## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
In Section 2, we previously discussed the partial pooling effect that can be attained by utilizing the hierarchical structure within this GLM framework. In short, the hierarchical GLM in Equation (14) explores unique parameter values for each individual (saturated cohort) and each cluster (reduced cohort), pooling them across multiple clusters rather than averaging the parameter values based on the available information [PERSON] and [PERSON] (2013). To achieve this, the GLM coefficients and dispersion parameters--\(\mathbf{\beta}_{j}^{F},\psi_{j},\mathbf{\beta}_{j}^{S},\sigma_{j}^{2}\)--should vary by each cluster (i.e., no-pooling). At the same time, the corresponding hyperparameters \(\mathbf{\beta}_{0}^{F},\Sigma_{\beta_{0}}^{F},u_{0}^{F},v_{0}^{F},\mathbf{\beta}_{0}^{ S},\Sigma_{\beta_{0}}^{S},u_{0}^{S},v_{0}^{S}\) should be updated by the entire data (i.e., complete pooling). The hyperpriors  
where \(1/\delta\) serves as a variance inflation factor; \(m_{0}\) and \(\Sigma_{\beta_{0}}\) denote the mean vector and variance-covariance matrix of the GLM coefficients, respectively; and \(q_{0}\) and \(\Lambda\) are the degrees of freedom and the scale matrix of an Inverse Wishart hyperprior to sample the variance-covariance matrix, respectively. As \(q_{0}\) increases, its scale matrix \(\Lambda\) becomes smaller, and thus the variance-covariance matrix \(\Sigma_{\beta_{0}}\) becomes more influential [25]. \(\rho_{u1}\) and \(\rho_{u2}\) are the shape and rate parameters of a Gamma hyperprior. Such distributional knowledge discussed here can be taken into account when we simulate the posterior parameter samples with the Gibbs sampler. The selection of hyperpriors in Equations (15)-(18) is informed by distributions that are conjugate to the priors outlined in Equation (14), significantly streamlining the Bayesian updating process. For the hyperpriors of \(u_{0}^{F}\) and \(u_{0}^{S}\) specified in Equations (16) and (18), we utilize analytically derived kernels from [PERSON] (1997) [26] due to their conjugacy with the Gamma and Inverse Gamma distributions of \(\psi_{j}\) and \(\sigma_{j}^{2}\) in Equations (14b) and (14c), which define their shape parameters. Consequently, the distributions of \(u_{0}^{F}\) and \(u_{0}^{S}\) will maintain their original kernel forms, even after incorporating updated values for \(\psi_{j}\) and \(\sigma_{j}^{2}\). To break down Equations (14)-(18), we consider three distinct layers involved in the parameter inferences as outlined below:  
* Data layer: \(N_{1}\), \(\{Y_{1(1)},\ldots Y_{1(N_{1})}\},\ldots,N_{H}\), \(\{Y_{H(1)},\ldots Y_{H(N_{H})}\}\mid\theta_{j}\)
* Parameter layer: \(\theta_{j}=\{\beta_{j}^{F},\psi_{j},\beta_{j}^{S},\sigma_{j}^{2}\mid\Phi\}\)
* Hyperparameter layers I and II: \(\Phi=\{\beta_{0}^{F},\Sigma_{\beta_{0}}^{F},u_{0}^{F},v_{0}^{F},\beta_{0}^{S },\Sigma_{\beta_{0}}^{S},u_{0}^{S},v_{0}^{S},\)  
\(\underline{m}_{0},\hat{\phi}\,\underline{q}_{0^{\prime}}\Delta\). \(\underline{\rho}_{u1^{\prime}}\rho_{u2^{\prime}}\rho_{v1^{\prime}}\rho_{v2^{ \prime}},m_{0},\delta,q_{0},\Lambda,\rho_{u1},\rho_{u2},\rho_{v1},\rho_{v2}\}\)  
It is important to note that the inference for the main parameter level relies on cluster-specific data, while hyperparameter inference uses the entire dataset. However, assuming that the hyperparameters \(\underline{m}_{0},\hat{\phi}\,\underline{q}_{0^{\prime}},\Delta\), \(\underline{\rho}_{u1^{\prime}},\underline{\rho}_{u2^{\prime}}\underline{ \rho}_{v1^{\prime}},\rho_{v2^{\prime}}\), \(m_{0},\delta,q_{0},\Lambda,\rho_{u1},\rho_{u2^{\prime}},\rho_{v1},\rho_{v2}\) are fixed makes their selection non-trivial. As a general guideline, flat hyperpriors--assigning equal probability to all potential parameter values--should be employed when specific knowledge about hyperparameter values is unavailable. Additional strategies for choosing hyperparameters can be found in the works of [PERSON] (1997) [26], [PERSON] (2001) [25], and [PERSON] (2008) [27], as well as their references.