Two-Stage Post-Estimation Elastic Net Shrinkage for Gaussian Copula Models: A Variable Selection Framework for Mixed Biomedical Outcomes

Abstract

This study introduces a pragmatic two-stage post-estimation shrinkage approach that combines Gaussian copula joint modeling with elastic net feature selection for mixed biomedical outcomes. The proposed methodology addresses key challenges in multivariate statistical analysis by first fitting an unpenalized Gaussian copula to capture complex dependence structures among mixed outcomes, and then applying elastic net proximal mapping to the estimated coefficients as a post-estimation shrinkage step. Our approach combines Lasso (L1) and Ridge (L2) penalties, achieving balanced regularization that reduces the number of non-zero regression coefficients by 37.5%, shrinking the coefficients of less important predictors to zero while retaining key features. All continuous predictors were standardized to z-scores prior to analysis, ensuring coefficient interpretability as a change per one standard deviation increase. Tuning parameters are selected via a two-step procedure combining grid search screening with bootstrap stability selection ( $B = 50$ ), confirming that the optimal configuration $\lambda _{1} = 0.1$ and $\lambda _{2} = 0.1$ yields robust and reproducible variable selection. Key clinical predictors, accelerations, and prolonged decelerations were retained in 100% of bootstrap replicates, with histogram mean retained in 100% and abnormal LTV in 90% of replicates. We validate our framework using the Cardiotocography (CTG) dataset, a clinically relevant biomedical application featuring fetal heart rate monitoring data. A 70/30 train-test evaluation demonstrates that the two-stage copula EN reduces ASTV prediction error by 87.1% over the unpenalized copula (MSE: 278.0 vs 2160.2) and achieves fetal health classification accuracy of 85.1% (Cohen’s Kappa = 0.614), compared to 75.7% (Kappa = 0.000) for the unpenalized model. The two-stage approach performs competitively with standard elastic net (accuracy = 87.9%, Kappa = 0.672) while additionally capturing joint dependence structure and providing interpretable sparse variable selection across mixed outcome types. The two-stage elastic net identifies clinically meaningful predictors (e.g., prolonged decelerations, accelerations, histogram features) and yields a sparse, interpretable model that may support clinical decision making.