Imai 2010 - LSEM Baron-Kenny Interaction and Estimation

Summary

Under sequential ignorability, Baron & Kenny’s product-of-coefficients ${\hat{\beta }}_2\hat{\gamma }$ is a valid ACME estimator, but implicitly assumes $\bar{\delta }(1)=\bar{\delta }(0)$ (Assumption 2). Adding a $T\times M$ interaction identifies $\bar{\delta }(t)={\beta }_2(\gamma +t\kappa )$. The paper also gives plug-in nonparametric estimators (Eq. 18-20) with asymptotic variance (Theorem 3), and clarifies that testing ${\beta }_1=0$ is unnecessary.

Overview

Baron & Kenny (1986) model, with optional pre-treatment covariates added to each equation:

$$Y_i={\alpha }_1+{\beta }_1{T}_i+{\epsilon }_{i1}(11),M_i={\alpha }_2+{\beta }_2{T}_i+{\epsilon }_{i2}(12),Y_i={\alpha }_3+{\beta }_3{T}_i+\gamma M_i+{\epsilon }_{i3}(13).$$

Substituting (12) into (13): $Y_i=({\alpha }_3+{\alpha }_2\gamma )+({\beta }_3+{\beta }_2\gamma )T_i+(\gamma {\epsilon }_{i2}+{\epsilon }_{i3})$ (14). So (11) is redundant, with ${\alpha }_1={\alpha }_3+{\alpha }_2\gamma$ and ${\beta }_1={\beta }_3+{\beta }_2\gamma$. Testing ${\beta }_1=0$ is unnecessary: the ACME can be nonzero while the total effect is zero (indirect offsets direct).

Main Content

Theorem 2 (identification under the LSEM)

In the LSEM (11)-(13) under Assumption 1, the ACME is identified and $\bar{\delta }(0)=\bar{\delta }(1)={\beta }_2\gamma$ (equality of the two is also assumed). The average natural direct effects are $\bar{\zeta }(0)=\bar{\zeta }(1)={\beta }_3$ and the total effect $\bar{\tau }={\beta }_3+{\beta }_2\gamma$.

Proof (Appendix B). Write potential outcomes $M_i(T_i)={\alpha }_2+{\beta }_2{T}_i+{\epsilon }_{i2}(T_i)$ and $Y_i(T_i,M_i(T_i))={\alpha }_3+{\beta }_3{T}_i+\gamma M_i(T_i)+{\epsilon }_{i3}(T_i,M_i(T_i))$ with $E{\epsilon }_{i2}(t)=E{\epsilon }_{i3}(t,m)=0$. Eq. (4) gives ${\epsilon }_{i2}(t)\perp T_i$ so $E({\epsilon }_{i2}|T_i)=0$; Eq. (5) gives ${\epsilon }_{i3}(t,m)\perp M_i|T_i=t$ so $E({\epsilon }_{i3}|T_i,M_i)=0$. Hence OLS on (12) and (13) is unbiased; applying Theorem 1 with $E(M|T)={\alpha }_2+{\beta }_{2}T$ and $E(Y|M,T)={\alpha }_3+{\beta }_{3}T+\gamma M$ yields ${\beta }_2\gamma$.

Assumption 2 (no interaction between treatment and the ACME)

$\bar{\delta }(1)=\bar{\delta }(0)$. Equivalent to $\bar{\zeta }(1)=\bar{\zeta }(0)$. It is implied by Robins’ no-interaction assumption (9) but is stated on the ACME, not on controlled direct effects, and is not required by Theorem 1.

With interaction. Extend (13) to $Y_i={\alpha }_3+{\beta }_3{T}_i+\gamma M_i+\kappa T_i{M}_i+{\epsilon }_{i3}$ (15). Under Assumption 1:

$$\bar{\delta }(t)={\beta }_2(\gamma +t\kappa ),\bar{\zeta }(t)={\beta }_3+\kappa ({\alpha }_2+{\beta }_{2}t),\bar{\tau }={\beta }_2\gamma +{\beta }_3+\kappa ({\alpha }_2+{\beta }_2).$$

Kraemer et al.’s proposal (establish mediation by testing $\gamma =0$ or $\kappa =0$) is “neither necessary nor sufficient” for $\bar{\delta }(t)=0$.

Estimation

Parametric. Fit (12) and (13) separately; $\hat{\delta }(t)={\hat{\beta }}_2\hat{\gamma }$. Standard errors: Delta/Sobel $\text{Var}(\hat{\delta })\approx {\beta }_2^2\text{Var}(\hat{\gamma })+{\gamma }^2\text{Var}({\hat{\beta }}_2)$; Goodman (1960) exact adds $\text{Var}(\hat{\gamma })\text{Var}({\hat{\beta }}_2)$. With interaction: $\text{Var}(\hat{\delta }(t))\approx (\gamma +t\kappa {)}^2\text{Var}({\hat{\beta }}_2)+{\beta }_2^2\{\text{Var}(\hat{\gamma })+t\text{Var}(\hat{\kappa })+2t\text{Cov}(\hat{\gamma },\hat{\kappa })\}$ (the printed formula uses $t$ for $t^2$, equal since $t\in \{0,1\}$).

