Imai 2010 - Causal Mediation Analysis - Overview

Summary

Imai, Keele & Yamamoto (Statistical Science 2010) prove that under a sequential ignorability assumption the average causal mediation effect (ACME) and average natural direct effects are nonparametrically identified, show that the Baron-Kenny product of coefficients ${\beta }_2\gamma$ is a causal ACME under that assumption plus no treatment-mediator interaction, give nonparametric estimators, and propose a sensitivity analysis indexed by the correlation $\rho$ between the mediator-model and outcome-model errors. Key result: the data contain no information about $\rho$ (bounds $(-1,1)$, ACME in $(-\infty ,\infty )$), so sensitivity analysis is mandatory.

Overview

Research question. When a treatment $T$ affects an outcome $Y$ partly through a mediator $M$, under what conditions can the indirect (through $M$) and direct (not through $M$) effects be identified, how should they be estimated, and how robust are they to unmeasured mediator-outcome confounding?

Contributions.

1. Theorem 1: ACME and natural direct effects identified under sequential ignorability that does not condition on post-treatment confounders; compared with Pearl (2001), Robins (2003), Petersen et al. (2006), Hafeman & VanderWeele, Rubin (2004) principal strata.
2. Theorem 2: LSEM (Baron-Kenny) gives $\bar{\delta }={\beta }_2\gamma$ under Assumptions 1-2; relaxing no-interaction gives $\bar{\delta }(t)={\beta }_2(\gamma +t\kappa )$.
3. Nonparametric estimators (Eq. 18-20), asymptotic variance (Theorem 3), simulation (Table 2).
4. Sensitivity analysis via $\rho$ (Theorem 4, Corollary 1) and via coefficients of determination ($R_M^{2\ast },R_Y^{2\ast }$ or ${\tilde{R}}_M^2,{\tilde{R}}_Y^2$); nonparametric version for binary $M,Y$ (Appendix E).
5. Application: Nelson, Clawson & Oxley (1997) media framing experiment, $n=136$.

Main Content

Source Map

Paper section	Content	Note
Sec. 1 (pp. 1-2)	motivation, Cochran’s eelworm example, contributions	this note
Sec. 2 (pp. 2-4), Table 1	media framing experiment	Imai 2010 - Media Framing Application and Simulation
Sec. 3.1-3.2, Eqs. 1-5, Thm 1	potential outcomes, ACME, sequential ignorability, identification	Imai 2010 - Potential Outcomes Mediation Effects and Sequential Ignorability
Sec. 3.3	comparison with Pearl, Robins, Petersen, Hafeman, Rubin	Imai 2010 - Identification Compared with Pearl Robins and Principal Strata
Sec. 3.4, Eqs. 11-17, Thm 2, Asm. 2	LSEM, Baron-Kenny, interaction	Imai 2010 - LSEM Baron-Kenny Interaction and Estimation
Sec. 4.1-4.2, Eqs. 18-20, Thm 3	parametric and nonparametric estimation, variance	Imai 2010 - LSEM Baron-Kenny Interaction and Estimation
Sec. 4.3, Table 2	Monte Carlo study	Imai 2010 - Media Framing Application and Simulation
Sec. 5.1, Eq. 24, Thm 4, Cor. 1	$\rho$ sensitivity	Imai 2010 - Sensitivity Analysis with Error Correlation Rho
Sec. 5.2	$R^2$ interpretations	Imai 2010 - Sensitivity Analysis with Error Correlation Rho
Sec. 5.3, Appendix E, Fig. 3	nonlinear / nonparametric sensitivity, sharp bounds, $\upsilon$ analysis	Imai 2010 - Sensitivity Analysis with Error Correlation Rho
Sec. 6, Tables 3, Figs. 1-2	application and sensitivity results	Imai 2010 - Media Framing Application and Simulation
Sec. 7	concluding remarks, future work	Imai 2010 - Bayesian Mediation Sketch for Halo Effects (limitations section)
Appendix A	proof of Thm 1	Imai 2010 - Potential Outcomes Mediation Effects and Sequential Ignorability
Appendix B	proof of Thm 2	Imai 2010 - LSEM Baron-Kenny Interaction and Estimation
Appendix C	$\rho =0$ under Asm. 1	Imai 2010 - Sensitivity Analysis with Error Correlation Rho
Appendix D	proof of Thm 4	Imai 2010 - Sensitivity Analysis with Error Correlation Rho

Motivating PyMC examples

From https://redam94.github.io/pymc-challenges/examples.html:

• E35 “Halo effects as Bayesian causal mediation: does Meta work through branded search?” (natural direct/indirect effects; joint mediator + outcome negative-binomial model; counterfactual propagation from posterior draws; product-of-coefficients check; “search as control” vs “search left out”).

Examples

A quick worked case (full details in the linked notes): with ${\beta }_2=0.8$, $\gamma =0.6$ the ACME under the LSEM is $0.48$. If an unobserved $U$ confounds mediator and outcome, the naive Baron-Kenny estimate can be $0.99$ with $\rho =0$ but Theorem 4 evaluated at the true $\rho =0.473$ returns $0.479$ (see Imai 2010 - Sensitivity Analysis with Error Correlation Rho).

Connections

• Principal strata alternative: Instrumental Variables and Principal Stratification.
• Potential outcomes background: Potential Outcomes Framework, DAGs and Causal Identification.
• OVB-style sensitivity for the outcome regression: Cinelli Hazlett 2020 - Making Sense of Sensitivity - Overview.
• Hierarchical/Bayesian mediation: Hierarchical Models.

See Also

Sensitivity Analysis in Observational Studies, Omitted Variables Bias