. 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 3.

Mean biases (“M.B.”), standard errors (“S.E.”) and mean squared errors (“MSE”) of the estimated coefficients from various methods at quantile levels 0.1, 0.5 and 0.9 under the setting S2–2 with sample size 500 and 200 Monte-Carlo replicates

Quantile levels		M.B.			S.E.			MSE
Quantile levels		0.1	0.5	0.9	0.1	0.5	0.9	0.1	0.5	0.9
x	${\hat{β}}_{1, F I}$	−0.009	−0.123	−1.968	0.024	0.266	0.998	0.001	0.086	4.868
	${\hat{β}}_{1, F I I P W}$	−0.007	−0.109	−1.333	0.023	0.286	1.854	0.001	0.094	5.213
	${\hat{β}}_{1, F I P}$	−0.008	−0.098	−1.280	0.023	0.296	1.844	0.001	0.097	5.037
	${\hat{β}}_{1, I P W}$	0.005	0.078	−1.032	0.075	1.657	2.483	0.006	2.752	7.231
	${\hat{β}}_{1, M I}$	−0.004	−0.086	−1.918	0.019	0.264	0.901	0.000	0.077	4.488
	${\hat{β}}_{1, M I I P W}$	−0.005	−0.075	−1.200	0.021	0.287	1.989	0.000	0.088	5.396
	${\hat{β}}_{1, M I P}$	−0.006	−0.087	−1.247	0.022	0.290	2.010	0.001	0.092	5.595
z	${\hat{β}}_{2, F I}$	0.008	0.071	0.896	0.023	0.212	1.011	0.001	0.050	1.826
	${\hat{β}}_{2, F I I P W}$	0.006	0.058	0.423	0.022	0.218	1.210	0.001	0.051	1.643
	${\hat{β}}_{2, F I P}$	0.007	0.052	0.434	0.022	0.220	1.191	0.001	0.051	1.608
	${\hat{β}}_{2, I P W}$	−0.002	−0.100	−0.654	0.079	1.384	2.452	0.001	1.926	6.440
	${\hat{β}}_{2, M I}$	0.003	0.057	0.882	0.019	0.209	0.903	0.000	0.047	1.593
	${\hat{β}}_{2, M I I P W}$	0.005	0.058	0.390	0.021	0.213	1.309	0.000	0.049	1.865
	${\hat{β}}_{2, M I P}$	0.005	0.062	0.433	0.021	0.217	1.313	0.000	0.051	1.912

FI stands for the proposed fast imputation algorithm, IPW stands for inverse probability weighting, FIIPW is the FI algorithm with IPW adjustment, FIP stands for the fast imputation algorithm using true weights calculated from the true density f (y|x, z); Likewise, MI, MIIPW and MIP respectively stand for the multiple imputation algorithm in Wei et al. (2012), and its adjustment with IPW and true weights