Multiply robust estimation for average treatment effect among treated

Figures & data

Table 1. Comparison of different estimators based on 1000 replications and n = 300. Each digit of the four-digit number inside an estimator's name, from left to right, indicates if ${\pi }^{\left(1\right)}\left({\mathit{\alpha }}^{\left(1\right)}\right)$, ${\pi }^{\left(2\right)}\left({\mathit{\alpha }}^{\left(2\right)}\right)$, $d_0^{\left(1\right)}\left({\mathit{\gamma }}^{\left(1\right)}\right)$ and $d_0^{\left(2\right)}\left({\mathit{\gamma }}^{\left(2\right)}\right)$ are used, respectively. The results have been multiplied by 100.

	${\pi }^{\left(1\right)}$						${\pi }^{\left(2\right)}$
	$(d_0^{\left(1\right)},d_1^{\left(1\right)})$			$(d_0^{\left(2\right)},d_1^{\left(2\right)})$			$(d_0^{\left(1\right)},d_1^{\left(1\right)})$			$(d_0^{\left(2\right)},d_1^{\left(2\right)})$
	rbias	rmse	mae	rbias	rmse	mae	rbias	rmse	mae	rbias	rmse	mae
IPW-1000	−6	99	60	−9	79	53	11	38	26	10	38	26
IPW-0100	69	93	64	−42	68	43	−1	44	28	−1	37	24
OR-0010	−8	57	37	−134	99	68	−2	35	23	−25	47	32
OR-0001	224	131	114	−7	51	34	−19	50	34	−2	36	24
AIPW-1010	−9	65	42	−16	80	54	−2	35	23	−34	51	39
AIPW-1001	−3	85	56	−8	65	41	34	53	38	−2	35	23
AIPW-0110	−9	62	40	−69	82	53	−2	35	23	−2	40	25
AIPW-0101	63	84	56	−8	61	39	−1	42	27	−2	35	24
MR-1000	−1	71	46	−10	76	50	14	40	27	12	39	26
MR-0100	51	77	52	−59	73	48	−1	49	32	−1	39	25
MR-0010	−8	57	37	−158	108	80	−2	35	23	−30	49	35
MR-0001	229	134	116	−6	52	34	21	47	32	−2	36	24
MR-1100	−24	76	49	−56	86	52	−3	39	26	−8	41	27
MR-1010	−10	65	42	−17	73	50	−1	35	23	−27	46	32
MR-1001	−1	73	50	−9	65	42	29	51	35	−2	35	23
MR-0110	−10	64	41	−68	80	52	−2	35	24	−2	38	24
MR-0101	60	79	53	−8	61	39	−2	40	26	−2	35	24
MR-0011	−13	83	51	−12	80	43	−2	35	24	−1	38	26
MR-1110	−9	65	42	−22	72	47	−2	36	24	−1	36	25
MR-1101	4	70	46	−9	65	42	−3	38	26	−2	35	24
MR-1011	−11	73	45	−10	73	44	−2	36	24	−2	36	24
MR-0111	−11	77	47	−10	78	48	−1	36	24	−2	35	24
MR-1111	−10	75	46	−10	75	46	−2	36	24	−2	35	24
EB-1	523	282	266	352	194	177	−317	321	311	−275	279	272
EB-2	−11	65	43	−19	71	46	−2	35	23	−31	49	36
MR-1	231	166	128	197	136	107	−318	323	313	−276	280	272
MR-2	−12	91	56	57	99	64	−1	34	23	−36	52	39

${\pi }^{\left(1\right)}$: $\pi \left(X\right)=\{1+\exp (0.8+0.5X-0.3X^2){\}}^{-1}$. ${\pi }^{\left(2\right)}$: $\pi \left(X\right)=1-\exp [-\exp \{0.5+0.5X-0.3\exp \left(X\right)\left\}\right]$. $(d_0^{\left(1\right)},d_1^{\left(1\right)})$: $\left\{d_0\right(X),d_1(X\left)\right\}=(1+2X+3X^2,2+3X+3X^2)$. $(d_0^{\left(2\right)},d_1^{\left(2\right)})$: $\left\{d_0\right(X),d_1(X\left)\right\}=\{1+2X+3\exp (X),2+3X+3\exp (X\left)\right\}$. rbias: relative bias, bias divided by the true value. rmse: root mean square error. mae: median absolute error. IPW: inverse probability weighting. OR: outcome regression. AIPW: augmented inverse probability weighting. MR: multiply robust. EB: entropy balancing. EB-1: EB with $\mathit{h}\left(X\right)=X$. EB-2: EB with $\mathit{h}\left(X\right)=(X,X^2)$. MR-1: MR with $\mathit{h}\left(X\right)=X$ and no working models. MR-2: MR with $\mathit{h}\left(X\right)=(X,X^2)$ and no working models.

Table 2. Comparison of different estimators based on 1000 replications and n = 800. Each digit of the four-digit number inside an estimator's name, from left to right, indicates if ${\pi }^{\left(1\right)}\left({\mathit{\alpha }}^{\left(1\right)}\right)$, ${\pi }^{\left(2\right)}\left({\mathit{\alpha }}^{\left(2\right)}\right)$, $d_0^{\left(1\right)}\left({\mathit{\gamma }}^{\left(1\right)}\right)$ and $d_0^{\left(2\right)}\left({\mathit{\gamma }}^{\left(2\right)}\right)$ are used, respectively. The results have been multiplied by 100.

