Imai 2010 - Potential Outcomes Mediation Effects and Sequential Ignorability

Summary

The causal mediation effect for unit $i$ is ${\delta }_i(t)=Y_i(t,M_i(1))-Y_i(t,M_i(0))$: hold treatment at $t$, shift the mediator from its control to its treated value. The ACME is nonparametrically identified by a mediator-distribution-weighted contrast of outcome regressions if treatment is ignorable given $X$ and the mediator is ignorable given $T$ and $X$ (Assumption 1, “sequential ignorability”). Total effect decomposes as $\tau =\delta (t)+\zeta (1-t)$.

Overview

Setup: simple random sample of size $n$; for unit $i$ observe $(T_i,M_i,X_i,Y_i)$. $T_i\in \{0,1\}$ treatment, $M_i$ mediator with support $\mathcal{M}$, $X_i$ pre-treatment covariates (support $\mathcal{X}$), $Y_i$ outcome. Potential mediator $M_i(t)$; potential outcome $Y_i(t,m)$; observed $M_i=M_i(T_i)$, $Y_i=Y_i(T_i,M_i(T_i))$. If $M$ takes $J$ values there are $2J$ potential outcomes, only one observed. The notation assumes the outcome depends only on the values of $(T,M)$, not on how they were realised (a no-”encouragement” condition: $Y_i(t,M_i(t))=Y_i(t,M_i(1-t))=Y_i(t,m)$ if $M_i(1)=M_i(0)=m$).

Main Content

Causal mediation effect, natural direct effect, total effect (Eqs. 1-3)

$${\delta }_i(t)=Y_i(t,M_i(1))-Y_i(t,M_i(0)),{\zeta }_i(t)=Y_i(1,M_i(t))-Y_i(0,M_i(t)),$$ $${\tau }_i=Y_i(1,M_i(1))-Y_i(0,M_i(0))={\delta }_i(t)+{\zeta }_i(1-t),t=0,1.$$

$\bar{\delta }(t)=E[{\delta }_i(t)]$ is the ACME, $\bar{\zeta }(t)=E[{\zeta }_i(t)]$ the average natural direct effect, $\bar{\tau }=\bar{\delta }(t)+\bar{\zeta }(1-t)$.

Terminology: Pearl (2001) calls ${\delta }_i(t)$ the natural indirect effect; Robins (2003) the pure indirect effect for ${\delta }_i(0)$ and total indirect effect for ${\delta }_i(1)$. The controlled direct effects $Y_i(t,m)-Y_i(t,m^{\prime })$ and $Y_i(1,m)-Y_i(0,m)$ differ from these: they fix the mediator at specific values rather than potential values. “The former examines whether $M$ mediates the relationship between $T$ and $Y$, the latter whether $T$ moderates the effect of $M$ on $Y$.” The counterfactual $Y_i(t,M_i(1-t))$ is never observable under any single treatment status (cross-world quantity); the paper notes disagreement on whether to infer such quantities (Rubin 2005 vs Pearl 2001).

Assumption 1: sequential ignorability (Eqs. 4-5)

For $t,t^{\prime }\in \{0,1\}$ and all $x\in \mathcal{X}$:

$$\{Y_i(t^{\prime },m),M_i(t)\}\perp \perp T_i\mid X_i=x(4)$$ $$Y_i(t^{\prime },m)\perp \perp M_i(t)\mid T_i=t,X_i=x(5)$$

with positivity $0<\mathrm{Pr}(T_i=t|X_i=x)$ and $0<p(M_i(t)=m|T_i=t,X_i=x)$. Treatment is first ignorable given pre-treatment $X$; then the mediator is ignorable given the observed treatment and $X$. Unlike Robins’ sequential ignorability, (5) must hold without conditioning on post-treatment confounders.

