## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
Equation (33a), along with the key insights (i), (ii), and (iii), establishes a crucial prior knowledge regarding the dispersion parameter of the linking component \(\boldsymbol{\mathfrak{T}}_{j}^{2}:V(\mathbf{x}^{*}|\mathbf{x})\). Together, they highlight that  
* \(\boldsymbol{\mathfrak{T}}_{j}^{2}=\widehat{\lambda}_{j}^{2}-\lambda_{j}^{2}\) i.e., \(V(\mathbf{x}^{*}|\mathbf{x})=V(\mathbf{x}^{*}|\mathbf{z})-V(\mathbf{x}| \mathbf{z})\) from Equation (33a). * \(\lambda_{j}^{2}=\zeta\times\widehat{\lambda}_{j}^{2}\) i.e., \(V(\mathbf{x}|\mathbf{z})=\zeta\times V(\mathbf{x}^{*}|\mathbf{z})\) from the findings (i), (ii), (iii). Ultimately, the relationships described above can be expressed through the equation \[\boldsymbol{\mathfrak{T}}_{j}^{2}=(1-\zeta)\widehat{\lambda}_{j}^{2}\ \ \text{or}\ \ V( \mathbf{x}^{*}|\mathbf{x})=(1-\zeta)\ V(\mathbf{x}^{*}|\mathbf{z})\] (34) which demonstrates that the dispersion parameter \(\boldsymbol{\mathfrak{T}}_{j}^{2}\) can be represented as a fraction \((1-\zeta)\) of the variance \(\widehat{\lambda}_{j}^{2}:V(\mathbf{x}^{*}|\mathbf{z})\). Thus, with an estimate for \(\widehat{\lambda}_{j}^{2}\), we can determine \(\boldsymbol{\mathfrak{T}}_{j}^{2}\) using the scaling factor \(0<\zeta<1\). In other words, given the availability of both \(\mathbf{x}^{*}\) and \(\mathbf{z}\), we can utilize \(\widehat{\lambda}_{j}^{2}:V(\mathbf{x}^{*}|\mathbf{z})\) as a proxy for estimating \(\boldsymbol{\mathfrak{T}}_{j}^{2}:V(\mathbf{x}^{*}|\mathbf{x})\). The scaling factor \(0<\zeta<1\) reflects our confidence in the adequacy of the known covariate \(\mathbf{z}\) as a substitute for the unobservable true covariate \(\mathbf{x}\). A higher value of \(\zeta\) indicates that \(\boldsymbol{\mathfrak{T}}^{2}\) relies less on \(V(\mathbf{x}^{*}|\mathbf{z})\). Conversely, a lower \(\zeta\) enhances the dependency on the observed variance \(V(\mathbf{x}^{*}|\mathbf{z})\) to inform our understanding of \(\boldsymbol{\mathfrak{T}}^{2}\). In this paper, we suggest utilizing the insights gained about \(\boldsymbol{\mathfrak{T}}_{j}^{2}\) as the prior knowledge for modeling the probability distribution of \(\mathbf{x}^{*}|\mathbf{x}\), which elucidates the cluster-specific relationship between the NDB covariate \(\mathbf{x}^{*}\) and the true covariate \(\mathbf{x}\). Additionally, we will implement a sensitivity analysis for \(\zeta\) and assess how changes in this scaling factor impact the estimates of \(\boldsymbol{\mathfrak{T}}_{j}^{2}\), in order to shed light on how the scaling factor \(\zeta\) influences the effectiveness of the error correction approach. This, in turn, will facilitate the identification of the ideal value for \(\boldsymbol{\mathfrak{T}}_{j}^{2}\) to improve estimation results impacted by the model risk associated with the NDB covariate \(\mathbf{x}^{*}\). **Gibbs sampler modification with the Gustafson correction:** We propose implementing the following enhancements (additional steps) to integrate the resulting system of equations from Equation (33) into the Gibbs sampler for our hierarchical GLM development, as described in Algorithm S2 in Part 2 of the Supplementary Materials:  
1. In line 12, assuming the NDB covariate value \(x_{k}^{*}\) in \(\mathbf{x}^{*}\) at observation \(h\) in the risk cluster \(j\), we add a step to sample the posterior parameters for the linking component in Equation (27) and the covariate model in Equation (29b). This sets the stage for parameter adjustment using the system of equations outlined in line 18: \[\boldsymbol{w}_{j}: