## 3 Results
### Effects of Initial Estimates on the Prediction Accuracy of NNG and ALASSO  
We investigated whether the choice of initial estimates (OLS, ridge, and lasso) influenced the prediction accuracy of NNG and ALASSO models. To explore this, we used a Bland-Altman plot [56] to evaluate the agreement between the predictions measured using RR. We observed that the effects of initial estimates on predictions for NNG and ALASSO were very similar; therefore, only the results for ALASSO are reported. Figure 1 compares the prediction errors of three ALASSO models: ALASSO (O, CV), ALASSO (R, CV), and ALASSO (L, CV). Specifically, it evaluates the agreement between the prediction errors of (i) ALASSO (O, CV) and ALASSO (R, CV), (ii) ALASSO (O, CV) and ALASSO (L, CV), and (iii) ALASSO (R, CV) and ALASSO (L, CV). The first and second rows compare the three models in scenarios with limited information: a small sample size (100), low SNR (0.25) (\(R^{2}\) of about 20%), and high correlation among covariates. These conditions provide little information for accurately estimating both the tuning parameters and initial estimates, leading all initial estimates to produce models with different predictions. This is indicated by a nonzero mean difference (horizontal solid line) and some differences falling outside the wider limits of agreement (LOA) (dashed lines) in the Bland-Altman plot. On average, ridge initial estimates produced models with the best predictions, followed by lasso estimates, while OLS estimates performed the worst. The third and fourth rows compare the three models in scenarios with moderate information: a small sample size (100), moderate SNR (1) (\(R^{2}\) of about 50%), and high correlation among covariates. In this setting, predictions across the three initial estimates differed, but not as substantially as in the limited-information scenarios. The fifth and sixth rows compare the three models in scenarios with sufficient information: a large sample size (400), high SNR (2.5) (\(R^{2}\) of about 71%), and low correlation among covariates. In this setting, the data contained adequate information to estimate the parameters accurately, resulting in nearly identical predictions across the three initial estimates. Differences were close to zero, and the LOAs were narrow, suggesting that the three initial estimates can be used interchangeably. Lasso initial estimates may be preferred, as they yielded simpler models (see Figure A1 in Appendix A). Overall, ridge initial estimates produced models with superior predictions. Therefore, we will compare the prediction accuracy of NNG and ALASSO using ridge initial estimates to those of other approaches (see Section 3.3). Figure 1: Effect of initial estimates (OLS, ridge, and lasso) on ALASSO predictions (measured by RR) for beta-type A. Rows 1 and 2 represent scenarios with limited information: small sample size (\(n=100\)), low SNR (0.25), and high correlation (C3). Rows 3 and 4 correspond to scenarios with moderate information: small sample size (\(n=100\)), moderate SNR (1), and high correlation (C3). Rows 5 and 6 show scenarios with sufficient information: large sample size (\(n=400\)), high SNR (2.5), and low correlation (C2). The odd-numbered rows display scatterplots with a line of identity (black), while the even-numbered rows show Bland-Altman plots. The horizontal solid lines (red) in the Bland-Altman plots indicate the mean difference, while the dashed lines (green) represent the lower and upper limits of agreement (green). Smoothed differences (black lines) with 95% pointwise confidence intervals (yellow) are also shown. **Left panel**: RR difference between ALASSO with OLS and ridge initial estimates. **Middle panel**: RR difference between ALASSO with OLS and lasso initial estimates. **Right panel**: RR difference between ALASSO with ridge and lasso initial estimates.