TABLE 3 Estimated causal effects of interest using the job search intervention study data
| Mym | Mye | Mem | Munion | ||
|---|---|---|---|---|---|
| Direct effect | Estimate | -0.0310 | -0.0310 | 0.0280 | -0.0409 |
| s.e.* | 0.0124 | 0.0620 | 0.0465 | 0.0217 | |
| Indirect effect | Estimate | -0.0160 | -0.0160 | -0.0750 | -0.0070 |
| s.e.* | 0.0372 | 0.0620 | 0.0434 | 0.0217 |
*Nonparametric bootstrap standard errors.
a continuous outcome measure Y of depressive symptoms based on the Hopkins Symptom Checklist [Imai, Keele and Tingley (2010)]. In the JOBS II data, a continuous measure of job search self-efficacy represented the hypothesized mediating variable M. The data also included baseline covariates X measured before administering the treatment including: pretreatment level of depression, education, income, race, marital status, age, sex, previous occupation, and the level of economic hardship.
Note that by randomization, the density of [E|X] was known by design not to depend on covariates, and therefore its estimation is not prone to modeling error. The continuous outcome and mediator variables were modeled using linear regression models with Gaussian error, with main effects for (E, M, X) included in the outcome regression and main effects for (E, X) included in the mediator regression. Table 3 summarizes results obtained using $\hat{\theta}_0^{\text{em}}$, $\hat{\theta}_0^{\text{ye}}$, $\hat{\theta}_0^{\text{ym}}$ and $\hat{\theta}_0^{\text{triply}}$ together with $\hat{\delta}_e^{\text{doubly}}$, $e = 0, 1$, to estimate the direct and indirect effects of the treatment.
Point estimates of both natural direct and indirect effects closely agreed under models $M_{ym}$ and $M_{ye}$, and also agreed with the results of Imai, Keele and Tingley (2010). We should note that inferences under our choice of $M_{ym}$ are actually robust to the normality assumption and, as in Imai, Keele and Tingley (2010), only require that the mean structure of [Y|E, M, X] and [M|E, X] is correct. In contrast, inferences under model $M_{em}$ require a correct model for the mediator density. This distinction may partly explain the apparent disagreement in the estimated direct effect under $M_{em}$ when compared to the other methods, also suggesting that the Gaussian error model for M is not entirely appropriate. The multiply robust estimate of the natural direct effect is consistent with estimates obtained under models $M_{ym}$ and $M_{ye}$, and is statistically significant, suggesting that the intervention may have beneficial direct effects on participants’ mental health; while the multiply robust approach suggests a much smaller indirect effect than all other estimators although none achieved statistical significance.
5. Improving the stability of $\hat{\theta}_0^{\text{triply}}$ when weights are highly variable. The triply robust estimator $\hat{\theta}_0^{\text{triply}}$ which involves inverse probability weights for the exposure and mediator variables, clearly relies on the positivity assumption, for good