Nonparametric, discrete $M$ (Eq. 18). With $J$ mediator values, $n_t={\sum }_{i}1\{T_i=t\}$:

$$\hat{\delta }(t)=\sum _{m=0}^{J-1}\left\{\frac{{\sum }_i{Y}_{i}1\{T_i=t,M_i=m\}}{{\sum }_{i}1\{T_i=t,M_i=m\}}\right\}\left\{\frac{1}{n_1}\sum _{i}1\{T_i=1,M_i=m\}-\frac{1}{n_0}\sum _{i}1\{T_i=0,M_i=m\}\right\}.$$

Theorem 3 (asymptotic variance, given realised $T$)

Under Assumption 1, with ${\nu }_{tm}=\mathrm{Pr}(M=m|T=t)$ and ${\mu }_{tm}=E(Y|M=m,T=t)$, the variance of the estimator in Eq. 18 is asymptotically approximated (Delta method) by a sum of three kinds of terms: (i) the conditional outcome variance $\text{Var}(Y|M=m,T=t)$ scaled by ${\nu }_{1-t,m}/(n_t{\nu }_{tm})$-type factors; (ii) the sampling variability of the estimated mediator probabilities in both arms, which carries the factors ${\nu }_{1-t,m}(1-{\nu }_{1-t,m}){\mu }_{tm}^2$ and the cross terms $-2{\sum }_{m<m^{\prime }}{\nu }_{1-t,m}{\nu }_{1-t,m^{\prime }}{\mu }_{tm}{\mu }_{t{m}^{\prime }}$, divided by $n_{1-t}$; and (iii) a $\text{Var}(Y|T=t)/n_t$ term. Population quantities are replaced by sample analogues and strata are aggregated by sample size.

Transcription note

The text extraction scrambles the multi-line displayed variance in Theorem 3, so only its structure (not every coefficient, nor the exact placement of $n_t$ and $n_{1-t}$) is described above. Use the bootstrap for production inference, which the paper also recommends for Eq. 19.

Covariate-adjusted (Eq. 19). With nonparametric fits ${\hat{\mu }}_{tm}(x)$, ${\hat{\nu }}_{tm}(x)$: $\hat{\delta }(t)=\frac{1}{n}{\sum }_i{\sum }_m{\hat{\mu }}_{tm}(X_i)\{{\hat{\nu }}_{1m}(X_i)-{\hat{\nu }}_{0m}(X_i)\}$; no simple variance, use bootstrap.

Continuous $M$ (Eq. 20). Draw $K$ mediator values ${\tilde{m}}_{ti}^{(k)}$ from the fitted conditional ${\hat{\psi }}_t(X_i)=p(M_i|T_i=t,X_i)$ and average ${\hat{\mu }}_{t{\tilde{m}}_{1i}^{(k)}}(X_i)-{\hat{\mu }}_{t{\tilde{m}}_{0i}^{(k)}}(X_i)$ over $i$ and $k$. This is the “counterfactual propagation” idea later used with posterior draws in Bayesian mediation; see Imai 2010 - Bayesian Mediation Sketch for Halo Effects.

Examples

ACME with and without interaction

Take ${\alpha }_2=-0.5,{\beta }_2=1,{\alpha }_3=0.5,{\beta }_3=-0.5,\gamma =0.5,\kappa =1.5$ in the paper’s Monte Carlo. With the interaction the ACME differs by treatment arm ($0.675$ vs $4.03$ on the lognormal outcome scale) so Assumption 2 is clearly violated; in the media framing data the test of $\bar{\delta }(0)=\bar{\delta }(1)$ has $p=0.238$ so no-interaction is tenable.

import numpy as np, statsmodels.api as sm
def lsem_acme(T, M, Y, interact=False):
    X2 = sm.add_constant(T); m2 = sm.OLS(M, X2).fit()
    cols = [T, M] + ([T*M] if interact else [])
    m3 = sm.OLS(Y, sm.add_constant(np.column_stack(cols))).fit()
    b2 = m2.params[1]; g = m3.params[2]; k = m3.params[3] if interact else 0.0
    return {t: b2*(g + t*k) for t in (0, 1)}, m2, m3

Connections

• The ACME as a product of two effects is the continuous-$M$ analogue of Eq. 17 in Imai 2010 - Potential Outcomes Mediation Effects and Sequential Ignorability.
• Sensitivity of ${\beta }_2\gamma$ to correlated errors: Imai 2010 - Sensitivity Analysis with Error Correlation Rho.

See Also

Imai 2010 - Causal Mediation Analysis - Overview, Hierarchical Models