## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
As a result, the integral solution in Equation (32), derived from the complete joint model, aligns with the incomplete joint model in Equation (30). The key point is that while these two solutions originate from different sources, they both describe the same incomplete joint model \(f(\tilde{Y}_{h},x_{h}^{*}\mid z_{h})\) in a practical framework, without needing to evaluate or sample the true covariate \(\mathbf{x}\). This allows for directly matching the parameters of the complete and incomplete joint models. This relationship can be formalized through a system of equations. Since the parameters of the incomplete joint model \(\widehat{\beta}_{j0},\widehat{\beta}_{j1},\widehat{\beta}_{j2},\widehat{ \sigma}_{j}^{2},\widehat{\lambda}_{j}^{2},\widehat{\kappa}_{j0},\widehat{ \kappa}_{j1}\) (with the \(\widehat{\cdot}\) denoting the parameter estimates based on NDB error) in Equation (30) are accessible, the parameters of the marginalized complete joint model \(\beta_{j0},\beta_{j1},\beta_{j2},\sigma_{j}^{2},\lambda_{j}^{2},\kappa_{j0}, \kappa_{j1}\) in Equation (32) can be expressed in terms of these counterparts. Solving this system of equations could provide a guideline for adjusting parameter estimates in the presence of the NDB covariate \(\mathbf{x}^{*}\) (RQ2). The full derivation and detailed explanation of the system of equations are provided in Part 3 of the Supplementary Materials, and the resulting system of equations for the complete joint model parameters is shown below:  
\[\lambda_{j}^{2} =\widehat{\lambda}_{j}^{2}-\overline{\mathbf{z}}_{j}^{2} \tag{33a}\] \[\kappa_{j0} =\widehat{\kappa}_{j0}\] (33b) \[\kappa_{j1} =\widehat{\kappa}_{j1}\] (33c) \[\beta_{j1} =\frac{\widehat{\beta}_{j1}\widehat{\lambda}_{j}^{2}}{\widehat{ \lambda}_{j}^{2}-\overline{\mathbf{z}}_{j}^{2}}\] (33d) \[\beta_{j0} =\widehat{\beta}_{j0}-\frac{\widehat{\beta}_{j1}\widehat{\kappa }_{j0}\overline{\mathbf{z}}_{j}^{2}}{\widehat{\lambda}_{j}^{2}-\overline{ \mathbf{z}}_{j}^{2}}\] (33e) \[\beta_{j2} =\widehat{\beta}_{j2}-\frac{\widehat{\beta}_{j1}\widehat{\kappa }_{j1}\overline{\mathbf{z}}_{j}^{2}}{\widehat{\lambda}_{j}^{2}-\overline{ \mathbf{z}}_{j}^{2}}\] (33f) \[\sigma_{j}^{2} =\widehat{\sigma}_{j}^{2}-\frac{\beta_{j1}^{2}\overline{\mathbf{z }}_{j}^{2}(\widehat{\lambda}_{j}^{2}-\overline{\mathbf{z}}_{j}^{2})}{\widehat{ \lambda}_{j}^{2}} \tag{33g}\]  
**Our novel prior knowledge on \(\overline{\mathbf{z}}_{j}^{2}\) and scaling factor \(\zeta\):** As shown in Equation (33), the adjustment of the parameter estimation in the incomplete joint model largely hinges on the value of \(\overline{\mathbf{z}}_{j}^{2}\). However, as mentioned previously, estimating \(\overline{\mathbf{z}}_{j}^{2}\) in Equation (27) is challenging until the relationship between the NDB covariate \(\mathbf{x}^{*}\) and the true covariate \(\mathbf{x}\) is clarified. The system of equations, particularly Equation (33a), now reveals that \(\overline{\mathbf{z}}_{j}^{2}\) can be expressed as \(\widehat{\lambda}_{j}^{2}-\lambda_{j}^{2}\). In other words, \(V(\mathbf{x}^{*}|\mathbf{x})=V(\mathbf{x}^{*}|\mathbf{z})-V(\mathbf{x}|\mathbf{z})\). From this, several key insights (our own findings) can be drawn:  
1. \(\widehat{\lambda}_{j}^{2}:V(\mathbf{x}^{*}|\mathbf{z})\) is always greater than \(\lambda_{j}^{2}:V(\mathbf{x}|\mathbf{z})\) according to Equation (33a). 2. Given that \(x^{*}|z\sim\mathbf{N}(\widehat{x}_{j0}+\widehat{x}_{j1}z,\ \widehat{\lambda}_{j}^{2})\) and \(x|z\sim\mathbf{N}(\kappa_{j0}+\kappa_{j1}z,\ \lambda_{j}^{2})\), it appears that \(\widehat{\kappa}_{j0}=\kappa_{j0}\) and \(\widehat{\kappa}_{j1}=\kappa_{j1}\), hence \(E[\mathbf{x}^{*}|\mathbf{z}]=E[\mathbf{x}|\mathbf{z}]\) according to Equations (33b) and (33c). 3. Given (i) and (ii), it is safe to say that the variance of the true covariate can be a scalar multiple of the variance of the NDB covariate: \(\lambda_{j}^{2}=\zeta\times\widehat{\lambda}_{j}^{2}\) where \(0<\zeta<1\) is a _scaling factor_.