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. Author manuscript; available in PMC: 2010 Aug 25.
Published in final edited form as: Psychol Methods. 2008 Dec;13(4):314–336. doi: 10.1037/a0014207

Table 9.

When None of the SEMBK and PSAIR Model Assumptions Hold

SEMBK model
 PSAIR model
 SEMMac(= PSIg) model
True value Estimate(SE) Coverage[Power] Estimate(SE) Coverage[Power] Estimate(SE) Coverage[Power]
SEM parameter
a 0.30 0.297 (0.042) 0.942 [1.000] 0.297 (0.042) 0.942 [1.000] 0.297 (0.042) 0.942 [1.000]
b 0.30 0.617 (0.096) 0.096 0.562 (0.319) 0.864 0.294 (0.139) 0.950 [0.580]
c −0.10 −0.016 (0.096) 0.840 0.000 −0.283 (0.126) 0.680
ab 0.09 0.183 (0.039) 0.304 0.167 (0.095) 0.860
d 0.30 0.000 0.000 0.607 (0.191) 0.654
α m 0.30 0.301 (0.030) 0.968 [1.000] 0.301 (0.030) 0.968 [1.000] 0.301 (0.030) 0.968 [1.000]
PS parameter
π n 0.30 0.402 (0.030) 0.066 0.402 (0.030) 0.066 0.402 (0.030) 0.066
π f 0.40 0.297 (0.042) 0.324 0.297 (0.042) 0.324 0.297 (0.042) 0.324
π b 0.10 0.000 0.000 0.000
π a 0.20 0.301 (0.030) 0.054 0.301 (0.030) 0.054 0.301 (0.030) 0.054
γ n −0.10 −0.016 (0.096) 0.840 0.000 −0.283 (0.126) 0.680
γ f 0.50 0.602 (0.114) 0.874 0.562 (0.319) 0.944 [0.442] 0.618 (0.112) 0.810
γ b −0.40 −0.633 (0.155) 0.664 −0.577 (0.154) 0.752
γ a 0.20 −0.016 (0.096) 0.392 0.000 0.324 (0.143) 0.864

Note. SEM = structural equation modeling; BK model = Baron and Kenny's (1986) model; Mac model = MacArthur model (Kraemer et al., 2008); PS = principal stratification; AIR model = Angrist, Imbens, and Rubin's (1996) model; Ig model = ignorability 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); αm = the mean of M in the control condition; πn = the proportion of never-improvers; πf = the proportion of forward-improvers; πb = the proportion of backward-improvers; πa = the proportion of always-improvers; γn = the average causal treatment effect for never-improvers; γf = the average causal treatment effect for forward-improvers; γb = the average causal treatment effect for backward-improvers; γa = the average causal treatment effect for always-improvers; Power is reported if the coverage rate is at least 0.9.