Table 2.

Ordinary Least Square and Standard Instrumental Variable Estimates for Compliers Average Causal Effect Without Covariates, With Covariates, and With Instrumental Propensity Scores, Using Simulated Data

Distribution	CACE	$\widehat{\mathit{CACE}}\mathbf{(}\mathbf{S}\mathbf{E}\mathbf{)}$
		OLS (With X)	SIV (Ignoring X)	SIV (With X)	SIV (IPS-Based)
Normal
Overall	2.808	2.338 (0.114)	4.295 (0.180)	2.905 (0.178)	2.866 (0.202)
k = 1	1				1.127 (0.390)
k = 2	2				2.063 (0.252)
k = 3	3				3.035 (0.300)
k = 4	4				4.022 (0.471)
k = 5	5				5.011 (1.299)
Log normal
Overall	53.7	160.1 (27.3)	263.7 (53.7)	45.9 (48.9)	61.4 (75.1)
k = 1	2.8				4.1 (6.3)
k = 2	10.5				12.9 (11.1)
k = 3	31.5				36.6 (33.1)
k = 4	88.4				100.2 (128.3)
k = 5	243.0				176.2 (1,091.6)
Poisson
Overall	2.808	2.335 (0.159)	4.295 (0.230)	2.902 (0.239)	2.862 (0.261)
k = 1	1				1.129 (0.504)
k = 2	2				2.065 (0.346)
k = 3	3				3.028 (0.403)
k = 4	4				4.022 (0.616)
k = 5	5				4.973 (1.723)

Abbreviations: CACE, true complier average causal effect; $\widehat{\mathit{CACE}}$, estimated complier average causal effect; IPS, instrumental propensity score; k, instrumental propensity score stratum; OLS, ordinary least square; SE, standard error; SIV, standard instrumental variable with 2-stage least square.