Quantile treatment effect estimation with dimension reduction

Figures & data

Table 1. Relative bias and standard deviation for simulation 1 with true or estimated $S_0$ and $S_1$.

		IPW				REG				AIPW
		Estimated		True		Estimated		True		Estimated		True
	$S_0,S_1$	RB	SD	RB	SD	RB	SD	RB	SD	RB	SD	RB	SD
n = 200
$\theta =9.6$	$S_T,S_T$	0.11	1.33	0.09	1.77	0.11	1.38	0.08	1.84	0.09	1.46	0.05	2.04
$\tau =0.25$	$S_{Y_0,T},S_{Y_1,T}$	0.09	1.18	0.10	1.08	0.06	1.19	0.08	1.03	0.04	1.26	0.06	1.04
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	0.09	1.09	0.10	1.08	0.09	1.09	0.08	1.03	0.07	1.07	0.06	1.04
	$S_{Y_0},S_{Y_1}$	0.06	0.96	0.05	0.87	0.05	0.91	0.04	0.85	0.02	0.84	0.01	0.76
$\theta =10.0$	$S_T,S_T$	0.12	1.14	0.08	1.51	0.09	1.19	0.06	1.57	0.08	1.25	0.04	1.72
$\tau =0.5$	$S_{Y_0,T},S_{Y_1,T}$	0.10	0.97	0.10	0.92	0.07	1.01	0.07	0.95	0.05	1.00	0.05	0.89
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	0.11	0.94	0.10	0.92	0.08	1.01	0.07	0.95	0.06	0.91	0.05	0.89
	$S_{Y_0},S_{Y_1}$	0.06	0.75	0.04	0.68	0.04	0.82	0.03	0.75	0.01	0.71	0.01	0.63
$\theta =10.4$	$S_T,S_T$	0.10	1.29	0.07	1.63	0.07	1.32	0.05	1.70	0.06	1.35	0.03	1.82
$\tau =0.75$	$S_{Y_0,T},S_{Y_1,T}$	0.08	1.12	0.09	1.08	0.06	1.10	0.06	1.01	0.03	1.05	0.04	0.97
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	0.09	1.10	0.09	1.08	0.06	1.06	0.06	1.01	0.05	0.98	0.04	0.97
	$S_{Y_0},S_{Y_1}$	0.05	0.83	0.03	0.75	0.03	0.86	0.02	0.81	0.01	0.74	0.01	0.68
n = 1000
$\theta =9.6$	$S_T,S_T$	0.07	0.73	0.06	0.94	0.04	0.83	0.04	1.03	0.02	0.95	0.02	1.17
$\tau =0.25$	$S_{Y_0,T},S_{Y_1,T}$	0.08	0.50	0.09	0.50	0.06	0.52	0.06	0.51	0.04	0.52	0.03	0.49
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	0.09	0.49	0.09	0.50	0.05	0.52	0.06	0.51	0.03	0.49	0.03	0.49
	$S_{Y_0},S_{Y_1}$	0.02	0.36	0.02	0.35	0.01	0.37	0.01	0.36	0.00	0.33	0.00	0.32
$\theta =10.0$	$S_T,S_T$	0.05	0.60	0.04	0.77	0.03	0.66	0.02	0.84	0.01	0.74	0.01	0.92
$\tau =0.5$	$S_{Y_0,T},S_{Y_1,T}$	0.07	0.42	0.07	0.42	0.04	0.44	0.04	0.44	0.02	0.42	0.02	0.40
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	0.07	0.42	0.07	0.42	0.03	0.44	0.04	0.44	0.01	0.41	0.02	0.40
	$S_{Y_0},S_{Y_1}$	0.02	0.30	0.02	0.29	0.01	0.31	0.01	0.30	0.00	0.29	0.00	0.27
$\theta =10.4$	$S_T,S_T$	0.04	0.67	0.04	0.80	0.02	0.73	0.02	0.84	0.01	0.80	0.01	0.93
$\tau =0.75$	$S_{Y_0,T},S_{Y_1,T}$	0.06	0.45	0.06	0.44	0.03	0.46	0.03	0.45	0.02	0.44	0.02	0.42
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	0.06	0.44	0.06	0.44	0.03	0.45	0.03	0.45	0.01	0.42	0.02	0.42
	$S_{Y_0},S_{Y_1}$	0.01	0.31	0.01	0.30	0.01	0.32	0.01	0.32	0.00	0.30	0.00	0.29

RB: relative bias; SD: standard deviation

Table 2. Relative bias and standard deviation for simulation 2 with true or estimated $S_0$ and $S_1$.

