Death Rate I: We can't tell, there's no good data yet.
This was inspired by a tweet by Ted Naiman, MD, whose Protein-to-Energy ratio analysis of food I credit for at least half of my weight loss (the other half I credit P. D. Mangan, for the clearest argument for intermittent fasting, which convinced me); so this is not about Dr Naiman's tweet, just that his was the tweet I saw with a variation of this proposition:
"COVID-19 is 'like the flu,' except the death rate is 30 to 50 times higher."
But here's the problem with that proposition: we don't have reliable data to determine that. Here are two simple arguments that cast some doubt on the proposition:
⬆︎ How the death rate could be higher: government officials and health organizations under-report the number of deaths in order to contain panic or to minimize criticism of government and health organizations; also possible that some deaths from COVID-19 are attributed to conditions that were aggravated by COVID-19, for example being reported as deaths from pneumonia.
⬇︎ How the death rate could be lower: people with mild cases of COVID-19 don't report them and treat themselves with over-the-counter medication (to avoid getting taken into forced quarantine, for example), hence there's a bias in the cases known to the health organizations, towards more serious cases, which are more likely to die.
How much we believe the first argument applies depends on how much we trust the institutions of the countries reporting, and... you can draw your own conclusions!
To illustrate the second argument, consider the incentives of someone with flu-like symptoms and let's rate their seriousness or aversiveness, $a$, as a continuous variable ranging from zero (no symptoms) to infinity (death). We'll assume that the distribution of $a$ is an exponential, to capture thin tails, and to be simple let's make its parameter $\lambda =1$.
Each sick patient will have to decide whether to seek treatment other than over-the-counter medicine, but depending on the health system that might come with a cost (being quarantined at home, being quarantined in "sick wards," for example); let's call that cost, in the same scale of aversiveness, $c$.
What we care about is how the average aversiveness that is reported changes with $c$. Note that if everyone reported their $a$, that average would be $1/\lambda = 1$, but what we observe is a self-selected subset, so we need $E[a | a > c]$, which we can compute easily, given the exponential distribution, as
\[
E[a | a > c]
=
\frac{\int_{c}^{\infty} a \, f_A(a) da }{1 - F_A(c)}
=
\frac{\left[ - \exp(-a)(a+1)\right]^{\infty}_{c}}{\exp(-c)}
= c + 1
\]
Note that the probability of being reported is $\Pr(a>c) = \exp(-c)$, so as the cost of reporting goes up, a vanishingly small percentage of cases are reported, but their severity increases [linearly, but that's an artifact of the simple exponential] with the cost. That's the self-selection bias in the second argument above.
A plot for $c$ between zero (everyone reports their problems) and 5 (the cost of reporting is so high that only the sickest 0.67% risk reporting their symptoms to the authorities):
Remember that for all cases in this plot the average aversiveness/seriousness doesn't change: it's fixed at 1, and everyone has the disease, with around 63% of the population having less than the average aversiveness/seriousness. But, if the cost of reporting is, for example, equal to twice the aversiveness of the average (in other words, people dislike being put in involuntary quarantine twice as much as they dislike the symptoms of the average seriousness of the disease), only the sickest 13.5% of people will look for help from the authorities/health organizations, who will report a seriousness of 3 (three times the average seriousness of the disease in the general population).*
With mixed incentives for all parties involved, it's difficult to trust the current reported numbers.
Death Rate II: Using the data from the Diamond Princess cruise ship.
A second endemic problem is arguing about small differences in the death rate, based on small data sets. Many of these differences are indistinguishable statistically, and to be nice to all flavors of statistical testing we're going to compute likelihood ratios, not rely on simple point estimate tests.
The Diamond Princess cruise ship is as close as one gets to a laboratory experiment in COVID-19, but there's a small numbers problem. In other words we'll get good estimates when we have large scale, high-quality data. Thanks to @Clarksterh on Twitter for the idea.
Using data from Wikipedia for Feb 20, there were 634 confirmed infections (328 asymptomatic) aboard the Diamond Princess and as of Feb 28 there were 6 deaths among those infections. The death rate is 6/634 = 0.0095.
(The ship's population isn't representative of the general population, being older and richer, but that's not what's at stake here. This is about fixating on the point estimates and small differences thereof. There's also a delay between the diagnosis and the death, so these numbers might be off by a factor of two or three.)
What we're doing now: using $d$ as the death rate, $d = 0.0095$ is the maximum likelihood estimate, so it will give the highest probability for the data, $\Pr(\text{6 dead out of 634} | d = 0.0095)$. Below, we calculate and plot the likelihood ratio between that probability and the computed probability of the data for other candidate death rates, $d_i$.**
\[LR(d_i) = \frac{\Pr(\text{6 dead out of 634} | d = 0.0095)}{\Pr(\text{6 dead out of 634} | d = d_i)}\]
We can't reject any rates between 0.5% and 1.5% with any confidence (okay, some people using single-sided point tests with marginal significance might narrow that a bit, but let's not rehash old fights here), and that's a three-fold range. And there are still a lot of issues with the data.
On the other hand...
It's easy to see that the COVID-19 death rate is much higher than that of the seasonal flu (0.1%): using the data from the Diamond Princess, the $LR(0.001) = 3434.22$, which should satisfy both the most strong-headed frequentists and Bayesians that these two rates are different. Note that $LR(0.03) = 510.01$, which also shows that with the data above the Diamond Princess invalidates the 3% death rate. (Again, noting that the numbers might be off by a factor of two or three in either direction due to the delay in diagnosing the infection and between diagnosis and recovery or death.)
As with most of these analyses, disaggregate clinical data will be necessary to establish these rates, which we're estimating from much less reliable [aggregate] epidemiological data.
Stay safe: wash hands, don't touch your face, avoid unnecessary contact with other people.
- - - - -
* A friend pointed out that there are some countries or subcultures where hypochondria is endemic and that would lead to underestimation of the seriousness of the disease; this model ignores that, but anecdotally I've met people who get doctor's appointments because they have DOMS and want the doctor to reassure them that it's normal, prescribe painkillers and anti-inflammatories, and other borderline psychotic behavior...
** We're just computing the binomial here, no assumptions beyond that:
$\Pr(\text{6 dead out of 634} | d = d_i) = C(634,6) \, d_i^6 (1-d_i)^{628}$,
and since we use a ratio the big annoying combinatorials cancel out.