	${\pi }^{\left(1\right)}$						${\pi }^{\left(2\right)}$
	$(d_0^{\left(1\right)},d_1^{\left(1\right)})$			$(d_0^{\left(2\right)},d_1^{\left(2\right)})$			$(d_0^{\left(1\right)},d_1^{\left(1\right)})$			$(d_0^{\left(2\right)},d_1^{\left(2\right)})$
	rbias	rmse	mae	rbias	rmse	mae	rbias	rmse	mae	rbias	rmse	mae
IPW-1000	2	58	39	1	46	32	14	26	18	13	26	17
IPW-0100	84	66	50	−32	41	28	0	27	18	0	23	15
OR-0010	2	35	25	−127	76	62	0	22	15	−24	34	26
OR-0001	244	126	119	2	32	22	−18	34	23	0	23	15
AIPW-1010	3	40	27	0	46	30	0	22	14	−33	40	33
AIPW-1001	4	52	35	3	40	27	37	44	35	0	22	15
AIPW-0110	3	38	26	−57	52	36	0	22	15	0	25	16
AIPW-0101	75	60	43	3	37	25	0	26	17	0	22	15
MR-1000	5	44	30	1	46	32	16	28	19	15	27	18
MR-0100	63	54	39	−50	46	32	0	30	19	0	24	16
MR-0010	2	35	25	−152	86	75	0	22	14	−28	37	29
MR-0001	246	127	120	2	32	22	24	35	26	0	23	15
MR-1100	−6	46	32	−18	49	32	−1	24	16	−3	24	15
MR-1010	3	40	28	0	43	30	0	22	15	−26	34	26
MR-1001	5	46	31	3	40	28	32	41	32	0	22	15
MR-0110	3	39	28	−56	51	35	0	22	15	0	23	15
MR-0101	73	57	41	3	37	25	0	25	16	0	22	15
MR-0011	5	55	36	8	68	30	0	22	15	1	24	15
MR-1110	3	40	27	−1	42	28	0	22	15	0	22	15
MR-1101	6	44	30	3	40	28	0	24	15	0	22	15
MR-1011	4	42	29	4	42	29	0	22	15	0	22	15
MR-0111	4	46	31	7	61	31	0	22	15	0	22	15
MR-1111	4	42	29	4	42	29	0	22	15	0	22	15
EB-1	564	281	277	383	193	189	−322	316	314	−279	274	271
EB-2	2	40	28	−1	42	29	0	22	15	−30	38	29
MR-1	251	142	126	220	121	112	−323	316	314	−279	274	272
MR-2	2	73	42	92	81	52	0	21	14	−35	41	34

${\pi }^{\left(1\right)}$: $\pi \left(X\right)=\{1+\exp (0.8+0.5X-0.3X^2){\}}^{-1}$. ${\pi }^{\left(2\right)}$: $\pi \left(X\right)=1-\exp [-\exp \{0.5+0.5X-0.3\exp \left(X\right)\left\}\right]$. $(d_0^{\left(1\right)},d_1^{\left(1\right)})$: $\left\{d_0\right(X),d_1(X\left)\right\}=(1+2X+3X^2,2+3X+3X^2)$. $(d_0^{\left(2\right)},d_1^{\left(2\right)})$: $\left\{d_0\right(X),d_1(X\left)\right\}=\{1+2X+3\exp (X),2+3X+3\exp (X\left)\right\}$. rbias: relative bias, bias divided by the true value. rmse: root mean square error. mae: median absolute error.IPW: inverse probability weighting. OR: outcome regression. AIPW: augmented inverse probability weighting. MR: multiply robust. EB: entropy balancing. EB-1: EB with $\mathit{h}\left(X\right)=X$. EB-2: EB with $\mathit{h}\left(X\right)=(X,X^2)$. MR-1: MR with $\mathit{h}\left(X\right)=X$ and no working models. MR-2: MR with $\mathit{h}\left(X\right)=(X,X^2)$ and no working models.

Table 3. Performance of the bootstrap method in calculating the standard errors for the proposed estimators in the setting of $\pi \left(X\right)=\{1+\exp (0.8+0.5X-0.3X^2){\}}^{-1}$ and $\left\{d_0\right(X),d_1(X\left)\right\}=(1+2X+3X^2,2+3X+3X^2)$. Results are summarized based on 1000 replications with n = 300 and the bootstrap resampling size is 200. Each digit of the four-digit number inside an estimator's name, from left to right, indicates if ${\pi }^{\left(1\right)}\left({\mathit{\alpha }}^{\left(1\right)}\right)$, ${\pi }^{\left(2\right)}\left({\mathit{\alpha }}^{\left(2\right)}\right)$, $d_0^{\left(1\right)}\left({\mathit{\gamma }}^{\left(1\right)}\right)$ and $d_0^{\left(2\right)}\left({\mathit{\gamma }}^{\left(2\right)}\right)$ are used, respectively.

	bias	se-emp	se-bp	cp-$95\mathrm{\%}$
MR-1000	0.05	0.71	0.67	92.50
MR-0100	0.31	0.72	0.69	89.00
MR-0010	−0.01	0.56	0.58	95.30
MR-0001	1.21	0.65	0.64	52.60
MR-1100	−0.08	0.77	0.74	94.10
MR-1010	0.01	0.64	0.62	93.60
MR-1001	0.05	0.74	0.70	92.40
MR-0110	0.00	0.62	0.61	94.00
MR-0101	0.36	0.74	0.70	87.50
MR-0011	0.03	0.81	0.75	93.30
MR-1110	0.01	0.63	0.61	93.60
MR-1101	0.08	0.70	0.67	92.00
MR-1011	0.03	0.72	0.70	94.40
MR-0111	0.03	0.76	0.73	93.90
MR-1111	0.04	0.76	0.73	94.10

se-emp: empirical standard error. se-bp: averaged bootstrap standard error. cp-$95\mathrm{\%}$: coverage percentage of the $95\mathrm{\%}$ confidence interval constructed based on the bootstrap standard error. MR: multiply robust.