Benchmarking the Covid-19 pandemic across countries and states in the USA under heterogeneous testing

Scientists and policymakers need to compare the incidence of Covid-19 across territories or periods with various levels of testing. Benchmarking based on the increase in total cases or case fatality rates is one way of comparing the evolution of the pandemic across countries or territories and could inform policy decisions about strategies to control coronavirus transmission. However, comparing cases and fatality rates across regions is challenging due to heterogeneity in testing and health systems. We show two complementary ways of benchmarking across territories and in time. First, we used multivariate regressions to estimate the test-elasticity of Covid-19 case incidence. Cases grow less than proportionally with testing when assessing weekly changes or looking across states in the USA. They tend to be proportional or even more than proportional when comparing the month-to-month evolution of an average country in the pandemic. Our results were robust to various model specifications. Second, we decomposed the growth in cases into test growth and positive test ratio growth to intuitively visualize the components of case growth. We hope these results can help support evidence-based decisions by public officials and help the public discussion when comparing across territories and in time.


Clarifications about the interpretation of regression coefficients for elasticity
Public Health officials worldwide are comparing various ratios and quantities that are not entirely independent of one another. Here, we briefly clarify the meaning of the coefficient we estimated for some of the policy-relevant amounts monitored.
If and are weekly tests and cases in week , the positive test rate PTR is / .
The change in PTR can be decomposed as [ / ] = − ; but the coefficient estimated in our paper says that = ; which replaced in the previous equation yields that In general, looking at the structure of this latter formula, any < 1 would imply a decrease in PTR when tests grow, while a > 1 would mean a growing PTR. As an example, a ≈ 0.2 like the one we estimated in Figure 1b for US states would mean that a thought experiment of 10% change in tests ( A final comment is that our estimate of for the average US state or average country during the current period is not a constant written in stone, but it could undoubtedly evolve.
Summing up, the coefficient estimated in the main section allows us to compare the evolution of cases for two countries with different levels of testing. Also, the estimated key coefficient helps to benchmark the change of the ratio of positive tests, which also depends on testing. This idea could complement the growth decomposition of cases, an analogous way to decompose case growth between testing intensity and PTR. The two methods are the same in the particular case when the test-elasticity is equal to one, and the R 2 of the regression is 100%.

Estimated elasticity and frequency of changes
To understand why a weekly elasticity might be smaller than the monthly elasticity, it is instructive to follow Hawawini (1983). 8 He deals with the change in the elasticity of two log-changes when periods change. Let's call , ≡ Δ and , ≡ Δ .
The changes are made every T periods (i.e., T=7 is weekly, T=30 if monthly). In a contemporaneous regression, one estimates an elasticity from = + .
In Hawawini's (1983) spirit, one could assume that the actual process of generating the data on cases could be correlated with testing not only in the current period but also potentially with one lag and one lead.
To summarize how significant are these inter-temporal correlations in comparison to the contemporaneous correlation Hawawini defines a ratio Then, Hawawini shows that the elasticity for a T period difference ( ) relates to the elasticity using oneperiod differences, (1), through the following expression Taking derivatives of the difference with respect to the period T, one gets This means that the elasticity ( ) grows with the period of the difference considered, T, when the above derivative is positive. That is the case when , > , , making the numerator positive. In other words, the elasticity increases from weekly to monthly when cases have relatively stronger lag-lead correlations with testing , vis-à-vis the autocorrelation of testing.

An alternative interpretation of β
A regression Δ ln = Δ ln + , to get the elasticity , could be combined with our exact decomposition, which we replace on the left-hand side of the regression, leading to Δ ln + Δ ln = Δ ln . Collecting terms related to testing on the right-hand side yields an implicit regression of positivity on testing would be Δ ln = ( − 1) Δ ln . That is why in some regression Tables, we will display both the standard test to see whether the elasticity is statistically different from zero, and also show a formal test of significance for ( − 1). For example, if ( − 1) were statistically zero, then it would be equivalent to claim that there is no significant correlation between changes in positivity and changes testing, validating the proportional approach in the exact decomposition used in Figure 3 and Figure 4, which is coherent to a = 1.

Use of decomposition for the predictability of future cases
To predict future cases, we could run a dynamic regression of current case growth on the lagged PTR and testing growth. Specifically, Given our exact decomposition, if β coefficients are the same, the previous equation would collapse to a standard first-order autocorrelation of case growth.
To illustrate the point, we can consider the case of India from April to Dec 2020. When predicting the growth in cases, the coefficient on the previous week's tests (Δ ln −1 ) is twice as large as the coefficient on lagged PTR (0.81 vs 0.42, p-value of difference = 0.05). Therefore, a 10% increase in testing would be associated with an 8% expected increase in cases next week. In contrast, a 10% increase in PTR would be associated with only a 4% expected increase in cases next week. Our exact decomposition therefore allows separating these two different predictions. Figure S1 shows week change in cases and tests (in logarithms) US states, similar to Figure 1b in the article's main text, but keeping differences between -1 and + 1.5 log points. Figure S1. Change in cases per capita relative to change in the number of tests for states in the USA, excluding outliers and focusing on the same range of testing growth as the global cross country sample. Data shown are changes in weekly reported cases and testing between April 4 through April 10, 2020. Sample of US states restricted to logarithmic week-on-week changes of testing rates between -1 and +1.5, as observed in the global sample. In Figure S1: β=0.54, p<0.0001, 95%CI: 0.36-0.71.   Case growth decomposition Figure S2 shows how the USA has been moving in a plot similar to Figure 2A. The USA is above the line in the two weeks in which the figure is plotted (exception vis-a-vis other countries). Figure S3 shows the results of growth decomposition, but with updated data from the week ending April 17 th and comparing the (logarithmic) growth vis-à-vis the previous week. Countries and territories tend to be in different positions compared to Figure 2 of the main text, which used data from one week earlier.

Elasticity regression results
For example, in Figure 2 (until April 11 th ), almost all US states were above the zero case growth line. In contrast, in Figure S3B we observe that many states have moved below with heterogeneous combinations of testing and PTR changes.   The regression for global data of weekly changes in the second part of the year 2020 displays a statistically significant coefficient of 1.1 (p-value < 0.001). The figure below (Fig S4) illustrates the practical magnitude of these coefficients. The figure compares the overall elasticity between a country in the 20 th percentile of Human Development Index and a country in the 80 th percentile. While the former has an overall elasticity of 0.44, the latter's elasticity is 0.71, with 95% confidence intervals that do not overlap. That means that quantitatively, during this period richer countries tended to have stronger elasticities. But qualitatively, both groups of countries have elasticities below one, which are less than proportional. This is our central claim for the week-to-week regressions in the main text.