Mediation Analysis

This module performs Debiased Machine Learning for mediation analysis, using joint estimation for longitudinal nonparametric parameters (in the Nested NPIV framework). It provides tools for estimating causal effects with mediation using a combination of machine learning models and instrumental variables techniques.

Different Estimands

Our DML framework allows for various types of estimands. Here, we describe each estimand in detail:

  • ATE (Average Treatment Effect): This estimand measures the average effect of treatment \(A\) on outcome \(Y\) across the entire population. It is defined as the difference in expected outcomes when the treatment is applied versus when it is not.

    \[E[Y(1)] - E[Y(0)]\]

  • Indirect Effect: This estimand captures the effect of the treatment \(A\) on the outcome \(Y\) that is mediated through the intermediate variable \(M\). It is defined as the expected difference in \(Y\) if all individuals received the treatment but the mediator was set to the level it would take without the treatment.

    \[E[Y(1, M(1)) - Y(1, M(0))]\]

  • Direct Effect: This estimand measures the direct effect of treatment \(A\) on outcome \(Y\) independent of the mediator \(M\). It is defined as the expected difference in \(Y\) if the treatment is applied versus not, while keeping the mediator at the level it would take without the treatment.

    \[E[Y(1, M(0)) - Y(0, M(0))]\]

  • E[Y1]: This estimand represents the expected outcome when the treatment is applied to the entire population.

    \[E[Y(1)]\]

  • E[Y0]: This estimand represents the expected outcome when the treatment is not applied to the entire population.

    \[E[Y(0)]\]

  • E[Y(1, M(0))]: This estimand captures the expected outcome if the treatment is applied, but the mediator is set to the level it would take without the treatment.

    \[E[Y(1, M(0))]\]

Localization

Let \(H\) be the uncentered score for the selected mediation estimand; for effects defined by treatment-arm contrasts, the contrast is formed before localization. When V is supplied, DML_mediated estimates

\[\theta_\lambda(v) =\frac{\mathbb{E}[K\{(V-v)/\lambda\}H]} {\mathbb{E}[K\{(V-v)/\lambda\}]}.\]

With \(\ell_{\lambda,v}=K/\mathbb{E}[K]\), the centered score contribution is \(\ell_{\lambda,v}\{H-\theta_\lambda(v)\}\). In particular, inference centers the localized score by \(\ell_{\lambda,v}\theta_\lambda(v)\), not by the unweighted scalar \(\theta_\lambda(v)\). The same ratio centering is used for pointwise and uniform confidence intervals.

All routes use this centering; see Localized Ratio Targets for the influence-function distinction and nuisance requirements. Use include_V=True for the usual conditional causal interpretation.

nnpiv.semiparametrics.DML_mediated(Y, D, M, W, Z)

Debiased Machine Learning for mediation analysis (DML-mediation) class with joint/sequential model fitting.

References