## 4 Empirical Study
### Model Implementation  
Using simulation data, we evaluate our hierarchical GLM-based approach for correcting NDB errors against the SIMEX error correction method discussed in Section 2.2. Both methodologies seek to simultaneously tackle two model risk issues--heterogeneity (RQ1) and NDB covariates (RQ2)--for risk premium development. As mentioned in Section 2.2, many existing error correction techniques rely on gold standard data (e.g., subsets of the true covariate) to estimate the true covariate by establishing relationships between mismeasured and true values. However, in practical applications, access to gold standard data is often limited or nonexistent. In this regard, this experiment aims to create methodologies that effectively address the NDB measurement errors without the need for gold standard data, leveraging the prior knowledge \(\mathbf{\Sigma}_{j}^{2}=(1-\zeta)\widehat{\lambda}_{j}^{2}\) derived from Section 3.4. We develop simulation data to explore the relationship between the scaling factor \(\zeta\) and the severity of NDB errors, represented by the error rate \(R_{ex}\), in the NDB covariate \(\mathbf{x}^{*}\). This process hinges on two elements: (a) gold-standard datasets with no errors, and (b) datasets with controlled error rates in the NDB covariate \(\mathbf{x}^{*}\). By comparing models from error-free data with those generated from datasets with varying NDB error levels, we assess the efficiency of [PERSON]'s correction approach, which is based on a hierarchical GLM, in relation to the error rate \(R_{ex}\) and the scaling factor \(0<\zeta<1\). Our ultimate goal is to establish a guideline for selecting an optimal scaling factor \(\zeta\) that accounts for the unknown variance \(\mathbf{\Sigma}_{j}^{2}=(1-\zeta)\widehat{\lambda}_{j}^{2}\) when the error rate \(R_{ex}\) is known. Once established, this guideline could aid in estimating the true covariate when gold-standard data are unavailable. **Design of simulation data:** We construct simulation datasets with different error rates \(R_{ex}\), derived from real LGPIF data, while preserving the original LGPIF dataset as the gold standard. These simulation datasets are generated by deliberately introducing controlled NDB errors into the true covariate \(\mathbf{x}^{S}\) at different error rates \(R_{ex}\). The method for creating NDB errors in the covariate \(\mathbf{x}^{5*}\), leading to variations in \(R_{ex}\), is summarized in Figure 3.  
Figure 3: Design of Non-Differential Berkson (NDB) error in \(\mathbf{x}^{*}\) and the induced heteroscedasticity varying by cluster \(j\). Note that the dotted lines and circles in the diagram form icons that represent heteroscedasticity. As defined in Section 2.2, the NDB error \(\epsilon_{j}\sim\mathbf{N}(0,\sigma_{je}^{2})\) is characterized by its independence from the outcome and other covariates, while being correlated with the latent factors. However, as indicated in Equation (34), the Gustafson correction method suggests that the variance of the linking component \(V(\mathbf{x}^{*}|\mathbf{x})\) can be approximated using the variance of the covariate component \(V(\mathbf{x}^{*}|\mathbf{z})\). Hence, to replicate this setting, we connect the NDB covariate \(\mathbf{x}^{S*}\) with the binary covariate \(\mathbf{z}^{S}\) as shown in Equation (29b) by conditionally introducing the NDB errors to the true covariate \(\mathbf{x}^{S}\) only when \(\mathbf{z}^{S}=1\). Additionally, as proposed by [PERSON] (2017) [29], the cluster-specific error variance \(\sigma_{je}^{2}\) is modeled using the random correlation \(-1<\rho_{j(x,x^{*})}<1\) between the true covariate \(\mathbf{x}^{S}\) and the NDB covariate \(\mathbf{x}^{S*}\), simulating the NDB error structure. This method utilizes the NDB noise generation technique introduced by [PERSON] et al.