## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
Instead, more practical terms available for implementation are:  
\[f_{LogN}(\bar{Y}_{h}|x_{h}^{*},z_{h})= \frac{1}{\bar{Y}_{h}\sqrt{2\pi\widehat{\sigma}_{j}^{2}}}\exp\{- \frac{1}{2}\Big{[}\frac{ln\bar{Y}_{h}-(\widehat{\beta}_{j0}+\widehat{\beta}_{ j1}x_{h}^{*}+\widehat{\beta}_{j2}z_{h})}{\widehat{\sigma}_{j}}\Big{]}^{2}\} \tag{29a}\] \[f_{N}(x_{h}^{*}|z_{h})= \frac{1}{\sqrt{2\pi\widehat{\lambda}_{j}^{2}}}\exp\{\frac{-(x_{h }^{*}-\{\widehat{\kappa}_{j0}+\widehat{\kappa}_{j1}z_{h}\})^{2}}{2\widehat{ \lambda}_{j}^{2}}\} \tag{29b}\]  
where \(\bar{Y}_{h}|x_{h}^{*},z_{h}\sim\mathbf{LogN}(\mathbf{X}^{*}\widehat{\beta}_{ j},\widehat{\sigma}_{j}^{2})\), \(\widehat{\sigma}_{j}^{2}:V(\bar{Y}|\mathbf{X}^{*})\), \(x_{h}^{*}|z_{h}\sim\mathbf{N}(\widehat{x}_{j0}+\widehat{\kappa}_{j1}z_{h},\ \widehat{\lambda}_{j}^{2})\), and \(\widehat{\lambda}_{j}^{2}:V(\mathbf{x}^{*}|\mathbf{z})\) (the notation \(\widehat{\cdot}\) is used to indicate that these parameters are derived from the covariate with NDB errors, prior to correction). By multiplying these two terms--the outcome model and the covariate model--in Equation (29), we obtain the _incomplete joint model_ in Equation (30), as per Equation (9), which serves as the practical available solution:  
\[f(\bar{Y}_{h},x_{h}^{*}\mid z)=\frac{1}{\bar{Y}_{h}(2\pi)\widehat{ \sigma}_{j}\widehat{\lambda}_{j}} \times\exp\Big{(}-\frac{1}{2\widehat{\sigma}_{j}^{2}}\Big{[}(ln \bar{Y}_{h}-\widehat{\beta}_{j0}-\widehat{\beta}_{j2}z_{h})-\widehat{\beta}_{ j1}x_{h}^{*}\Big{]}^{2}\Big{)} \tag{30}\] \[\times\exp\Big{(}-\frac{1}{2\widehat{\lambda}_{j}^{2}}\Big{[}x_{ h}^{*}-(\widehat{\kappa}_{j0}+\widehat{\kappa}_{j1}z_{h})\Big{]}^{2}\Big{)}\]  
However, this joint model has a notable limitation: it is developed without the true covariate \(\mathbf{x}\). Fortunately, connecting the complete joint model in Equation (25) with the incomplete joint model in Equation (30) is a straightforward process. As noted in Section 3.2, we can obtain another incomplete model by marginalizing the complete joint model in Equation (25) over the true covariate \(\mathbf{x}\) using the following integral:  
\[\int f(\bar{Y}_{h},x_{h}^{*},x_{h}\mid z_{h})\ dx_{h} \tag{31}\] \[\qquad=\int_{\mathbf{x}} \frac{1}{\sigma_{j}\bar{Y}_{h}\sqrt{2\pi}}\exp\{-\frac{1}{2} \Big{[}\frac{ln\bar{Y}_{h}-(\beta_{j0}+\beta_{j1}x_{h}+\beta_{j2}z_{h})}{ \sigma_{j}}\Big{]}^{2}\}\] \[\qquad\qquad\times\frac{1}{\sqrt{2\pi\mathbf{x}_{j}^{2}}}\exp\{ \frac{-(x_{h}^{*}-x_{h})^{2}}{2\mathbf{x}_{j}^{2}}\}\times\frac{1}{\sqrt{2\pi \lambda_{j}^{2}}}\exp\{\frac{-(x_{h}-\{\kappa_{j0}+\kappa_{j1}z_{h}\})^{2}}{2 \lambda_{j}^{2}}\}\,dx_{h}\]The evaluation process for the integral in Equation (31) is detailed in Part 3 of the Supplementary Materials, and the resulting solution is presented below:  
\[ \begin{split}&\int f(\tilde{Y}_{h},x_{h}^{*},x_{h}\mid z_{h})\ dx_{h}\\ &=\frac{1}{\tilde{Y}_{h}(2\pi)\sigma_{j}\overline{\mathbf{z}}_{j} \lambda_{j}}\Big{(}\frac{\sigma_{j}^{2}\lambda_{j}^{2}\overline{\mathbf{z}}_{ j}^{2}}{\beta_{h}^{2}\overline{\mathbf{z}}_{j}^{2}\lambda_{j}^{2}+\sigma_{j}^{2} \lambda_{j}^{2}+\sigma_{j}^{2}\overline{\mathbf{z}}_{j}^{2}}\Big{)}^{1/2}\\ &\quad\times\exp\bigg{(}-\frac{1}{2}\Big{(}\frac{\lambda_{j}^{2}+ \overline{\mathbf{z}}_{j}^{2}}{\beta_{h}^{2}\overline{\mathbf{z}}_{j}^{2} \lambda_{j}^{2}+\sigma_{j}^{2}\lambda_{j}^{2}+\sigma_{j}^{2}\overline{\mathbf{z }}_{j}^{2}}\Big{)}\Big{[}(ln\tilde{Y}_{h}-\beta_{j0}-\beta_{j2}z_{h})-\frac{ \beta_{j1}\big{(}\frac{x_{h}^{*}}{\overline{\mathbf{z}}_{j}^{2}}+\frac{\kappa _{j0}+\kappa_{j1}z_{h}}{\lambda_{j}^{2}}\big{)}^{2}\Big{)}}\\ &\quad\times\exp\bigg{(}-\frac{1}{2}\Big{(}\frac{1}{\overline{ \mathbf{z}}_{j}^{2}+\lambda_{j}^{2}}\Big{)}\Big{[}\Big{(}x_{h}^{*}-(\kappa_{ j0}+\kappa_{j1}z_{h})\Big{)}^{2}\Big{]}\bigg{)}=f(\tilde{Y}_{h},x_{h}^{*}\mid z_{h} )\end{split}  \tag{32}\]  
In Equation (32), the unobservable true covariate term \(\mathbf{x}\) is completely eliminated through this marginalization process.