## 4 Empirical Study
### Results with LGPIF (H = 1679)
Applying Gustafson corrections in Model (C) produces LPDP values of \(-16\),\(770.85\) with \(\zeta=0.6\) for an error rate \(0.01\), \(-16\),\(370.44\) with \(\zeta=0.6\) for an error rate \(0.10\), and \(-16\),\(188.31\) with \(\zeta=0.5\) for an error rate \(0.40\), closely aligning with the LPDP of the error-free gold standard model, Model (A). This result aligns with other metrics, such as SSPE and SAPE, across different error rate scenarios. The erroneous model with model risk, Model (B), consistently exhibits the highest SSPE and SAPE values, reflecting its weaker predictive performance. In contrast, the Gustafson correction model, Model (C), achieves significantly lower SSPE and SAPE values, closely approximating those of the gold standard model, Model (A). Additionally, as the error rate increases, Model (C) increasingly outperforms the SIMEX method, Model (D), underscoring its robustness. The effectiveness of the Gustafson correction technique is further substantiated by \(D_{KL}\), which quantifies the distance between the estimated distribution produced by Model (B) and Model (C) and the target (gold standard) distribution, Model (A). A smaller \(D_{KL}\) value indicates a more accurate error correction. Notably, as shown in Tables 5-7, within the range of \(0.5\leq\zeta\leq 0.7\), Model (C) exhibits a reduction in \(D_{KL}\) from \(0.61\) to \(0.33\) as the error rate increases from \(R_{ex}=0.01\) to \(R_{ex}=0.40\). This trend suggests that the Gustafson correction effectively mitigates the impact of the NDB covariate, thereby enhancing the model's fidelity to the true data distribution as error rates rise. It is important to note that Model (D), a GLM with SIMEX, is excluded from this analysis due to its frequentist nature, which does not yield the L PPD values necessary for this divergence assessment. In the Conditional Tail Expectation (CTE) analysis, the predictive distribution generated by the Gustafson correction in Model (C) exhibits thicker tails compared to Model (D), with higher CTE values of \(CTE\)\(95\%=269\),\(656\) at \(R_{ex}=0.01\), \(CTE\)\(95\%=281\),\(371\) at \(R_{ex}=0.10\), \(CTE\)\(95\%=278\),\(099\) at \(R_{ex}=0.40\). Notably, the CTE values of Model (C)do not demonstrate a consistent trend across the varying error rate scenarios. However, the higher CTE values in Model (C) suggest that while the Gustafson correction enhances the model's overall predictive accuracy, the underlying hierarchical GLM may effectively address extreme values or outliers. This indicates a potential improvement in the model's applicability in contexts where accurately capturing outliers is essential. Upon inspection of Figures 6-8, it is clear that the Gustafson correction (Model (C), blue curve) effectively mitigates the model risk from the NDB covariate (Model (B), dotted curve) because the corrected hierarchical GLM curves (Model (C), blue) align closely with that of the gold standard curve  
scenarios with higher error rates: \(R_{e_{x}}=0.40\) in Figure 8. However, a slight gap still persists between Model (A) and -(C), indicating that the correction, while beneficial, has limitations. In contrast, Model (B) demonstrates significant distortions, including extreme variations and multiple peaks that worsen with increasing error rates. Although both Model (B) and -(C) are derived from the same LGPIF dataset, the Gustafson correction in Model (C) appears to effectively restore the properties of the original distribution, particularly pronounced at higher error rates such as \(R_{e_{x}}=0.40\). This improvement is observed with scaling factors in the range \(0.5\leq\zeta\leq 0.7\), suggesting that the relationship between \(V(\mathbf{x}^{*}|\mathbf{x})\) and \(V(\mathbf{x}^{*}|\mathbf{z})\) in the LGPIF dataset is characterized by this range of \(\zeta\). Consequently, further investigation may be warranted using different datasets where \(V(\mathbf{x}^{*}|\mathbf{x})\) and \(V(\mathbf{x}^{*}|\mathbf{z})\) have relationships defined by a different range of scaling factors.