. Author manuscript; available in PMC: 2019 Mar 14.

Published in final edited form as: Comput Stat. 2018 May 15;33(4):1589–603. doi: 10.1007/s00180-018-0813-z

Table 5.

Relative biases (“R.B.”), standard errors (“S.E.”) and mean squared errors (“MSE”) of the estimated coefficients at quantile levels 0.1 and 0.5 from 200 Monte-Carlo simulations in Model (7)

Quantile levels			M = 10		M = 20		M = 50		M = 100
Quantile levels			0.1	0.5	0.1	0.5	0.1	0.5	0.1	0.5
S1–1	R.B. (%)	${\hat{β}}_{1, F I}$	1.000	−2.600	0.300	−1.800	2.400	0.500	2.100	0.400
	R.B. (%)	${\hat{β}}_{2, F I}$	−0.500	2.200	−0.700	1.300	−1.800	−0.200	−1.500	−0.300
	S.E.	${\hat{β}}_{1, F I}$	0.364	0.275	0.366	0.276	0.362	0.283	0.358	0.283
	S.E.	${\hat{β}}_{2, F I}$	0.342	0.253	0.341	0.258	0.344	0.262	0.346	0.260
	MSE	${\hat{β}}_{1, F I}$	0.133	0.076	0.134	0.077	0.131	0.080	0.129	0.080
	MSE	${\hat{β}}_{2, F I}$	0.117	0.064	0.116	0.067	0.119	0.069	0.120	0.068
S1–2	R.B. (%)	${\hat{β}}_{1, F I}$	−2.800	−12.800	−3.600	−13.400	−4.100	−13.600	−4.400	14.200
	R.B. (%)	${\hat{β}}_{2, F I}$	4.700	7.600	5.900	9.700	6.600	10.200	7.000	11.500
	S.E.	${\hat{β}}_{1, F I}$	4.700	0.248	0.069	0.265	0.079	0.277	0.085	0.282
	S.E.	${\hat{β}}_{2, F I}$	0.078	0.240	0.098	0.247	0.110	0.253	0.115	0.259
	MSE	${\hat{β}}_{1, F I}$	0.003	0.078	0.006	0.088	0.008	0.095	0.009	0.100
	MSE	${\hat{β}}_{2, F I}$	0.008	0.063	0.013	0.070	0.016	0.074	0.018	0.100

Relative bias is defined as the ratio between the bias and the true value. Here S1–1 means the missingness is independent with Y and e_i is normal. S1–2 means the missingness is independent with Y and e_i is chi-square. FI stands for the proposed fast imputation algorithm. The estimated coefficients at quantile level 0.9 are just similar to the case at quantile level 0.1