## 2 Background and Conventional Approaches Related to RQ1 and RQ2
### NDB Errors in Covariates  
Before delving into the methods for correcting measurement errors in covariates, we first provide an overview of the various types of measurement error. Consider \(\mathbf{X}^{\mathrm{S}}=\{\mathbf{x}^{\mathrm{S}},\ \boldsymbol{\varepsilon}^{ \mathrm{S}}\}\), where only \(\mathbf{x}^{\mathrm{S}}\), a continuous covariate, is subject to mismeasurement, while \(\boldsymbol{\varepsilon}^{\mathrm{S}}\), a binary covariate, is complete. For simplicity, we omit the superscript \(s\) from the covariate matrix \(\mathbf{X}^{\mathrm{S*}}\) and the continuous covariate vector \(\mathbf{x}^{\mathrm{S*}}\), referring to them as \(\mathbf{X}^{\mathrm{*}}\) and \(\mathbf{x}^{\mathrm{*}}\), respectively. The measurement error \(\boldsymbol{\epsilon}\) and the error-prone observed covariate \(\mathbf{x}^{\mathrm{*}}\) are assumed to follow \(\boldsymbol{\epsilon}\sim\mathbf{N}(0,\ \sigma_{\boldsymbol{\varepsilon}}^{2})\) and \(\mathbf{x}^{\mathrm{*}}\sim\mathbf{N}(\mathbf{x},\ \boldsymbol{\Xi}^{2})\). The approach to handling mismeasurement covariates depends on the nature of \(\boldsymbol{\epsilon}\). Now, the different types of measurement errors \(\boldsymbol{\epsilon}\) are outlined as follows:  
* Additive vs. multiplicative [14]: With _additive_ error, the mismeasurement covariate \(\mathbf{x}^{\mathrm{*}}\) can be expressed as \(\mathbf{x}^{\mathrm{*}}=\mathbf{x}+\boldsymbol{\epsilon}\), while the _multiplicative_ error can be represented as \(\mathbf{x}^{\mathrm{*}}=\mathbf{x}\cdot\boldsymbol{\epsilon}\). * Differential vs. non-differential [15]: The error \(\boldsymbol{\epsilon}\) can be classified as _differential_ or _non-differential_ based on its relationship with other variables. Differential error occurs when the mismeasurement covariate \(\mathbf{x}^{\mathrm{*}}\) is correlated with the outcome, represented as \(\mathbf{x}^{\mathrm{*}}\sim\ Y|\ \mathbf{z},\mathbf{x}\). Conversely, non-differential error arises when the error \(\boldsymbol{\epsilon}\) in \(\mathbf{x}^{\mathrm{*}}\) provides no additional information about the outcome beyond what is already contained in \(\mathbf{x}\), leading to \(\mathbf{x}^{\mathrm{*}}\bot\ Y|\ \mathbf{x},\mathbf{z}\). This implies that \(f(Y|\mathbf{x}^{\mathrm{*}},\mathbf{x},\mathbf{z})=f(Y|\mathbf{x},\mathbf{z})\) and \(f(\mathbf{x}^{\mathrm{*}}|Y,\mathbf{x},\mathbf{z})=f(\mathbf{x}^{\mathrm{*}}| \mathbf{x})\). * Classical vs. Berkson [16]: _Classical_ error occurs when the error \(\boldsymbol{\epsilon}\) is independent of the true covariates, leading to \(\boldsymbol{\epsilon}\bot\ Y,\mathbf{x},\mathbf{z}\), \(E[\boldsymbol{\epsilon}|\mathbf{x}^{\mathrm{*}},\mathbf{z}]=0\) and \(\boldsymbol{\Xi}^{2}=V(\mathbf{x})+V(\boldsymbol{\epsilon})>V(\mathbf{x})\). In contrast, if the error \(\boldsymbol{\epsilon}\) is independent of the observed covariates but associated with other latent factors with multiple levels, _Berkson_ error applies, where \(\boldsymbol{\epsilon}_{j}\perp\ Y,\mathbf{x}^{\mathrm{*}},\mathbf{z}\)resulting in \(E[\mathbf{e}_{j}|\mathbf{x}^{*},\mathbf{z}]=0\) and \(\mathbf{\Xi}_{j}^{2}=V(\mathbf{x}_{j})+V(\mathbf{e}_{j})<V(\mathbf{x})\) for \(j=1,\ldots,J\). This implies that the error \(\mathbf{e}\) can display varying levels of heteroscedasticity across different risk clusters due to the latent factors. We focus on the case in which the mismeasured covariate \(\mathbf{x}^{*}\) arises from additive measurement error in a manner that is non-differential (blind to the outcome and other covariates) and Berkson (correlated to the latent factors), allowing us to address error structures that may vary across risk clusters. Throughout this paper, we refer to it as the 'NDB covariate'. Regression Calibration (RC) is a widely used method for addressing the NDB covariate in the regression framework. RC intuitively corrects errors by leveraging the relationship between the NDB covariate and true covariate, regressing on the expected true covariates given the NDB ones [17]. [PERSON] et al.