Sensitivity analysis

Recall: mediation models

For unit $i$ in a random sample of size $n$, we denote $T_i\in\{0,1\}$ as the treatment indicator, $X_i$ is pre-treatment covariate, $M_i$ is the mediator, $\mu_i\in\mathcal{R}^{K}$ is transformed from an compositional outcome with $K$ entries. Then the mediation model assumes (1) and (2), where $\alpha$’s and $\beta$’s are parameters, $\varepsilon_i^m$ and $\varepsilon^i_{\mu}$ are errors.

$$\begin{equation} m_i = \alpha_0 + \alpha_T T_i + \alpha_X^TX_i + \varepsilon_i^{m} \tag{1} \end{equation}$$

$$\begin{equation} \mu_{i} = \beta_0 + \beta_T T_i + \beta_X^TX_i + \beta_M^TM_i + \varepsilon_i^{\mu} \tag{2} \end{equation}$$

Sensitivity in terms of correlations

Let $U_i$ be unmeasured confounder, then when the sequential ignorability assumption is violated, we can rewrite error terms for taxon $k$ as

$$\begin{align*} \varepsilon_{ik}^{\mu} &= \lambda^{\mu}_k U_{ik} + \tilde{\varepsilon}_{ik}^{\mu}\\ \varepsilon_{ik}^m &= \lambda^m_k U_{ik} + \tilde{\varepsilon}_{ik}^m, \end{align*}$$ where $\text{Cov}(\tilde{\varepsilon}_{ik}^{\mu}, \tilde{\varepsilon}_{ik}^{m})=0$, $\lambda_{k}^{\mu}$ and $\lambda_k^m$ are unknown parameters.

We define the sensitivity parameter $\rho_k$ to be their correlation

$$\begin{equation} \rho_k = \text{Cor}(\varepsilon_{ik}^{\mu},\varepsilon_{ik}^m), \tag{3} \end{equation}$$

which after some calculation, can be formulated as

$$\begin{align*} \rho_k &= \text{sgn}(\lambda^{\mu}_k\lambda_k^m)\left\{1-\dfrac{\text{var}(\tilde{\varepsilon_{ik}^m})}{\text{var}(\tilde{\epsilon}_{ik}^m)}\right\}\left\{1-\dfrac{\text{var}(\tilde{\varepsilon}_{ik}^{\mu})}{\text{var}(\tilde{\epsilon}_{ik}^{\mu})}\right\}. \end{align*}$$

Reference

A general approach to causal mediation analysis Identification, Inference and Sensitivity Analysis for Causal Mediation Effects