30 March 2020
Why daily growth rates, and how to calculate them
I’ve focused so far on the evolution of confirmed case doubling times (i.e., the number of days it takes for the number of confirmed cases in a given locale to double) as a way of looking for early signs of progress, as do a number of epidemiologists and statisticians. There are some technical reasons to use doubling times: primarily that they are constant-slope at logarithmic scale, as you’ll have seen on the justly famous chart comparing doubling time by country.
However, there are several disadvantages of using doubling times to measure progress. They’re volatile: lumpy data (which is frequently the case in this crisis) makes them jump around in a way that can suggest more near-term progress or deterioration than is likely the case in reality. As presented, they’re coarse-grained: while it’s technically meaningful to talk about (say) a “4.2-day doubling time”, neither I nor most other sources present them in that way as it’s hardly intuitive. And finally, I’ve found that most people find doubling times hard to understand.
Today, I’m going to present a different, but mathematically equivalent, way of thinking about progress. I’ll replace doubling times with the Compound Daily Growth Rate (CDGR). I’d love to get your feedback about whether this is a more intuitive way for you to understand what’s going on.
What does CDGR mean? Anyone working in finance, and in many other businesses, will be familiar with CAGR, or Compound Annual Growth Rate. It’s simply a way of stating the annual growth rate in some figure (revenue, customers, etc.) in a mathematically useful way. (The “compound” part simple means that instead of taking the average of the growth rates over the relevant period, we calculate the growth rate that, if applied from the first period and allowed to compound, like interest, each year, gets us to the actual figure in the last period.)
Since COVID-19 is spreading so quickly, we need to talk about the daily growth rate rather than the annual rate. CDGR is a simple calculation. (Skip the rest of this paragraph if the math doesn’t interest you.) Say we’re interested in the CDGR of the cumulative confirmed case count in France over the last week, 22-29 March 2020. We divide the figure for the 29th by the figure for the 22st, take the seventh root (because it covers seven elapsed days) of that ratio, and subtract 1: (37575/14459)^(1/7)-1=15%. More generally, to calculate the CDGR for a period of d days, with c1 the count at the first period and c2 at the second, the formula is: (c2/c1)^(1/d)-1.
The relationship between doubling time and CDGR looks like this:
Obviously, high daily growth rates are bad. For many, that’s more intuitive than doubling time, where high doubling times are good and low doubling times bad.
One more technical point. For CDGR, we have to choose what period of time to use. The CDGR for a given locale will be different calculated over one day, one week, or one month. Given that the data are lumpy, we want to balance getting the benefit of some smoothing by looking over at least a few days, with wanting to see whether we’re making progress in the near-term. Somewhat arbitrarily, I’m going to use one week unless specified otherwise below.
What do daily growth rates tell us?
So what do the daily growth rates look like? First let’s look at some of the most impacted countries as well as the total for the world. I’ve highlighted US and World:
The good news is that most lines slope downwards. That means that in most countries, the rate of growth is slowing over time, exactly what control measures are meant to do. South Korea obviously cracked this early and has managed to get the growth rate almost to zero, which we all need to do. But Italy, France, Spain, and Germany are all making progress.
The news is still fairly bad. The growth rate for the world as a while is steadily increasing, not decreasing: that means that the pace of the disease’s spread is accelerating. The US’s growth rate, while decreasing recently, is still stubbornly high. And importantly, until these rates get to zero, we’re not out of the woods. A 10% daily growth rate is still very fast exponential growth! At 10% daily growth, cases double roughly every 8 days; triple every 13 days; increase 15-fold in a month; 100-fold in 50 days; and 1,000-fold in 74 days.
Is the picture different at the state level in the US?
A number of friends have also asked about what is happening in the US at the state level. Fortunately, the New York Times has aggregated data at a state level from various sources and made the data available in a GitHub repository. (As always, we should regard the data with a great deal of caution, since they likely suffer from the same limitations as country data: inconsistent standards, data lags, low / changing testing rates, etc.).
Here’s what the data say for the worst-impacted 10 states, which accounted for 79% of all reported cases in the US as of 28 March:
While it’s good news that these lines are generally sloping downwards, only Washington (the earliest state to be significantly hit) has been consistently below 20%; and per the point about compounding several paragraphs ago, sitting around 15% is not a good place to be.
More generally, most US states are still at a significantly earlier stage of dealing with the epidemic than even Italy, France, Germany, Spain, or even the UK. All of these European countries now have daily average growth rates consistently below 20% and consistently falling; by contrast, only Washington has achieved that milestone.
The news in the other 40 states isn’t much better. Yes, they have (or at least are reporting) significantly fewer confirmed cases. But only 11 are at or below 20% a 7-day CDGR, another 31 between 20% and 30%, and 9 between 30% and 40% (this includes the capitol, Washington DC).
I also looked to see if there was a relationship between the number of confirmed cases and the growth rate at a state level:
Interestingly, there isn’t an obvious relationship. That’s different than at the country level, where in general countries that have many cases are showing slowing growth rates: rapid growth leads countries to implement control measures which slows growth. I suspect that the lack of a relationship here says one or both of two things, neither that insightful. Either the confirmed case counts are so low in many states that they haven’t felt it necessary to take drastic action yet; or the underrtesting and underreporting is severe and inconsistent, so we really don’t know how many cases are in each state. Likely both are true.