## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
Given two covariate matrices \(\mathbf{X}^{F}:\{\mathbf{x}^{F},\mathbf{z}^{F}\}\) for the negative binomial outcome \(N_{h}\) and \(\mathbf{X}^{S}:\{\mathbf{z}^{S},\mathbf{x}^{S}\}\) for the log-normal outcome \(\bar{Y}_{h}\) respectively, we consider the case where the continuous covariate \(\mathbf{x}^{S}\) contains mismeasurements, classified as Non-Differential Berkson (NDB) measurement errors. Since the covariates for the negative binomial outcome \(N_{h}\) are assumed to be complete, this section particularly focuses on the log-normal outcome \(\bar{Y}_{h}\) and its covariate \(\mathbf{x}^{S}\) within the hierarchical framework. For simplicity, we will omit the superscript \(s\) and refer to \(\mathbf{x}^{S}\) as simply \(\mathbf{x}\) in the remainder of the discussion. In Section 3.2, we briefly introduce [PERSON]'s error correction method to address model risk associated with the NDB covariate. The key idea is to estimate or sample the parameter values for the true covariate using a specialized joint model. This model incorporates prior knowledge of \(\theta_{j}\) through the linking component \(f(\mathbf{x}^{*}|\mathbf{x},\theta_{j})\), which captures the cluster-wise relationship between the true covariate \(\mathbf{x}\) and the NDB covariate \(\mathbf{x}^{*}\)[8]. The main objective of this section is to accurately estimate the parameter values \(\theta_{j}\) for the joint model. To elaborate, the complete joint model that encompasses the outcome \(\bar{Y}_{h}\), the NDB covariate \(\mathbf{x}^{*}\), the true covariate \(\mathbf{x}\), and the additional covariate \(\mathbf{z}\) is expressed as per Equation (8) as:  
\[\beta_{0}^{F}\mid\underline{m_{0}},\underline{\measuredangle},N,\mathbf{X}^{F}, \ \Sigma_{\beta_{0}}^{F}\sim\ \mathbf{MVN}\Bigg{(}\frac{\hat{\xi}}{\hat{\xi}+1} \underline{m_{0}}+\frac{1}{\hat{\xi}+1}\beta^{F},\ \frac{\Sigma_{\beta_{0}}^{F}}{\hat{\xi}+1}\Bigg{)}\]2  
where the true covariate \(\mathbf{x}\) is included and is unknown in real life. As outlined in Section 3.2, the full joint distribution can be simplified to the form in Equation (25) because the NDB covariate \(\mathbf{x}^{*}\) is uncorrelated with any other variables except the true covariate \(\mathbf{x}\) itself. To integrate [PERSON]'s complete joint model for the NDB covariate from Equation (25) into the hierarchical GLM framework, we redefine the risk premium modeling structure along with the priors originally specified in Equation (14) as shown below:\[\beta_{0}^{F}\mid\underline{m_{0}},\underline{\measuredangle},N,\mathbf{X}^{F}, \ \Sigma_{\beta_{0}}^{F}\sim\ \mathbf{MVN}\Bigg{(}\frac{\hat{\xi}}{\hat{\xi}+1} \underline{m_{0}}+\frac{1}{\hat{\xi}+1}\beta^{F},\ \frac{\Sigma_{\beta_{0}}^{F}}{\hat{\xi}+1}\Bigg{)}\]3 \[\text{For }N_{h} \begin{cases}\beta_{j}^{F}\mid\beta_{0}^{F}, \Sigma_{\beta_{0}}^{F}\sim\mathbf{MVN}(\beta_{0}^{F},\ \Sigma_{\beta_{0}}^{F})\\ \psi_{j}\mid u_{0}^{F},v_{0}^{F}\sim\mathbf{Ga}(\frac{u_{0}^{F}}{2},\frac{v_{0 }^{F}}{2})\end{cases} \] (26b) \[\text{For }\bar{Y}_{h} \begin{cases}\beta_{j}^{S}\mid\beta_{0}^{S}, \Sigma_{\beta_{0}}^{S}\sim\mathbf{MVN}(\beta_{0}^{S},\ \Sigma_{\beta_{0}}^{S})\\ \sigma_{j}^{2}\mid u_{0}^{S},v_{0}^{S}\sim\mathbf{InvGa}(\frac{u_{0}^{S}}{2}, \frac{v_{0}^{S}}{2})\end{cases} \] (26c) \[\text{For }\mathbf{X}^{F} \begin{cases}x_{j(h)}^{F}\sim\mathbf{N}(E[ \mathbf{x}_{j}^{F}],\ \lambda_{j}^{2F})\\ \lambda_{j}^{2F}\sim\ \mathbf{InvGa}(c_{0}^{F},d_{0}^{F})\\ z_{j(h)}^{F}\sim\mathbf{Bernoulli}(\pi_{j}^{F})\\ \pi_{j}^{F}\sim\ \mathbf{Beta}(g_{0}^{F},h_{0}^{F})\\ \end{cases} \] (26d) \[\text{For }\mathbf{X}^{S*}