prevention. There appear to have been no studies against coronaviruses in humans, nor Ebola, herpes, or hepatitis. There appear to have been no studies of a nasal-spray version in humans (and I couldn’t even find any in animals, just studies of tissue samples in a lab). It’s not clear that a nasal spray would work even if it worked — eg, is preventing infection via the nose enough, or do you need to worry about the mouth.

Researchers should absolutely be trying all these things, but making claims of demonstrated effectiveness is not on.  We don’t want busy journalists having to ask Dr Bloomfield if we should stick seaweed up our noses.

With NZ’s good-so-far control of the coronavirus, there has been discussion on the internets as to whether New Zealand has high or low population density, and also on whether March/April is summer here or not.  The second question is easy. It’s not summer here. The first is harder.

New Zealand’s average population density is very low.  It’s lower than the USA. It’s lower than the UK. It’s lower than Italy even if you count the sheep.  On the other hand, a lot of New Zealand has no people in it, so the density in places that have people is higher.  Here are a couple of maps: “Nobody Lives Here” by Andrew Douglas-Clifford, showing the 78% of the country’s land area with no inhabitants, and a 3-d map of population density by Alasdair Rae (@undertheraedar)

We really should be looking at the population density of inhabited areas. That’s harder than it sounds, because it makes a big difference where you draw the boundaries. Take Dunedin. The local government area goes on for ever in all directions. You can get on an old-fashioned train at the Dunedin station, and travel 75km through countryside and scenic gorges to the historic town of Middlemarch, and you’ll still be in Dunedin. The average population density is 40 people per square kilometre.  If you look just within the StatsNZ urban boundary, the average population density is 410/square kilometre — ten times higher.

A better solution is population-weighted density, where you work out the population density where each person lives and average them. Population-weighted density tells you how far away is the average person’s nearest neighbour; how far you can go within bumping into someone else’s bubble.The boundary problem doesn’t matter as much: hardly anyone lives in 90% of Dunedin, so it gets little weight — including or excluding the non-urban area doesn’t affect the average.  What does matter is the resolution.

If you work out population weighted densities using 1km squares you will get a larger number than if you use 2km squares, because the 1km square ignores density variation within 1km, and the 2km squares ignore density variation within 2km. If you use census meshblocks you will get a larger number than if you use electorates, and so on.  That can be an issue for international comparisons

However, this is a graph of population-weighted density with 1km squares across a wide range of European and Australian cities using 1km square grids, in units of people per hectare:

If you compare the Australian and NZ cities using meshblocks, Auckland slots in just behind Melbourne, with Wellington following, and Christchurch is a little lower than Perth. The New York Metropolitan Area is at about 120.  Greater LA, the San Francisco Bay Area, and Honolulu, are in the mid-40s, based on Census Bureau data. New York City is at about 200.  I couldn’t find data for any Asian cities, but here’s a map of Singapore showing that a lot of people live in areas with population density well over 2000 people per square kilometre, or 200/hectare.

So, yes, New Zealand is more urban than foreigners might think, and Auckland is denser than many US metropolitan areas. But by world standards even Auckland and Wellington are pretty spacious.

Scientists often speak their own language. They sometimes use strange words, and they sometimes use normal words but mean something different by them.  Toby Morris & Siouxsie Wiles have an animation of some examples.

The goal of scientific language is usually to be precise, to make distinctions that aren’t important in everyday speech. Scientists aren’t trying to confuse you or keep you out, though those effects can happen  — and they aren’t always unwelcome.  I’ve written on my blog about two examples: bacteria vs virus (where the scientists are right) and organic (where they need to get over themselves).

This week’s example of conflict between trying to be approachable and trying to be precise is the phrase “false positive rate”.  When someone gets a COVID test, whether looking for the virus itself or looking for antibodies they’ve made in reaction to it, the test could be positive or negative.  We can also divide people up by whether they really have/had COVID infection or no infection. This gives four possibilities

• True positives:  positive test, have/had COVID
• True negatives: negative test, really no COVID
• False positives: positive test, really no COVID
• False negatives: negative test, have/had COVID

If you encounter something called the “false positive rate”, what is it? It obvious involves the false positives, divided by something, but it could be false positives as a proportion of all positive tests, or false positives as a proportion of people who don’t have COVID, or even false positives as a proportion of all tests.  It turns out that the first two of these definitions are both in common use.

Scientists (statisticians and epidemiologists) would define two pairs of accuracy summaries

• Sensitivity:  true positives divided by people with COVID
• Specificity: true negatives divided by people without COVID
• Positive Predictive Value(PPV): true positives divided by all positives
• Negative Predictive Value(NPV): true negatives divided by all negatives

The first ‘false positive rate’ definition is 1-PPV NPV; the second is 1-specificity.

If you write about the antibody studies carried out in the US, you can either use the precise terms, which will put off people who don’t know anything, or use the vague terms, and people who know a bit about the topic may misunderstand and think you’ve got them wrong.

This isn’t precisely statistics, but Bloomberg Opinion may have set some kind of record for the density of easily checkable false statements in a single column that’s still making a basically sound argument.  Joe Nocera was arguing in favour of strict lockdown policy like the one in New Zealand.

Just looking at the tweets we have

1. New Zealanders aren’t allowed to drive except in emergencies and can only be out of the house for an hour a day, to get exercise or to buy essentials. This one-hour limit is enforced by the police. The one hour limit isn’t true, and so isn’t being enforced by police. The ‘only in emergencies’ is at best misleading: you can drive to buy essentials.
2. At the pharmacy, only one person is allowed in at a time, and clerks retrieve the goods so customers never touch anything until they return home. The wait to get in a grocery store is around an hour. If you don’t have a mask and gloves, you won’t get in. True as to pharmacies.  False as to the wait to get in a grocery store — which also contradicts the alleged one-hour limit.  False as to masks and gloves: the use of masks is encouraged by some groups but they are not required and users are only barely a majority, if that. Gloves aren’t even recommended.
3. In New Zealand…
   Every restaurant is closed.

