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[**Appendix for “Propensity score-based estimators with multiple error-prone covariates"**]{}

Hwanhee Hong, David A. Aaby, Juned Siddique, and Elizabeth A. Stuart

Appendix 1. Total Confounding
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In our first simulation study, the true outcome model is $$\label{eq:true_reg}
Y = \tau A + \beta_1 X_1 + \beta_2 X_2 + \epsilon.$$ Here, we assume the intercept is zero without loss of generality. If I omit $X_1$ and $X_2$, the model is rewritten as $$\label{eq:omitted_reg}
Y = \tau^* A + \epsilon^*.$$ The estimated $\tau^*$ under  is $$\label{eq:beta_hat_star}
\hat{\tau^*} = (A^T A)^{-1} A^T Y.$$ To calculate the bias of $\hat{\tau^*}$ under the true outcome model , let’s first plug in  to . $$\begin{aligned}
 \notag
\hat{\tau^*} = & (A^T A)^{-1} A^T Y \\ \notag
                = & (A^T A)^{-1} A^T (\tau A + \beta_1 X_1 + \beta_2 X_2 + \epsilon) \\ \notag
                = & (A^T A)^{-1} A^T A \tau + (A^T A)^{-1} A^T X_1 \beta_1 + (A^T A)^{-1} A^T X_2 \beta_2 + (A^T A)^{-1} A^T \epsilon\end{aligned}$$ Then, $$\begin{aligned}
 \notag
E(\hat{\tau^*} {\,|\,}A) = & \tau + (A^T A)^{-1} E(A^T X_1 {\,|\,}A) \beta_1 + (A^T A)^{-1} E(A^T X_2 {\,|\,}A) \beta_2 \\ \notag
= & \tau + bias.        \end{aligned}$$ We define this bias as our total confounding. We can rewrite $(A^T A)^{-1} E(A^T X_1 {\,|\,}A) = E(X_1 {\,|\,}A=1)-E(X_1 {\,|\,}A=0)$ because the left hand side is the estimated coefficient of a regression model of $X_1 {\sim} A$. Then, the total confounding can be rewritten as $$\notag
\beta_1[E(X_1 {\,|\,}A=1)-E(X_1 {\,|\,}A=0)] + \beta_2 [E(X_2 {\,|\,}A=1)-E(X_2 {\,|\,}A=0)].$$

Appendix 2. Additional results
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![Bias of the IPTW estimator in Simulation 1 when data are generated using negative values for $\rho^x$ and ${ \mbox{\boldmath $ \alpha $} }$. We vary ${ \mbox{\boldmath $ \beta $} }$ for this plots to have the same total confounding across all settings.[]{data-label="table:negative"}](case1_TC_fix_edit.pdf){height="17cm"}

  Sex       Measurement    $\widehat{ATE}_{DR}$ (95% CI)    SE     P-value
  --------- ------------- ------------------------------- ------- ---------
  Males     Biomarker          0.553 (-0.538, 1.645)       0.557    0.322
            24HR               1.03 (-0.106, 2.166)        0.58     0.077
            FFQ                1.084 (-0.038, 2.207)       0.573    0.059
  Females   Biomarker         -0.476 (-1.951, 0.999)       0.752    0.528
            24HR              -0.787 (-2.311, 0.737)       0.777    0.313
            FFQ               -0.872 (-2.354, 0.611)       0.756    0.25

  : DR estimates from OPEN data analyses. Reliabilities and SMDs of biomarkers are omitted because they are the same as in Table 3 in the main manuscript.[]{data-label="table:OPENresults"}