		IPW				REG				AIPW
		Estimated		True		Estimated		True		Estimated		True
	$S_0,S_1$	RB	SD	RB	SD	RB	SD	RB	SD	RB	SD	RB	SD
n = 200
$\theta =12.7$	$S_T,S_T$	−0.07	1.61	−0.06	1.74	−0.06	1.71	−0.04	1.89	−0.05	1.75	−0.03	1.90
$\tau =0.25$	$S_{Y_0,T},S_{Y_1,T}$	−0.06	1.31	−0.07	1.38	−0.05	1.29	−0.06	1.34	−0.03	1.25	−0.05	1.24
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	−0.05	1.25	−0.07	1.36	−0.03	1.22	−0.05	1.31	−0.02	1.15	−0.04	1.19
	$S_{Y_0},S_{Y_1}$	−0.02	1.04	−0.04	1.10	−0.01	1.09	−0.02	1.12	0.01	0.97	−0.01	0.96
$\theta =12.0$	$S_T,S_T$	−0.08	1.47	−0.08	1.64	−0.07	1.54	−0.06	1.79	−0.06	1.58	−0.05	1.81
$\tau =0.5$	$S_{Y_0,T},S_{Y_1,T}$	−0.05	1.16	−0.08	1.27	−0.04	1.16	−0.06	1.22	−0.02	1.16	−0.05	1.15
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	−0.05	1.14	−0.08	1.23	−0.03	1.14	−0.05	1.21	−0.02	1.04	−0.04	1.09
	$S_{Y_0},S_{Y_1}$	−0.02	0.95	−0.04	1.03	−0.01	1.00	−0.02	1.01	0.01	0.90	−0.01	0.88
$\theta =11.3$	$S_T,S_T$	−0.09	1.63	−0.09	1.81	−0.08	1.68	−0.07	1.94	−0.07	1.75	−0.05	2.03
$\tau =0.75$	$S_{Y_0,T},S_{Y_1,T}$	−0.06	1.29	−0.09	1.41	−0.04	1.28	−0.06	1.33	−0.03	1.27	−0.06	1.29
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	−0.06	1.26	−0.08	1.37	−0.04	1.25	−0.06	1.32	−0.02	1.20	−0.05	1.27
	$S_{Y_0},S_{Y_1}$	−0.02	1.16	−0.04	1.23	−0.01	1.10	−0.02	1.12	0.00	1.06	−0.01	1.07
n = 1000
$\theta =12.7$	$S_T,S_T$	−0.05	0.70	−0.05	0.84	−0.03	0.76	−0.03	0.92	−0.02	0.79	−0.02	0.93
$\tau =0.25$	$S_{Y_0,T},S_{Y_1,T}$	−0.03	0.55	−0.06	0.62	−0.01	0.57	−0.04	0.62	0.00	0.53	−0.03	0.55
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	−0.02	0.50	−0.06	0.61	0.00	0.52	−0.03	0.59	0.01	0.47	−0.02	0.50
	$S_{Y_0},S_{Y_1}$	0.00	0.43	−0.02	0.47	0.01	0.45	−0.01	0.48	0.02	0.42	0.00	0.42
$\theta =12.0$	$S_T,S_T$	−0.07	0.60	−0.07	0.76	−0.05	0.66	−0.04	0.84	−0.03	0.68	−0.03	0.88
$\tau =0.5$	$S_{Y_0,T},S_{Y_1,T}$	−0.03	0.50	−0.07	0.54	−0.02	0.50	−0.04	0.55	0.00	0.48	−0.03	0.48
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	−0.02	0.42	−0.07	0.51	−0.01	0.44	−0.03	0.51	0.01	0.41	−0.02	0.45
	$S_{Y_0},S_{Y_1}$	0.00	0.38	−0.02	0.40	0.01	0.40	−0.01	0.42	0.01	0.37	0.00	0.38
$\theta =11.3$	$S_T,S_T$	−0.08	0.72	−0.08	0.87	−0.06	0.76	−0.06	0.96	−0.04	0.81	−0.04	1.00
$\tau =0.75$	$S_{Y_0,T},S_{Y_1,T}$	−0.04	0.60	−0.08	0.64	−0.02	0.59	−0.05	0.61	0.00	0.58	−0.04	0.60
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	−0.03	0.53	−0.08	0.62	−0.01	0.53	−0.04	0.58	0.01	0.50	−0.03	0.55
	$S_{Y_0},S_{Y_1}$	0.00	0.45	−0.03	0.50	0.01	0.46	−0.01	0.48	0.01	0.43	0.00	0.44

RB: relative bias; SD: standard deviation

Figure 1. Boxplots of observed annualised payment amount (in thousands) for overall, CCM group T = 1, and non-CCM group T = 0.

Table 3. Estimates and standard errors (SE) for the University of Wisconsin Health ACO data.

		IPW		REG		AIPW
	$S_0,S_1$	Estimate	SE	Estimate	SE	Estimate	SE
25% QTE	$S_T$	0.16	1.00	0.52	1.10	0.39	1.15
	$S_{Y_0,Y_1}$	0.47	0.91	0.57	0.84	0.47	0.89
	$S_{Y_0}$,$S_{Y_1}$	0.47	0.92	−0.10	0.81	0.47	0.88
50% QTE	$S_T$	2.82	2.67	1.59	2.76	3.31	2.64
	$S_{Y_0,Y_1}$	3.21	2.52	2.34	2.63	3.27	2.46
	$S_{Y_0},S_{Y_1}$	3.27	2.41	3.57	2.42	3.27	2.37
75% QTE	$S_T$	−6.09	4.74	−5.31	4.65	−6.33	4.62
	$S_{Y_0,Y_1}$	−8.51	3.92	−5.14	3.90	−8.38	3.82
	$S_{Y_0},S_{Y_1}$	−8.38	3.83	−8.65	3.72	−8.14	3.76
ATE	$S_T$	−2.36	3.72	−2.54	3.83	−2.11	3.53
	$S_{Y_0,Y_1},S_{Y_0,Y_1}$	−5.68	3.09	−3.41	2.93	−6.03	3.05
	$S_{Y_0},S_{Y_1}$	−5.42	2.93	−5.89	2.78	−5.24	2.87

SE: standard error by bootstrapping

Figure A1. Five choices of $S_k$ in the space of all linear combinations of X. For $(S_{Y_0},S_{Y_1})$ and $(S_{Y_0,T},S_{Y_1,T})$, the first row are $S_0$ for estimating $Y_0$ characteristics, the second row are $S_1$ for estimating $Y_1$ characteristics.

Figure A2. Relative efficiencies of estimators. Solid arrow from A to B means that A is more asymptotically efficient than B. Dashed arrow from A to B means that empirically A is more efficient than B.