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