   There’s no take-out.
   There are no deliveries.
   E-commerce has been halted. Food-processing companies still operate, but virtually every other form of blue-collar work is shut down. True on restaurant closures (until tomorrow).  It’s ambiguous whether ‘deliveries’ goes with the previous bullet or the following one, but e-commerce is certainly not halted. You can’t get prepared food delivered, but groceries, wine and spirits, some office equipment, and a range of other things are being delivered.  I have bought an office chair, because I’m working from home until at least the end of July.  The blue-collar work thing is true-ish — he may underestimate how much of NZ’s economy is in food production for export, but construction and forestry are shut down.  
4. Citizens are surviving financially with emergency checks from the government. Essential workers in New Zealand are truly essential. Although there are Covid-19 clusters — a church; a rest home; a wedding party — workplaces have largely been virus-free.  The Marist College cluster and the World Hereford Conference definitely count as ‘workplace’.  The government is primarily sending payments to businesses so they can keep employees or provide them sick leave, and to self-employed people, though there has obviously been an increase in the number of people getting unemployment benefits and applying for hardship grants. And, of course, we don’t use checks for all this.

It’s probably best to think of the column as ‘about New Zealand’ in the same way Gilbert and Sullivan’s Mikado is ‘about Japan’ — it’s really just a background setting to criticise the writer’s own country.

There have been three fairly large American seroprevalence studies recently.  These are studies that sample people from the community and test their blood for antibodies to the COViD-19 virus. Most people who have been infected, even if they recovered weeks ago, will have antibodies to the virus.  And people who have not been infected will not have antibodies to the COVID-19 virus, though the test isn’t perfect and will occasionally be confused by antibodies to something else.

The two studies in Los Angeles and in Santa Clara County (Silicon Valley) estimated that a few percent of people had been exposed to the virus. The New York study estimated far higher rates — in New York City itself, 21%.  The Californian studies have been widely criticised because of questions about the representativeness of the sample and the accuracy of the statistical calculations.  The New York study has had much less pushback.

One difference between the New York study and the others is that it’s not being pushed as a revolutionary change in our understanding of coronavirus, so people aren’t putting as much effort into looking at the details. Much more important, though, is that it is far easier to get a prevalence of 20% roughly right than to get a prevalence of 2-3% roughly right. If you make extraordinary claims based on a prevalence estimate of 2-3%, you need data and analysis of extraordinary quality (in a good way).  If your prevalence estimate of 20% is consistent with the other empirical data and models for coronavirus, it doesn’t need to stand on its own to the same extent.

Getting a good estimate of a prevalence of 2-3% is hard because the number of people who really have been exposed is going to be about the same as the number where the test gets confused and gives the wrong answer.  If you aren’t precisely certain of the accuracy of the test (and you’re not), the uncertainty in the true prevalence can easily be so large as to make the effort meaningless. On top of that, the quality of the sampling is critical:  even a little bit of over-volunteering by people who have been sick and want reassurance can drive up your estimate to be larger than the truth.  You can easily end up with an estimate saying the prevalence is much higher than most people expect, but only very weak evidence for that claim.

It looks as though the antibody test used in New York was less accurate than the one used in Santa Clara; the New York State lab that ran the testing says only that they are confident the rate of positive tests in truly unexposed people is less than 7%; their best-guess estimate will presumably be around 2-3%, in contrast with the best-guess estimate of 0.5% for test used in Santa Clara. But even, worst-case,  if 7% of tests were false positives, that still leaves 14% that were genuine. And since the test will miss some people who were truly exposed, the true prevalence will be higher than 14%. Suppose, for example, that the test picks up antibodies in 90% of people who really have been exposed. The 14% we’re seeing is only 90%80% of the truth, so the truth would be about 16%, and with a less-sensitive test, the truth would have to be higher.  So, even though the test is imperfect, somewhere between one in five and one in seven people tested had been exposed to the virus.  That’s a narrow enough range to be useful.  You still have to worry about sampling: it’s not clear whether sampling people trying to shop will give you an overestimate or an underestimate relative to the whole population, but the bias would have to be quite large to change the basic conclusions of the study.

The estimate for New York fits reasonably well with the estimate of roughly 0.1% for the proportion of the New York City population that have died because of the pandemic, and the orthodox estimate of around 1% for the proportion of infected people who will die of COViD.  These all have substantial uncertainty: I’ve talked about the prevalence estimate already. The infection fatality rate estimate is based on a mixture of data sets, all unrepresentative in different ways. And the excess mortality figure itself is fairly concrete, but it includes, eg, people who died because they didn’t try to get to hospital for heart attacks, and in the other direction, road deaths that didn’t happen.  It is still important that these three estimates fit together, and it correctly gives researchers more confidence in all the numbers.  The Californian studies imply only about 0.1% of infections are fatal, and that doesn’t fit the excess mortality data or the standard fatality rate estimate at all.

There’s an analogy that science is like a crossword¹. But it’s not a cryptic crossword, where correct answers are obvious once you see the trick. It’s the other sort of crossword, where potential answers to different clues support or contradict each other.  If the clue is “Controversial NZ party leader (7)” and using “Bridges” would force another clue to be a seven letter word ending in “b”, you might pencil in “Seymour” instead and await further evidence.

¹: Independently invented by multiple people including Susan Haack and Chad Orzel.