Causal mediation analysis: selection with asymptotically valid inference

Researchers are often interested in learning not only the effect of treatments on outcomes, but also the mechanisms that transmit these effects. A mediator is a variable that is affected by treatment and subsequently affects outcome. Existing methods for penalized mediation analyses may lead to ignoring important mediators and either assume that finite-dimensional linear models are sufficient to remove confounding bias, or perform no confounding control at all. In practice, these assumptions may not hold. We propose a method that considers the confounding functions as nuisance parameters to be estimated using data-adaptive methods. We then use a novel regularization method applied to this objective function to identify a set of important mediators. We consider natural direct and indirect effects as our target parameters. We then proceed to derive the asymptotic properties of our estimators and establish the oracle property under specific assumptions. Asymptotic results are also presented in a local setting, which contrast the proposal with the standard adaptive lasso. We also propose a perturbation bootstrap technique to provide asymptotically valid postselection inference for the mediated effects of interest. The performance of these methods will be discussed and demonstrated through simulation studies.