. Author manuscript; available in PMC: 2017 May 10.

Published in final edited form as: Stat Med. 2015 Dec 2;35(10):1637–1653. doi: 10.1002/sim.6820

Table 1.

Performance of the proposed ( ${\hat{R}}_{S}$ ) and model-based estimators ( ${\hat{R}}_{S}^{M}$ and ${\hat{R}}_{S}^{F}$ ) in terms of bias, empirical standard deviation (ESD), average standard deviation (ASD) obtained using the proposed perturbation-resampling approach, mean squared error (MSE) and coverage of the 95% confidence intervals obtained using both the quantiles of the perturbed estimates (Quantile) and Fieller's method (Fieller), in Setting (i) where there is a single surrogate marker and the models assumed by the flexible model-based estimator, ${\hat{R}}_{S}^{M}$ , are correctly specified, and Setting (ii) where there is a single surrogate marker and the specified models are not correct with n_T = n_S = 200 and with n_T = n_S = 1000.

Setting (i) Model Correctly Specified
n_T = 200, n_C = 200
	R_S = 0.2			R_S = 0.5			R_S = 0.9

	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$

Bias	−0.0023	−0.0005	−0.0072	−0.0066	−0.0021	−0.0189	−0.0072	0.0009	−0.0295
ESD	0.0260	0.0259	0.0251	0.0457	0.0445	0.0432	0.0559	0.0506	0.0488
ASD	0.0262	0.0258	0.0250	0.0466	0.0450	0.0434	0.0561	0.0513	0.0494
MSE	0.0007	0.0007	0.0007	0.0021	0.0020	0.0022	0.0032	0.0026	0.0033
Coverage (Quantile)	0.946	0.941	0.938	0.945	0.941	0.914	0.926	0.942	0.885
Coverage (Fieller)	0.951	0.947	0.942	0.946	0.947	0.915	0.934	0.944	0.885

n_T = 1000, n_C = 1000
	R_S = 0.2			R_S = 0.5			R_S = 0.9

	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$

Bias	−0.0003	0.0003	−0.0065	−0.0014	−0.0001	−0.0169	−0.0031	−0.0007	−0.031
ESD	0.0118	0.0115	0.0111	0.0207	0.0196	0.0188	0.0241	0.0214	0.0205
ASD	0.0117	0.0115	0.0111	0.0203	0.0196	0.0189	0.0238	0.0216	0.0208
MSE	0.0001	0.0001	0.0002	0.0004	0.0004	0.0006	0.0006	0.0005	0.0014
Coverage (Quantile)	0.948	0.947	0.899	0.939	0.946	0.840	0.936	0.934	0.676
Coverage (Fieller)	0.948	0.950	0.900	0.940	0.951	0.835	0.941	0.941	0.682

Setting (ii) Model Misspecified
n_T = 200, n_C = 200
	R_S = 0.2			R_S = 0.5			R_S = 0.9

	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$

Bias	−0.0022	−0.0141	−0.0241	−0.0064	−0.0364	−0.0612	−0.007	−0.0629	−0.0844
ESD	0.0250	0.0290	0.0275	0.0404	0.0521	0.0492	0.0161	0.0436	0.0404
ASD	0.0255	0.0292	0.0277	0.0417	0.0525	0.0497	0.0176	0.0441	0.0414
MSE	0.0006	0.0010	0.0013	0.0017	0.0040	0.0062	0.0003	0.0059	0.0088
Coverage (Quantile)	0.946	0.912	0.842	0.948	0.883	0.737	0.937	0.562	0.246
Coverage (Fieller)	0.947	0.908	0.832	0.947	0.873	0.715	0.933	0.537	0.224

n_T = 1000, n_C = 1000
	R_S = 0.2			R_S = 0.5			R_S = 0.9

	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$	R̂_S	${\hat{R}}_{S}^{M}$	${\hat{R}}_{S}^{F}$

Bias	−0.0008	−0.0134	−0.0234	−0.0022	−0.0335	−0.0585	−0.0018	−0.0587	−0.0805
ESD	0.0116	0.0133	0.0126	0.0185	0.0236	0.0223	0.0071	0.0192	0.0177
ASD	0.0116	0.0133	0.0126	0.0185	0.0236	0.0224	0.0072	0.0191	0.0177
MSE	0.0001	0.0004	0.0007	0.0003	0.0017	0.0039	0.0001	0.0038	0.0068
Coverage (Quantile)	0.950	0.826	0.528	0.944	0.681	0.214	0.936	0.079	0.005
Coverage (Fieller)	0.951	0.818	0.521	0.948	0.673	0.201	0.942	0.067	0.002