Imai 2010 - Identification Compared with Pearl Robins and Principal Strata
Summary
Assumption 1 gives the same ACME expression as Pearl’s (2001) conditions, is easier to interpret, and needs no no-interaction assumption, but cannot condition on observed post-treatment confounders. Robins’ FRCISTG model can condition on them but then needs (no interaction). Petersen et al. weaken (4) at the cost of another assumption; Rubin’s principal strata define direct effects only for a latent subpopulation.
Overview
Section 3.3 compares Theorem 1 with five existing identification routes. The trade-off table below collects them.
Main Content
| Source | Assumptions | Post-treatment confounders | Extra assumption | Remark |
|---|---|---|---|---|
| This paper (Asm. 1) | (4)-(5) | not allowed | none | any mediator type; any treatment regime |
| Pearl (2001, Thm 2) | (6) $p(Y(t,m) | X=x)p(M(t^*) | X=x)Y_i(t,m)\perp M_i(t^*) | X_i=x$ |
| Robins (2003) FRCISTG | (4) plus (8) $Y_i(t,m)\perp M_i(t) | T_i=t,Z_i=z,X_i=x$ | allowed (observed) | (9) with independent of (no interaction) |
| Petersen, Sinisi & van der Laan (2006) | keep (5); replace (4) with (10) $Y_i(t,m)\perp T_i | X_iM_i(t)\perp T_i | X_i$ | not allowed |
| Hafeman & VanderWeele (2010) | weaker set for binary | - | - | not shown to generalise to non-binary |
| Rubin (2004) principal strata | direct effect $E{Y(1,M(1))-Y(0,M(0)) | M(1)=M(0)}$ | - | - |
Key trade-off. “Allowing for conditioning on observed post-treatment confounders requires an additional assumption for the identification of the ACME.”
Sobel (2008) (also Albert 2008; Jo 2008) uses an instrumental-variables approach to mediation in an LSEM but assumes the direct effect is zero for every unit a priori, which is undesirable when the natural direct effect is itself of interest.
Examples
Choosing an identification route for a marketing funnel
Treatment: a randomised geo-test of Meta spend. Mediator: branded-search clicks. Observed post-treatment confounder: e.g. a Google brand-campaign spend change triggered by the test. Asm. 1 would be violated (the post-treatment confounder is not conditioned on); Robins’ route allows conditioning on it but needs no spend-search interaction in the outcome. The analyst must pick which assumption is more defensible, or run the sensitivity analysis of Imai 2010 - Sensitivity Analysis with Error Correlation Rho on top.
Connections
- For principal-strata ideas in Bayesian causal inference see Instrumental Variables and Principal Stratification.
- Graphical treatment of mediators and colliders: DAGs and Causal Identification, Summary Causal DAGs.
See Also
Imai 2010 - Causal Mediation Analysis - Overview, Potential Outcomes Framework