Robust Inference for Mediated Effects in Partially Linear Models
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We consider mediated effects of an exposure, X on an outcome, Y, via a single mediator, M, under the assumption of no unmeasured confounding in the setting where models for the conditional expectation of the mediator and outcome are partially linear. In this setting the indirect effect is defined through a product of model coefficients, and the direct effect by a single model coefficient. We propose G-estimators for the direct and indirect effect and demonstrate consistent asymptotically normality for indirect effects when models for M, or X and Y are correctly specified, and for direct effects, when models for Y, or X and M are correct. This marks an improvement over previous `triple' robust methods, which do not assume partially linear mean models, but instead require correct specification of any pair of: the conditional expectation of Y and the conditional densities of M and X. Testing of the no-mediation hypothesis is inherently problematic due to the composite nature of the test (either X has no effect on M or M no effect on Y), leading to low power when both effect sizes are small. We use Generalized Methods of Moments (GMM) results to construct a new score testing framework, which includes as special cases the no-mediation and the no-direct-effect hypotheses. The proposed tests rely on a bias-reduction strategy for estimating parameters in nuisance confounder models. Simulations show that the GMM based tests perform better in terms of power and small sample performance compared with traditional tests in the partially linear setting, with drastic improvement under model misspecification. New methods are illustrated in a mediation analysis of data from the COPERS trial, a randomized trial on the effect of a non-pharmacological intervention of patients suffering from chronic pain. An accompanying R package implementing these methods can be found at github.com/ohines/plmed.
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