## 3 Research Method
### Modeling Approach  
The residual sum of squares (RSS) for the \(k\)th model is defined as follows:  
\[SSE_{k}=\parallel Y-\hat{\mu}_{k}\parallel^{2}\]  
where \(\hat{\mu}_{k}=X_{k}\hat{\beta}_{k}\) represents the estimated mean vector, and \(\hat{\beta}_{k}\) is the least-squares estimate of the parameter vector \(\hat{\beta}_{k}\). In the context of nested models, the residual sum of squares, denoted as \(SSE_{k}=\parallel Y-X_{k}\hat{\beta}_{k}\parallel^{2}\), tends to decrease as more parameters are added. However, simply minimizing \(SSE_{k}\) across different models does not necessarily lead to the optimal model, as it may result in overfitting. Instead, a more informative metric is  
\[D_{k}=\mathrm{E}\Big{[}\parallel\mu-\hat{\mu}_{k}\parallel^{2}\Big{]}\]  
which represents the expected squared distance between the true mean \(\mu\) and its estimate \(\hat{\mu}_{k}\). This measure provides a reliable indication of how accurately \(\hat{\mu}_{k}\) approximates \(\mu\). Although increasing the number of parameters typically leads to a lower \(SSE_{k}\), it does not guarantee a better model. Therefore, model selection should balance goodness of fit and generalizability by considering both \(SSE_{k}\) and \(D_{k}\). The optimal model is ideally the one that minimizes \(D_{k}\), the expected squared distance between the true mean vector, and its estimate. However, because \(D_{k}\) depends on unknown population parameters, it cannot be computed directly. Therefore, an estimate \(\hat{D}_{k}\) must be obtained, and the model corresponding to the smallest \(\hat{D}_{k}\) (for \(k=1,\ldots,K\)) is selected. If the minimum is achieved \(k=\hat{k}\), then the corresponding estimate \(\hat{\mu}_{k}=X_{k}\hat{\beta}_{k}\) is considered the most accurate approximation of the mean vector. To obtain a reliable estimate of \(D_{k}\), criteria such as Akaike's Final Prediction Error (FPE) and Mallows \(C_{p}\) are commonly used. After fitting multiple models to a given dataset, evaluating their performance, and considering their complexity, the final step in model selection is to choose the one that balances goodness of fit with parsimony. This principle applies to both supervised machine learning models (e.g., regression or classification) and unsupervised learning models (e.g., clustering). Extensive research has explored numerous strategies for model selection, drawing upon contributions from various academic disciplines. Among the most widely used approaches are the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). These criteria incorporate factors such as the model's log-likelihood, the number of estimated parameters, and the principle of minimum description length (MDL) derived from information theory. The BIC is formally defined as  
\[BIC(k)=nlog\hat{\sigma}_{k}^{2}+log(n)p_{k} \tag{12}\]  
where \(\hat{\sigma}_{k}^{2}\) is the estimated variance of the residuals, \(n\) is the number of observations, and \(p_{k}\) is the number of parameters in the kth model. Lower BIC values indicate a more optimal balance between model fit and complexity. The AIC is defined as  
\[AIC(k)=nlog\hat{\sigma}_{k}^{2}+2p_{k} \tag{13}\]  
where \(\hat{\sigma}_{k}^{2}\) is the estimated variance of residuals, \(n\) represents the number of observations, and \(p_{k}\) is the number of parameters in the \(k\)th model. Lower AIC values indicate models that achieve a favorable trade-off between goodness of fit and complexity. An important consideration in our model development was interpretability, particularly due to the clinical nature of the application. GLMs were chosen not only for their flexibility and statistical grounding, but also because their coefficients provide direct insights into the direction and strength of association between covariates and the outcome variable. In the final model, each covariate's estimated coefficient was reviewed for clinical plausibility, and the significance levels were interpreted in the context of existing domain knowledge.