## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2  
Assuming additive Non-Differential Berkson (NDB) errors in a covariate, we examine the Bayesian framework to account for its cluster-wise error structure (i.e., heteroscedasticity) probabilistically. The strength of a Bayesian approach to address NDB covariate errors has been well documented by [PERSON] et al. (2021) [8]. From a Bayesian perspective, deviations from true values can be corrected through the incorporation of the prior knowledge that captures the relation between the unobservable true covariate \(\mathbf{x}\) and the observed NDB covariate \(\mathbf{x}^{*}\). When the NDB error structure introduces cluster-wise heteroscedasticity, the framework specifies structural components to capture varying variances across risk clusters. This cluster-wise inference can also be achieved by leveraging the manageable joint product and Gibbs sampling [23]. The model component specification plays a crucial role in formulating the strategy, as it determines the parameterization when incorporating the necessary parameter knowledge. For the NDB covariate problem, [PERSON] (2008) [23] begins by specifying the full joint density of the relevant variables as:  
\[f(Y,\mathbf{x}^{*},\mathbf{x},\mathbf{z})= f(Y|\mathbf{x}^{*},\mathbf{x},\mathbf{z})\cdot f(\mathbf{x}^{*}| \mathbf{x},\mathbf{z})\cdot f(\mathbf{x}|\mathbf{z})\cdot f(\mathbf{z}) \tag{7a}\] \[f(Y,\mathbf{x}^{*},\mathbf{x}|\mathbf{z})= f(Y|\mathbf{x}^{*},\mathbf{x},\mathbf{z})\cdot f(\mathbf{x}^{*}| \mathbf{x},\mathbf{z})\cdot f(\mathbf{x}|\mathbf{z}) \tag{7b}\]  
Figure 1: As a Bayesian parametric example with \(J=4\) clusters, this diagram depicts a typical Bayesian hierarchical model with the bias–variance trade-off through the partial pooling. For prediction, the class membership \(j\) of each data point \(\{Y_{hi},\mathbf{X}_{hi}\}\) should be known beforehand. in which the term for the precisely measured covariate \(\mathbf{z}\) is factored out for the sake of simplicity. Due to the assumption of the non-differential error, the conditional joint density in Equation (7b) can be further reduced to:  
\[f(Y,\mathbf{x}^{*},\mathbf{x}|\mathbf{z})=f(Y|\mathbf{x}^{*},\mathbf{x}, \mathbf{z})\cdot f(\mathbf{x}^{*}|\mathbf{x},\mathbf{z})\cdot f(\mathbf{x}| \mathbf{z}) \tag{8}\]  
where the cancellations are based on the definition of NDB error as described in Section 2.2. This conditional joint density in Equation (8) is termed as the _complete joint_ model [23]. As the true covariate \(\mathbf{x}\) is not observable, the complete joint model is technically inaccessible or theoretical. However, in the construction of the complete joint model in Equation (8), three main components that make up the relationship between the true covariate \(\mathbf{x}\) and the observed covariate \(\mathbf{x}^{*}\) can be identified:  
* Outcome component \(f(Y|\mathbf{x},\mathbf{z})\);
* Linking (measurement [23]) component \(f(\mathbf{x}^{*}|\mathbf{x})\);
* Covariate (exposure [23]) component \(f(\mathbf{x}|\mathbf{z})\). In particular, the linking component incorporates the measurement error mechanism of \(\mathbf{x}_{j}^{*}\sim\mathbf{N}(\mathbf{x}_{j},\ \ \mathbf{\Xi}_{j}^{2})\) into the analysis, allowing control over the simulation process [8]. The linking (measurement) component term and \(\mathbf{\Xi}_{j}^{2}\) also enable investigation of the NDB error's effect on the unknown risk clusters. In contrast, the conditional joint model available in reality is referred to as the _incomplete joint_ model [23], which can be represented as:  
\[f(Y,\mathbf{x}^{*}|\mathbf{z})=f(Y|\mathbf{x}^{*},\mathbf{z})\cdot f(\mathbf{ x}^{*}|\mathbf{z}) \tag{9}\]  
where the outcome term \(f(Y|\mathbf{x}^{*},\mathbf{z})\) and the exposure term \(f(\mathbf{x}^{*}|\mathbf{z})\) are fully known. Note that the incomplete joint model in Equation (9) can be obtained from marginalizing the complete joint model in Equation (8) over the unobservable true covariate \(\mathbf{x}\).