. Author manuscript; available in PMC: 2020 Jan 15.

Published in final edited form as: Stat Med. 2018 Sep 12;38(1):100–114. doi: 10.1002/sim.7966

TABLE 2.

Estimate, bias, variance, median of estimated variance $(Med (\hat{Var}))$ and coverage of 95% confidence interval (CI) of AUC and pAUC (t₀, t₁) estimators using the empirical method, IPW method with the estimated sampling probabilities ${\hat{p}}^{Str}$ and ${\hat{p}}^{Aux}$ , for scenarios where ρ_XV* = 0.5, ρ_XW = 0.5, ρ_WV* = 0.1, μ_{X_D̄} = 0, and μ_{X_D} = 1 in finite-population stratified sampling.

Method	n_D = n_D̄	Estimate	Bias × 100	Var × N	$Med (\hat{Var}) \times N$	Coverage of 95% CI
AUC = 0.7602
${\hat{AUC}}_{em}$	100	0.7258	−3.4428	2.1767	2.4639	83.7%
	250	0.7265	−3.3782	2.1105	2.4791	65.8%
	500	0.7264	−3.3857	2.2695	2.4807	38.7%
${\hat{AUC}}_{IPW}^{Str}$	100	0.7597	−0.0564	2.0841	2.1019	95.1%
	250	0.7604	0.0105	2.0643	2.0993	95.5%
	500	0.7603	0.0057	2.2095	2.1025	94.3%
${\hat{AUC}}_{IPW}^{Aux}$	100	0.7600	−0.0244	1.6382	1.6100	95.0%
	250	0.7605	0.0223	1.6015	1.6149	94.9%
	500	0.7603	0.0014	1.7089	1.6146	94.5%
pAUC(0,0.1) = 0.0244
$p \hat{A U} C_{em}$	100	0.0198	−0.4535	0.0632	0.0685	95.0%
	250	0.0203	−0.4043	0.0663	0.0695	87.8%
	500	0.0204	−0.4002	0.0654	0.0701	75.5%
$p \hat{A U} C_{IPW}^{Str}$	100	0.0230	−0.1351	0.0688	0.0642	94.9%
	250	0.0236	−0.0721	0.0716	0.0681	94.8%
	500	0.0238	−0.0566	0.0711	0.0695	94.7%
$p \hat{A U} C_{IPW}^{Aux}$	100	0.0229	−0.1464	0.0629	0.0613	95.4%
	250	0.0236	−0.0713	0.0630	0.0641	95.5%
	500	0.0238	−0.0591	0.0618	0.0656	95.5%
pAUC(0,0.2) = 0.0726
$p \hat{A U} C_{em}$	100	0.0619	−1.0735	0.2692	0.2975	92.6%
	250	0.0625	−1.0053	0.2787	0.2990	81.2%
	500	0.0626	−0.9949	0.2785	0.3016	62.8%
$p \hat{A U} C_{IPW}^{Str}$	100	0.0698	−0.2757	0.2874	0.2728	94.5%
	250	0.0711	−0.1515	0.2899	0.2809	95.0%
	500	0.0714	−0.1212	0.2965	0.2841	94.6%
$p \hat{A U} C_{IPW}^{Aux}$	100	0.0696	−0.2949	0.2489	0.2544	95.9%
	250	0.0711	−0.1496	0.2478	0.2615	95.8%
	500	0.0713	−0.1253	0.2459	0.2634	95.5%