Imai 2010 - Sensitivity Analysis with Error Correlation Rho

Summary

Randomising $T$ guarantees Eq. (4) but not Eq. (5), so unmeasured pre-treatment variables can confound $M$ and $Y$. The sensitivity parameter $\rho =\text{Corr}({\epsilon }_{i2},{\epsilon }_{i3})$ (zero under sequential ignorability) indexes the violation. Theorem 4 gives the ACME as a closed-form function of $\rho$; Corollary 1 shows the data cannot bound $\rho$ ($(-1,1)$) or the ACME ($(-\infty ,\infty )$). The ACME changes sign exactly at $\rho =\tilde{\rho }$, the residual correlation from the reduced-form equations.

Overview

The proof of Theorem 2 shows that under Eq. (4) alone, ${\epsilon }_{i2}\perp T_i$ and ${\epsilon }_{i3}\perp T_i$ but ${\epsilon }_{i2}/\perp {\epsilon }_{i3}$ unless Eq. (5) also holds. The sensitivity parameter

$$\rho \equiv \text{Corr}({\epsilon }_{i2},{\epsilon }_{i3}),-1<\rho <1(24)$$

is nonzero when omitted variables related to both the observed mediator and the potential outcomes remain after conditioning on $T_i$ and $X_i$. These variables must causally precede $T_i$; post-treatment confounders are not covered. (Roy, Hogan & Marcus 2008 used the same strategy of evaluating a quantity over an unidentifiable sensitivity parameter.)

Main Content

Theorem 4 (identification given $\rho$)

In the LSEM (11)-(13) with Eq. (4) holding and $\rho$ given, with ${\sigma }_j^2=\text{Var}({\epsilon }_{ij})$ for $j=1,2$ (reduced-form errors from (11) and (12)) and $\tilde{\rho }=\text{Corr}({\epsilon }_{i1},{\epsilon }_{i2})$:

$$\bar{\delta }(0)=\bar{\delta }(1)=\frac{{\beta }_2{\sigma }_1}{{\sigma }_2}\left\{\tilde{\rho }-\rho \sqrt{\frac{1-{\tilde{\rho }}^2}{1-{\rho }^2}}\right\}.$$

Proof (Appendix D). Estimable: $({\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2)$ from equation-by-equation OLS of (11), (12), plus ${\sigma }_1^2,{\sigma }_2^2,\tilde{\rho }$ from the residuals. Write ${\epsilon }_{i1}=\gamma {\epsilon }_{i2}+{\epsilon }_{i3}$ (from (14)). Then $\tilde{\rho }{\sigma }_1{\sigma }_2=\gamma {\sigma }_2^2+\rho {\sigma }_2{\sigma }_3$ and ${\sigma }_1^2={\gamma }^2{\sigma }_2^2+{\sigma }_3^2+2\gamma \rho {\sigma }_2{\sigma }_3$. If $\rho =0$: $\gamma =\tilde{\rho }{\sigma }_1/{\sigma }_2$. Otherwise substitute ${\sigma }_3=(\tilde{\rho }{\sigma }_1-\gamma {\sigma }_2)/\rho$ to get the quadratic ${\gamma }^2-2\gamma \tilde{\rho }{\sigma }_1/{\sigma }_2+{\sigma }_1^2({\tilde{\rho }}^2-{\rho }^2)/\{{\sigma }_2^2(1-{\rho }^2)\}=0$; the root with ${\sigma }_3\ge 0$ is $\gamma =\frac{{\sigma }_1}{{\sigma }_2}\{\tilde{\rho }-\rho \sqrt{(1-{\tilde{\rho }}^2)/(1-{\rho }^2)}\}$. Multiply by ${\beta }_2$.

Remarks on Theorem 4:

1. Estimation: covariance matrix of $({\epsilon }_{i1},{\epsilon }_{i2})$ from OLS residuals of (11)-(12). For standard errors the paper recommends iterative feasible GLS for seemingly unrelated regressions (Zellner 1962); this also handles interactions and covariates.
2. The ACME is monotone in $\rho$ (increasing or decreasing depending on $\text{sign}({\beta }_2)$) and symmetric about $(\rho ,\bar{\delta })=(0,{\beta }_2\tilde{\rho }{\sigma }_1/{\sigma }_2)$.
3. $\bar{\delta }=0⟺\rho =\tilde{\rho }$. So if $\hat{\delta }={\hat{\beta }}_2\hat{\gamma }<0$ the true ACME is negative for all $\rho <\tilde{\rho }$ (the opposite sign convention if positive).

Corollary 1 (no information about $\rho$)

Under the LSEM with Eq. (4) but not necessarily Eq. (5), the sharp bounds on $\rho$ and the ACME are $(-1,1)$ and $(-\infty ,\infty )$.

(The first statement follows from the proof of Theorem 4; the second by taking limits of $\bar{\delta }$ as $\rho \to \pm 1$.)

Appendix C: Assumption 1 implies $\rho =0$

Under Eqs. (4)-(5), $E({\epsilon }_{i2}(T_i)|T_i)=0$ and $E({\epsilon }_{i3}(T_i,M_i(T_i))|T_i,M_i)=0$. Then $0=E[{\epsilon }_{i3}{M}_i]=E[{\epsilon }_{i3}({\alpha }_2+{\beta }_2{T}_i+{\epsilon }_{i2})]=E[{\epsilon }_{i3}{\epsilon }_{i2}]$, so $\rho =0$; equivalently $\rho \ne 0$ implies a violation of Assumption 1.

