Table A1.

Sensitivity, specificity, and probability of correct classification for classifying hospitals using the external threshold c1.

Method	Accuracy measures
DIR	Sensitivity: $\frac{1}{M\mathrm{\Phi }\left(\frac{\mu -c_1}{\tau }\right)}{\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}>\frac{c_1-\mu }{\tau /{\rho }_i},Z_{2i}>\frac{c_1-\mu }{\tau },{\rho }_i\right)$
	Specificity: $\frac{1}{M\mathrm{\Phi }\left(\frac{\mu -c_1}{\tau }\right)}{\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}<\frac{c_1-\mu }{\tau /{\rho }_i},Z_{2i}<\frac{c_1-\mu }{\tau },{\rho }_i\right)$
	PCC: $\frac{1}{M}\sum _i{\text{Φ}}_2\left(Z_{1i}>\frac{c_1-\mu }{\tau /{\rho }_i},Z_{2i}>\frac{c_1-\mu }{\tau },{\rho }_i\right)+{\text{Φ}}_2\left(Z_{1i}<\frac{c_1-\mu }{\tau /{\rho }_i},Z_{2i}<\frac{c_1-\mu }{\tau },{\rho }_i\right)$
SHR	Sensitivity: $\frac{1}{M\mathrm{\Phi }\left(\frac{\mu -c_1}{\tau }\right)}{\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}>\frac{c_1-\mu }{\tau /{\rho }_i},Z_{2i}>\frac{c_1-\mu }{\tau },{\rho }_i\right)$
	Specificity: $\frac{1}{M\mathrm{\Phi }\left(\frac{\mu -c_1}{\tau }\right)}{\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}<\frac{c_1-\mu }{\tau /{\rho }_i},Z_{2i}<\frac{c_1-\mu }{\tau },{\rho }_i\right)$
	PCC: $\frac{1}{M}{\sum }_i{\mathrm{\Phi }}_2\left(Z_{1i}>\frac{c_1-\mu }{\tau /{\rho }_i},Z_{2i}>\frac{c_1-\mu }{\tau },{\rho }_i\right)+{\mathrm{\Phi }}_2\left(Z_{1i}<\frac{c_1-\mu }{\tau /{\mathcal{l}}_i},Z_{2i}<\frac{c_1-\mu }{\tau },{\rho }_i\right)$

DIR, direct method; SHR, shrinkage method; PCC, probability of correct classification.