Moderating the Mediation Bootstrap for Causal Inference

Abstract: Mediation analysis is a form of causal inference that investigates indirect effects and causal mechanisms. Confidence intervals for indirect effects play a central role in conducting inference. The problem is non-standard leading to coverage rates that deviate considerably from their nominal level. The default inference method in the mediation model is the paired bootstrap, which resamples directly from the observed data. However, a residual bootstrap that explicitly exploits the assumed causal structure (X->M->Y) could also be applied. There is also a debate whether the bias-corrected (BC) bootstrap method is superior to the percentile method, with the former showing liberal behavior (actual coverage too low) in certain circumstances. Moreover, bootstrap methods tend to be very conservative (coverage higher than required) when mediation effects are small. Finally, iterated bootstrap methods like the double bootstrap have not been considered due to their high computational demands. We investigate the issues mentioned in the simple mediation model by a large-scale simulation. Results are explained using graphical methods and the newly derived finite-sample distribution. The main findings are: (i) conservative behavior of the bootstrap is caused by extreme dependence of the bootstrap distribution's shape on the estimated coefficients (ii) this dependence leads to counterproductive correction of the the double bootstrap. The added randomness of the BC method inflates the coverage in the absence of mediation, but still leads to (invalid) liberal inference when the mediation effect is small.