Estimation of Causal Effect Measures in the Presence of Measurement Error in Confounders

Abstract

The odds ratio, risk ratio, and the risk difference are important measures for assessing comparative effectiveness of available treatment plans in epidemiological studies. Estimation of these measures, however, is often challenged by the presence of error-contaminated confounders. In this article, by adapting two correction methods for measurement error effects applicable to the noncausal context, we propose valid methods which consistently estimate the causal odds ratio, causal risk ratio, and the causal risk difference for settings with error-prone confounders. Furthermore, we develop a bootstrap-based procedure to construct estimators with improved asymptotic efficiency. Numerical studies are conducted to assess the performance of the proposed methods.

Appendix: Proof of Theorem

Let $\Delta _{i}(k)= X^*_{ik}+\{A_i(k)-1/2\} {\varvec{\Sigma }}_{\epsilon k}{{\varvec{\gamma }}}_{\mathrm{X}k}$ and $G_i(k)= 1+\exp [\{-\gamma _{0k}-{{\varvec{\gamma }}}_{\mathrm{A}k}^T\bar{A}_i(k-1)-{{\varvec{\gamma }}}_{\mathrm{Z}k}^TZ_i(k)-{{\varvec{\gamma }}}_{\mathrm{X}k}^T\Delta _i(k)\}\{2A_i(k)-1\}]$. We first show $E\left\{ YI(\bar{A}=\bar{a})\prod _{k=0}^K G_k \right\} =E(Y_{\bar{a}})$.

$$\begin{aligned}&E\left\{ YI(\bar{A}=\bar{a})\prod _{k=0}^K G_k\right\} \\&\quad =P({\bar{A}}={\bar{a}})\iiiint y_{\bar{a}}\prod _{k=0}^K g_kf(y_{\bar{a}},{\bar{z}}, {\bar{x}},{\bar{x}}^*|{\bar{A}}={\bar{a}})d{\bar{x}}^*d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}} \left\{ \int \prod _{k=0}^K g_kf({\bar{x}}^*|{\bar{z}}, {\bar{x}},y_{\bar{a}},{\bar{A}}={\bar{a}})d{\bar{x}}^*\right\} f({\bar{z}}, {\bar{x}}, y_{\bar{a}}|{\bar{A}}={\bar{a}}) d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}} \left\{ \prod _{k=0}^K \int g_k f\{x^*_k| x(k)\}d x^*_k \right\} f({\bar{z}}, {\bar{x}}, y_{\bar{a}}|{\bar{A}}={\bar{a}}) d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}}\prod _{k=0}^K\frac{1}{P\{a(k)|,\bar{a}(k-1), z(k), x(k)\}} f({\bar{z}}, {\bar{x}}, y_{\bar{a}}|{\bar{A}}={\bar{a}})d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =P({\bar{A}}={\bar{a}})\iiint y_{\bar{a}}\prod _{k=0}^K\frac{1}{P\{a(k)|,\bar{a}(k-1), z(k), x(k)\}} \dfrac{P({\bar{A}}={\bar{a}}|{\bar{z}}, {\bar{x}}, y_{\bar{a}})f({\bar{z}}, {\bar{x}}, y_{\bar{a}})}{P({\bar{A}}={\bar{a}})}d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =\iiint y_{\bar{a}}\dfrac{1}{P({\bar{A}}={\bar{a}}|{\bar{z}}, {\bar{x}})} {P({\bar{A}}={\bar{a}}|{\bar{z}}, {\bar{x}})f({\bar{z}}, {\bar{x}}, y_{\bar{a}})}d{\bar{z}}d{\bar{x}}dy_{\bar{a}}\\&\quad =E(Y_{{\bar{a}}}). \end{aligned}$$

Using similar arguments, it can then be shown that $E\left\{ I(\bar{A}=\bar{a})\prod _{k=0}^K G_k \right\} =1.$ Therefore, the causal mean $E(Y_{\bar{a}})$ can be consistently estimated by

$$\begin{aligned} \dfrac{\sum _{i=1}^n {\hat{w}}_iY_iI(\bar{A}_i=\bar{a})}{n}\bigg /\dfrac{\sum _{i=1}^n {\hat{w}}_iI(\bar{A}_i=\bar{a})}{n}= \dfrac{\sum _{i=1}^n {\hat{w}}_iY_iI(\bar{A}_i=\bar{a})}{\sum _{i=1}^n {\hat{w}}_iI(\bar{A}_i=\bar{a})}. \end{aligned}$$