Theorem 1: nonparametric identification

Under Assumption 1, for $t=0,1$,

$$\bar{\delta }(t)=\iint E(Y_i\mid M_i=m,T_i=t,X_i=x)\{d{F}_{M_i|T_i=1,X_i=x}(m)-d{F}_{M_i|T_i=0,X_i=x}(m)\}d{F}_{X_i}(x),$$ $$\bar{\zeta }(t)=\iint \{E(Y_i|M_i=m,T_i=1,X_i=x)-E(Y_i|M_i=m,T_i=0,X_i=x)\}d{F}_{M_i|T_i=t,X_i=x}(m)d{F}_{X_i}(x).$$

Proof sketch (Appendix A; the paper’s chain has seven equalities, condensed here). Eq. (4) implies $Y_i(t^{\prime },m)\perp T_i\mid M_i(t)=m^{\prime },X_i=x$ (Eq. 25). Then

$$E[Y_i(t,M_i(t^{\prime }))|X_i=x]=\int E[Y_i(t,m)\mid M_i(t^{\prime })=m,X_i=x]d{F}_{M_i(t^{\prime })|X_i=x}(m)$$ $$=\int E[Y_i(t,m)\mid M_i(t^{\prime })=m,T_i=t^{\prime },X_i=x]d{F}_{M_i(t^{\prime })|T_i=t^{\prime },X_i}(m)(\text{by 4})$$ $$=\int E[Y_i(t,m)\mid T_i=t^{\prime },X_i=x]dF(\cdot )=\int E[Y_i(t,m)|T_i=t,X_i=x]dF(\cdot )$$

(Eq. 5 removes the conditioning on $M_i(t^{\prime })$; Eq. 4 then lets $T_i$ be switched between $t^{\prime }$ and $t$), then Eq. 5 again with $M_i(t)=m$, $T_i=t$, and consistency $M_i=M_i(T_i)$, $Y_i=Y_i(T_i,M_i(T_i))$ give $\int E(Y_i|M_i=m,T_i=t,X_i=x)d{F}_{M_i|T_i=t^{\prime },X_i=x}(m)$. Integrating over $X$ and differencing in $t^{\prime }$ yields $\bar{\delta }(t)$; $\bar{\zeta }(t)$ follows from $\bar{\tau }=\bar{\zeta }(t)+\bar{\delta }(t^{\prime })$.

Remarks

• Holds for any (continuous or discrete) mediator and treatment regime; eq. (5) can be weakened to mean independence.
• Post-treatment confounders. If (5) only holds after conditioning on post-treatment confounders $Z_i$, natural effects are not identified without extra assumptions (Avin, Shpitser & Pearl 2005; Robins 2003). Sometimes pre-treatment conditioning is enough to address the main source of confounding.
• Randomised treatment. Eq. (4) holds by design; Eq. (5) does not (mediators are typically not randomised) - motivating Imai 2010 - Sensitivity Analysis with Error Correlation Rho.
• The data generating process contains no information about whether Eq. 5 holds.

Examples

Binary $T$, binary $M$ (Eqs. 16-17)

For discrete $M$: $\bar{\delta }(t)={\sum }_{m}E(Y|M=m,T=t,X)\{\mathrm{Pr}(M=m|T=1,X)-\mathrm{Pr}(M=m|T=0,X)\}$. For $J=2$:

$$\bar{\delta }(t)=\{\mathrm{Pr}(M=1|T=1,X)-\mathrm{Pr}(M=1|T=0,X)\}\{E(Y|M=1,T=t,X)-E(Y|M=0,T=t,X)\}.$$

The ACME is the product of the effect of $T$ on $M$ and the effect of $M$ on $Y$ (holding $T=t$). Numerical check: $\mathrm{Pr}(M=1|T=1)=0.7,\mathrm{Pr}(M=1|T=0)=0.3$, $E(Y|M=1,T=t)-E(Y|M=0,T=t)=2$ gives $\bar{\delta }(t)=0.4\times 2=0.8$.

import numpy as np
def acme_discrete(p1, p0, mu_t):          # p1[m]=Pr(M=m|T=1), p0[m]=Pr(M=m|T=0), mu_t[m]=E[Y|M=m,T=t]
    return np.sum(np.asarray(mu_t)*(np.asarray(p1)-np.asarray(p0)))
print(acme_discrete([0.3,0.7],[0.7,0.3],[1.0,3.0]))   # 0.8

Connections

• Marketing: $T$ = Meta spend (or exposure), $M$ = branded-search clicks, $Y$ = new customers; the ACME is the “halo through search” and the natural direct effect is Meta’s effect with search held at its no-Meta value; see Imai 2010 - Bayesian Mediation Sketch for Halo Effects.
• Compared with other identification results: Imai 2010 - Identification Compared with Pearl Robins and Principal Strata.

See Also

Imai 2010 - Causal Mediation Analysis - Overview, Potential Outcomes Framework, DAGs and Causal Identification