$\hat{\delta}_1^{\text{doubly}} - \hat{\theta}0^{\text{triple}}$ are semiparametric efficient in model $\mathcal{M}{\text{nonpar}}$ at the intersection submodel $\mathcal{M}_a \cap \mathcal{M}_b \cap \mathcal{M}_c$.
- A simulation study of estimators of direct effect. In this section, we report a simulation study which illustrates the finite sample performance of the various estimators described in previous sections. We generated 1000 samples of size $n = 600, 1000$ from the following model:
We then evaluated the performance of the following four estimators of the natural direct effect $\hat{\theta}_0^{\text{em}} - \hat{\delta}_0^{\text{doubly}}$, $\hat{\theta}_0^{\text{ye}} - \hat{\delta}_0^{\text{doubly}}$, $\hat{\theta}_0^{\text{ym}} - \hat{\delta}_0^{\text{doubly}}$ and $\hat{\theta}_0^{\text{triple}} - \hat{\delta}_0^{\text{doubly}}$. Note that the doubly robust estimator $\hat{\delta}_0^{\text{doubly}}$ was used throughout to estimate $\delta_0 = E(Y_0)$. To assess the impact of modeling error, we evaluated these estimators in four separate scenarios. In the first scenario, all models were correctly specified, whereas the remaining three scenarios respectively mis-specified only one of Model E, Model M and Model Y. In order to mis-specify Model E and Model M, we respectively left out the $X_1X_3$ interaction when fitting each model, and we assumed an incorrect log-log link function. The incorrect model for $Y$ simply assumed no EM interaction.
Tables 1 and 2 summarize the simulation results which largely agree with the theory developed in the previous sections. Mainly, all proposed estimators performed well at both moderate and large sample sizes in the absence of modeling error. Furthermore, under the partially mis-specified model in which Model.Y was incorrect, both estimators, $\hat{\theta}_0^{\text{ye}} - \hat{\delta}_0^{\text{doubly}}$ and $\hat{\theta}_0^{\text{ym}} - \hat{\delta}_0^{\text{doubly}}$, showed significant bias irrespective of sample size, while $\hat{\theta}_0^{\text{em}} - \hat{\delta}_0^{\text{doubly}}$ and $\hat{\theta}_0^{\text{triple}} - \hat{\delta}_0^{\text{doubly}}$ both performed well. Similarly when Model M was incorrect, the estimators $\hat{\theta}_0^{\text{em}} - \hat{\delta}_0^{\text{doubly}}$ and $\hat{\theta}_0^{\text{ym}} - \hat{\delta}_0^{\text{doubly}}$ resulted in large bias, when compared to the relatively small bias of $\hat{\theta}_0^{\text{ye}} - \hat{\delta}_0^{\text{doubly}}$ and $\hat{\theta}_0^{\text{triple}} - \hat{\delta}_0^{\text{doubly}}$. Finally, mis-specifying Model E lead to estimators $\hat{\theta}_0^{\text{ye}} - \hat{\delta}_0^{\text{doubly}}$ and $\hat{\theta}_0^{\text{em}} - \hat{\delta}_0^{\text{doubly}}$ that were significantly more biased than the estimators $\hat{\theta}_0^{\text{ym}} - \hat{\delta}_0^{\text{doubly}}$ and $\hat{\theta}_0^{\text{triple}} - \hat{\delta}_0^{\text{doubly}}$. Interestingly, the efficiency loss of the multiply robust estimator remained relatively small when compared to the consistent nonrobust estimator under the various scenarios, suggesting that, at least in this simulation study, the benefits of robustness appear to outweigh the loss of efficiency.