Table I.

Sensitivity and specificity functions for classifying top 100(1 − c)% of hospitals.

Method	Accuracy measures
DIR	Cutoff equation: ${\sum }_i\mathrm{\Phi }\left(\frac{C_{\text{DIR}}^{*}-\mu }{\tau /{\rho }_i}\right)=Mc$
	Sensitivity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}>\frac{C_{\text{DIR}}^{*}-\mu }{\tau /{\rho }_i},Z_{2i}>{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(M(1-c))$
	Specificity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}<\frac{C_{\text{DIR}}^{*}-\mu }{\tau /{\rho }_i},Z_{2i}<{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(Mc)$
SHR	Cutoff equation: ${\sum }_i\mathrm{\Phi }\left(\frac{C_{\text{SHR}}^{*}-\mu }{\tau {\rho }_i}\right)=Mc$
	Sensitivity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}>\frac{C_{\text{SHR}}^{*}-\mu }{\tau {\rho }_i},Z_{2i}>{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(M(1-c))$
	Specificity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}<\frac{C_{\text{SHR}}^{*}-\mu }{\tau {\rho }_i},Z_{2i}<{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(Mc)$
PROB1	Cutoff equation: ${\sum }_i\mathrm{\Phi }\left(\frac{C_{\text{prob}}^{*}-\left(\mu -{\mathrm{\Phi }}^{-1}\left(P_{\text{prob}}\right){\rho }_i\sqrt{\frac{{\sigma }^2}{n_i}}\right)}{\tau {\rho }_i}\right)=Mc$
	Sensitivity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}>\frac{C_{\text{prob}}^{*}-\left(\mu -{\mathrm{\Phi }}^{-1}\left(P_{\text{prob}}\right){\rho }_i\sqrt{\frac{{\sigma }^2}{n_i}}\right)}{\tau {\rho }_i},Z_{2i}>{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(M(1-c))$
	Specificity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}<\frac{C_{\text{prob}}^{*}-\left(\mu -{\mathrm{\Phi }}^{-1}\left(P_{\text{prob}}\right){\rho }_i\sqrt{\frac{{\sigma }^2}{n_i}}\right)}{\tau {\rho }_i},Z_{2i}<{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(Mc)$
PROB2	Cutoff equation: ${\sum }_i\mathrm{\Phi }\left(\frac{C_{\text{prob}}-\left(\mu -{\mathrm{\Phi }}^{-1}{\left(P_{\text{prob}}\right)}^{*}{\rho }_i\sqrt{\frac{{\sigma }^2}{n_i}}\right)}{\tau {\rho }_i}\right)=Mc$
	Sensitivity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}>\frac{C_{\text{prob}}-\left(\mu -{\mathrm{\Phi }}^{-1}{\left(P_{\text{prob}}\right)}^{*}{\rho }_i\sqrt{\frac{{\sigma }^2}{n_i}}\right)}{\tau {\rho }_i},Z_{2i}>{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(M(1-c))$
	Specificity: ${\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}<\frac{C_{\text{prob}}-\left(\mu -{\mathrm{\Phi }}^{-1}{\left(P_{\text{prob}}\right)}^{*}{\rho }_i\sqrt{\frac{{\sigma }^2}{n_i}}\right)}{\tau {\rho }_i},Z_{2i}<{\mathrm{\Phi }}^{-1}(c),{\rho }_i\right)/(Mc)$

Note: ${\rho }_i=\sqrt{\frac{{\tau }^2}{{\tau }^2+{\sigma }^2/n_i}}$, $C_{\text{DIR}}^{*}$, $C_{\text{SHR}}^{*}$, $C_{\text{prob}}^{*}$ and ${\mathrm{\Phi }}^{-1}{\left(P_{\text{prob}}\right)}^{*}$ are the cutoff points.

DIR, direct method; SHR, shrinkage method.