Table 3.

Standardized mean differences for measuring covariate balance.

			Data generation scenario
			1	2	3	4	5
		Description	Standard	Random sample	Selection depends on x and t	Weight scales differ	Non-linear treatment
		Selection	s ~ x	s ⊥ x	s ~ (x, t)	s ~ x∣t	s ~ x
		Treatment	t ~ x	t ~ x	t ~ x	t ~ x	t ~ x2
	Fit PS model
	Sampling weights	PS model	PATE
1	None	t ~ 1	1.18	1.06	1.10	1.17	0.13
2	None	t ~ x	0.10	0.03	0.03	0.09	2.00
3	Covariate	t ~ x + sw	0.06	0.03	2.28	1.39	0.89
4	Weight	t ~ x	0.04	0.03	0.03	0.04	0.01
			PATT
5	None	t ~ 1	1.18	1.06	1.10	1.17	0.13
6	None	t ~ x	0.17	0.03	0.08	0.16	0.74
7	Covariate	t ~ x + sw	0.12	0.03	1.57	0.44	1.10
8	Weight	t ~ x	0.12	0.02	0.06	0.11	0.08

Notes: Covariate balance measured with absolute standardized mean differences. All models fit use sampling weights when computing these differences, but vary on whether or how those sampling weights were used in the propensity score model.