Table 1.

Relative bias (RB), standard error (SE) and mean squared error (MSE) of the estimated quantile coefficients under Model (2) at quantile levels of 0.5 and 0.9. In Model (2), x_i = u_i,1 + u_i,2 where u_i,1 and u_i,2 are iid bernoulli random variables with p = 0.3, z_i ~ N(0, 1), and $e_i\sim {\chi }_1^2/\sqrt{2}$ “‘QR_controls” stands for unadjusted quantile regression using controls only. “QR_cases” stands for unadjusted quantile regression using cases only. “QR_controls” are unadjusted quantile regression using both case and control samples. IPW is the estimates using inverse probability weighting; “KS” is the KS estimates using kernel smoothing. SICO(m) is the SICO estimates with m replicate.

n	Methods	τ = 0.5			τ = 0.9
		RB (%)	SE	MSE × n	RB (%)	SE	MSE × n
2000	QR_controls	−1.7	0.025	1.3	−22.3	0.102	23.2
	QR_cases	−28.8	0.086	17.4	−111.9	0.184	130.0
	QR_case-control	39.1	0.035	7.3	50.6	0.141	52.6
	IP	2.0	0.025	1.3	0.2	0.099	19.6
	KS	2.2	0.026	1.4	−0.9	0.100	19.8
	SICO (m=1)	2.7	0.028	1.6	0.0	0.108	23.1
	SICO (m=10)	2.0	0.026	1.3	0.2	0.098	19.3
	SICO (m=100)	2.1	0.025	1.3	0.5	0.096	18.5
500	QR_controls	−2.5	0.048	1.2	−19.0	0.197	19.8
	QR_cases	−34.7	0.184	17.9	−102.2	0.372	82.1
	QR_case-control	36.6	0.072	3.7	57.9	0.291	46.6
	IPW	1.3	0.049	1.2	4.3	0.201	20.2
	KS	2.6	0.051	1.3	5.0	0.204	20.7
	SICO (m=1)	2.2	0.054	1.5	8.0	0.224	25.1
	SICO (m=10)	1.5	0.048	1.2	3.5	0.198	19.5
	SICO (m=100)	1.6	0.048	1.1	2.8	0.194	18.9