Regularized (bridge) logistic regression for variable selection based on ROC criterion

It is well known that the bridge regression (with tuning parameter less or equal to 1) gives asymptotically unbiased estimates of the nonzero regression parameters while shrinking smaller regression parameters to zero to achieve variable selection. Despite advances in the last several decades in developing such regularized regression models, issues regarding the choice of penalty parameter and the computational methods for models fitting with parameter constraints even for bridge linear regression are still not resolved. In this article, we first propose a new criterion based on an area under the receiver operating characteristic (ROC) curve (AUC) to choose the appropriate penalty parameter as opposed to the conventional generalized cross-validation criterion. The model selected by the AUC criterion is shown to have better predictive accuracy while achieving sparsity simultaneously. We then approach the problem from a constrained parameter model and develop a fast minorization-maximization (MM) algorithm for non-linear optimization with positivity constraints for model fitting. This algorithm is further applied to bridge regression where the regression coefficients are constrained with l(p)-norm with the level of p selected by data for binary responses. Examples of prognostic factors and gene selection are presented to illustrate the proposed method.