TABLE 2.

Results of all 4 methods applied to PBC UDCA data set. All SEs and CIs are based on the nonparametric bootstrap

Method		Estimate	Bootstrap SE	95% CI	Time (mins)
(A) Unadjusted		$-0.7503$	0.2181	[$-1.1852,-0.3347$]	0.05
(B) IPTW		$-0.7290$	0.2159	[$-1.1862,-0.3232$]	0.18
(C) Adjusted marginal	$2m=200$	$-0.5706$	0.2888	[$-1.4054,-0.2200$]	0.83
(our proposal)	$2m=2000$	$-0.8192$	0.2018	[$-1.2246,-0.4269$]	2.18
	$2m=4000$	$-0.8874$	0.1973	[$-1.1994,-0.4122$]	3.80
	$2m=10,000$	$-0.7631$	0.1994	[$-1.1817,-0.3976$]	7.81
	$2m=20,000$	$-0.7376$	0.1991	[$-1.1918,-0.3871$]	15.46
	$2m=40,000$	$-0.7787$	0.2004	[$-1.1980,-0.3929$]	28.75
	$2m=100,000$	$-0.7914$	0.1956	[$-1.1758,-0.4005$]	72.18
(D) Conditional		$-0.8643$	0.2253	[$-1.3305,-0.4574$]	0.05

Notes. $2m$ is the sample size for the simulated data set ($m$ for each exposure level) used within the proposal—see Section 4.2). For all methods, the estimated SE was obtained using the nonparametric bootstrap with 1000 bootstrap samples, and the 95% CI obtained from the 2.5th and 97.5th percentile of the distribution of nonparametric bootstrap estimates.