Table 7.

When Only Exclusion Restriction Is Violated

		SEMBK model				PSAIR model
True value		Estimate	SE	Coverage	Power	Estimate	SE	Coverage	Power
SEM parameter
a	0.40	0.397	0.040	0.936	1.000	0.397	0.040	0.936	1.000
b	0.30	0.296	0.100	0.956	0.826	0.805	0.235	0.396
c	0.20	0.201	0.098	0.938	0.528	0.000
ab	0.12	0.118	0.042	0.938	0.824	0.318	0.090	0.408
d	0.00	0.000				0.000
α m	0.20	0.201	0.028	0.966	1.000	0.201	0.028	0.966	1.000
PS parameter
π n	0.40	0.402	0.028	0.930	1.000	0.402	0.028	0.930	1.000
π f	0.40	0.397	0.040	0.936	1.000	0.397	0.040	0.936	1.000
π b	0.00	0.000				0.000
π a	0.20	0.201	0.028	0.966	1.000	0.201	0.028	0.966	1.000
γ n	0.20	0.201	0.098	0.938	0.528	0.000
γ f	0.50	0.497	0.108	0.944	0.996	0.805	0.235	0.766
γ b		−0.095	0.166
γ a	0.20	0.201	0.098	0.938	0.528	0.000

Note. SEM = structural equation modeling; BK model = Baron and Kenny's (1986) model; PS = principal stratification; AIR model = Angrist, Imbens, and Rubin's (1996) model; a = the effect of treatment assignment X on the mediator M; b = the effect of M on outcome Y conditioning on X; c = the effect of X on Y conditioning on M; ab = the indirect effect (= a × b); d = the interaction effect (how much the effect of X increases [or decreases] as the value of M increases by one unit); α_m = the mean of M in the control condition; π_n = the proportion of never-improvers; π_f = the proportion of forward-improvers; π_b = the proportion of backward-improvers; π_a = the proportion of always-improvers; γ_n = the average causal treatment effect for never-improvers; γ_f = the average causal treatment effect for forward-improvers; γ_b = the average causal treatment effect for backward-improvers; γ_a = the average causal treatment effect for always-improvers.