Table 1.

Estimators used for finding unobserved quantities (see Table S1). Bold numbers indicate the order of calculation. The estimators for the total number of infections at the state and county levels [(7) and (10); ${\hat{\iota }}_s$ and ${\hat{\iota }}_c$] are defined implicitly and are found by solving the given equations numerically. These estimators and $\hat{\omega }$ are based on the approximation for the mean of the Wallenius’ noncentral hypergeometric distribution given in Expression (1). Time (t), state (j), and county (i) subscripts are omitted for readability, but all estimators are time-, state-, and county-specific.

Region	Infection fatality ratio	Infections	Tests	Odds ratio
Country	φ = 5 (observed)	(4)$\hat{\iota }=\frac{1000D}{\varphi }$	(3)$\hat{T}=\sum {\hat{T}}_s$	(5)$\hat{\omega }=\frac{\log \left(1-\frac{C}{\hat{\iota }}\right)}{\log \left(1-\frac{\hat{T}-C}{P-\hat{\iota }}\right)}$
State	(8)${\hat{\phi }}_s=\frac{1000D_s}{{\hat{\iota }}_s}$	(7)$\frac{C_S}{{\widehat{\iota }}_S}+{\left(1-\frac{{\widehat{T}}_S-C_S}{P_S-{\widehat{\iota }}_S}\right)}^{{\hat{\omega }}_S}=1$	(2)${\hat{T}}_s=\sum {\hat{T}}_c$	(6)${\hat{\omega }}_s=\hat{\omega }$
County	(11)${\hat{\phi }}_c=\frac{1000D_c}{{\hat{\iota }}_c}$	(10)$\frac{C_c}{{\hat{\iota }}_c}+{\left(1-\frac{{\hat{T}}_c-C_c}{P_c-{\hat{\iota }}_c}\right)}^{{\hat{\omega }}_c}=1$	(1)${\hat{T}}_c$ (linear model)	(9)${\hat{\omega }}_c=\hat{\omega }$