Interpretation through omitted-variable $R^2$ (Sec. 5.2)

Decompose ${\epsilon }_{ij}={\lambda }_j{U}_i+{\epsilon }_{ij}^{\prime }$, $j=2,3$, with $U_i$ an unobserved pre-treatment confounder, sequential ignorability holding given $(U_i,T_i)$, and ${\epsilon }_{ij}^{\prime }\perp U_i$. Define the proportions of previously unexplained variance explained by $U$:

$$R_M^{2\ast }=1-\frac{\text{Var}({\epsilon }_{i2}^{\prime })}{\text{Var}({\epsilon }_{i2})},R_Y^{2\ast }=1-\frac{\text{Var}({\epsilon }_{i3}^{\prime })}{\text{Var}({\epsilon }_{i3})},$$

and of the original variance: ${\tilde{R}}_M^2=(1-R_M^2)R_M^{2\ast }$, ${\tilde{R}}_Y^2=(1-R_Y^2)R_Y^{2\ast }$, where $R_M^2,R_Y^2$ are the $R^2$ of regressions (12), (13). (The tilde versions are bounded above by $\text{Var}({\epsilon }_{i2})/\text{Var}(M_i)$ and $\text{Var}({\epsilon }_{i3})/\text{Var}(Y_i)$.) Then

$${\rho }^2=R_M^{2\ast }{R}_Y^{2\ast }=\frac{{\tilde{R}}_M^2{\tilde{R}}_Y^2}{(1-R_M^2)(1-R_Y^2)},\rho =\text{sgn}({\lambda }_2{\lambda }_3)R_M^{\ast }{R}_Y^{\ast }.$$

So the analyst specifies the two $R^2$ values and the sign of ${\lambda }_2{\lambda }_3$. This is the mediation counterpart of the partial-$R^2$ parameterisation in Cinelli Hazlett 2020 - Partial R2 Parameterization and Bias Formulas (cf. Imbens 2003).

Extensions (Sec. 5.3, Appendix E)

• Probit models for discrete mediator/outcome: Imai, Keele & Tingley (2009).
• Nonparametric, binary $M$ and $Y$. Sharp bounds without Eq. 5 (Eq. 29), written in terms of $P_{ymt}=\mathrm{Pr}(Y=y,M=m|T=t)$; for $\bar{\delta }(1)$ the lower bound is $\max \{-P_{001}-P_{011},-P_{000}-P_{001}-P_{100},-P_{011}-P_{010}-P_{110}\}$ and the upper bound $\min \{P_{101}+P_{111},P_{000}+P_{100}+P_{101},P_{010}+P_{110}+P_{111}\}$. These always contain zero, so the sign of the ACME is unidentified without more assumptions. The sensitivity parameter $\upsilon$ is the maximum absolute difference in the proportion of any principal stratum between those with different mediator values under the same treatment (Eq. 31); for fixed $\upsilon$ the sharp bounds solve a linear programme over the stratum probabilities ${\pi }_{y_{11}{y}_{10}{y}_{01}{y}_{00}}^{m_1{m}_0}$ with constraints (32)-(35) and sum to one (constraint (35) ties the stratum probabilities to the observed $P_{ymt}$ via ignorability of treatment; $\upsilon \in [0,1]$ and $\upsilon =0$ iff sequential ignorability holds).
  • Fig. 3 (media framing, mediator and treatment dichotomised at sample medians). The sharp upper and lower bounds on $\bar{\delta }(1)$ and $\bar{\delta }(0)$ widen with $\upsilon$ until they flatten at the no-assumption bounds (Eqs. 29-30). The upper bounds cross zero at very small $\upsilon$ ($0.019$ for $\bar{\delta }(1)$, $0.022$ for $\bar{\delta }(0)$), so the nonparametric analysis finds the ACME rather sensitive to violations of sequential ignorability, in contrast to the apparent robustness of the parametric LSEM analysis; the paper cautions that the two are not directly comparable because the parameters differ.

Examples

Checking Theorem 4 by simulation

$T\sim \text{Bernoulli}(0.5)$, $U\sim N(0,1)$, $M=0.5+0.8T+0.7U+0.7e_1$, $Y=1+0.4T+0.6M+0.9U+e_2$, $n=4\times 10^5$. True ACME $=0.8\times 0.6=0.48$. The Baron-Kenny estimate ignoring $U$ is $0.993$ ($\rho =0$ assumed). The residual correlation $\tilde{\rho }=0.720$. The structural $\rho =\frac{0.7\cdot 0.9}{\sqrt{0.98}\sqrt{1.81}}=0.473$. Theorem 4 at $\rho =0.473$ gives $0.479\approx 0.48$. Values along the curve: $\rho =-0.5\to 1.546$; $0\to 0.993$; $0.3\to 0.692$; $0.6\to 0.275$; $0.72\to 0$.

import numpy as np
def acme_given_rho(beta2, s1, s2, rho_tilde, rho):
    return beta2*s1/s2*(rho_tilde - rho*np.sqrt((1-rho_tilde**2)/(1-rho**2)))
rho_grid = np.linspace(-0.95, 0.95, 77)
# usage: curve = acme_given_rho(b2_hat, s1_hat, s2_hat, rt_hat, rho_grid); zero crossing at rho = rho_tilde

Connections

• Reading the curve: how correlated must unmeasured mediator-outcome causes be to flip the sign of the indirect effect? In the marketing halo example (E35) the “search as control” vs “search left out” contrast is a coarser version of the same question; see Imai 2010 - Bayesian Mediation Sketch for Halo Effects.
• OVB analogue for the outcome regression: Cinelli Hazlett 2020 - Robustness Value and Extreme Scenarios.

See Also

Imai 2010 - Causal Mediation Analysis - Overview, Sensitivity Analysis in Observational Studies