Table 2.

Performance of the proposed (${\widehat{R}}_{\mathbf{S}}$) and model-based estimators (${\widehat{R}}_{\mathbf{S}}^M$ and ${\widehat{R}}_{\mathbf{S}}^F$) 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 (iii) where there are three surrogate markers and the models assumed by the flexible model-based estimator, ${\widehat{R}}_{\mathbf{S}}^M$, are correctly specified, and Setting (iv) where there are three surrogate markers and the specified models are not correct with n_T = n_S = 200 and with n_T = n_S = 1000.

Setting (iii) Model Correctly Specified
nT = 200, nC = 200
	RS = 0.20			RS = 0.50			RS = 0.90
	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$
Bias	−0.0023	−0.0003	−0.0045	−0.0049	0.000	−0.0103	−0.0047	0.0041	−0.0014
ESD	0.0201	0.0189	0.0182	0.0423	0.0385	0.0367	0.0846	0.0772	0.0745
ASD	0.0200	0.0191	0.0183	0.0412	0.0389	0.0367	0.0814	0.0781	0.0746
MSE	0.0004	0.0004	0.0003	0.0018	0.0015	0.0015	0.0072	0.006	0.0055
Coverage (Quantile)	0.946	0.952	0.937	0.931	0.946	0.926	0.924	0.943	0.934
Coverage (Fieller)	0.950	0.951	0.941	0.936	0.949	0.932	0.929	0.942	0.938

nT = 1000, nC = 1000
	RS = 0.20			RS = 0.50			RS = 0.90
	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$
Bias	−0.0001	0.0004	−0.0039	−0.0001	0.0011	−0.0096	0.0005	0.0027	−0.0036
ESD	0.0091	0.0086	0.0081	0.0190	0.0175	0.0164	0.0364	0.0338	0.0323
ASD	0.0089	0.0085	0.0082	0.0184	0.0173	0.0163	0.0357	0.0338	0.0322
MSE	0.0001	0.0001	0.0001	0.0004	0.0003	0.0004	0.0013	0.0011	0.0011
Coverage (Quantile)	0.948	0.947	0.922	0.945	0.946	0.907	0.948	0.949	0.943
Coverage (Fieller)	0.950	0.948	0.923	0.948	0.947	0.908	0.948	0.950	0.945

Setting (iv) Model Misspecified
nT = 200, nC = 200
	RS = 0.20			RS = 0.50			RS = 0.90
	RQ = 0.206		RF = 0.211	RQ = 0.523		RF = 0.546	RQ = 0.924		RF = 0.926
	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$
Bias	−0.0031	0.0190	−0.0169	−0.0095	0.0462	−0.0941	−0.0138	0.0887	−0.0850
ESD	0.0259	0.0360	0.0283	0.0458	0.0742	0.0528	0.0298	0.0952	0.0664
ASD	0.0260	0.0351	0.0277	0.0463	0.0720	0.0517	0.0313	0.0934	0.0655
MSE	0.0007	0.0017	0.0011	0.0022	0.0076	0.0116	0.0011	0.0169	0.0116
Coverage (Quantile)	0.943	0.909	0.888	0.935	0.894	0.501	0.912	0.826	0.699
Coverage(Fieller)	0.941	0.924	0.884	0.936	0.907	0.480	0.944	0.847	0.704

nT = 1000, nC = 1000
	RS = 0.20			RS = 0.50			RS = 0.90
	RQ = 0.206		RF = 0.211	RQ = 0.523		RF = 0.546	RQ = 0.924		RF = 0.926
	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$	R̂ S	${\widehat{R}}_{\mathbf{S}}^M$	${\widehat{R}}_{\mathbf{S}}^F$
Bias	−0.0008	0.0186	−0.0171	−0.0024	0.0468	−0.0930	−0.0035	0.0875	−0.0859
ESD	0.0114	0.0155	0.0121	0.0196	0.0316	0.0224	0.0125	0.0411	0.0289
ASD	0.0117	0.0159	0.0125	0.0203	0.0323	0.0232	0.0126	0.0411	0.0289
MSE	0.0001	0.0006	0.0004	0.0004	0.0032	0.0091	0.0002	0.0094	0.0082
Coverage (Quantile)	0.956	0.788	0.719	0.960	0.701	0.012	0.926	0.443	0.142
Coverage(Fieller)	0.957	0.808	0.714	0.960	0.715	0.010	0.939	0.464	0.134