OSCAR: Optimal subset cardinality regression using the L0-pseudonorm with applications to prognostic modelling of prostate cancer

In many real-world applications, such as those based on electronic health records, prognostic prediction of patient survival is based on heterogeneous sets of clinical laboratory measurements. To address the trade-off between the predictive accuracy of a prognostic model and the costs related to its clinical implementation, we propose an optimized L0-pseudonorm approach to learn sparse solutions in multivariable regression. The model sparsity is maintained by restricting the number of nonzero coefficients in the model with a cardinality constraint, which makes the optimization problem NP-hard. In addition, we generalize the cardinality constraint for grouped feature selection, which makes it possible to identify key sets of predictors that may be measured together in a kit in clinical practice. We demonstrate the operation of our cardinality constraint-based feature subset selection method, named OSCAR, in the context of prognostic prediction of prostate cancer patients, where it enables one to determine the key explanatory predictors at different levels of model sparsity. We further explore how the model sparsity affects the model accuracy and implementation cost. Lastly, we demonstrate generalization of the presented methodology to high-dimensional transcriptomics data.Author summary: Feature subset selection has become a crucial part of building biomedical models, due to the abundance of available predictors in many applications, yet there remains an uncertainty of their importance and generalization ability. Regularized regression methods have become popular approaches to tackle this challenge by balancing the model goodness-of-fit against the increasing complexity of the model in terms of coefficients that deviate from zero. Regularization norms are pivotal in formulating the model complexity, and currently L1-norm (LASSO), L2-norm (Ridge Regression) and their hybrid (Elastic Net) dominate the field. In this paper, we present a novel methodology that is based on the L0-pseudonorm, also known as the best subset selection, which has largely gone overlooked due to its challenging discrete nature. Our methodology makes use of a continuous transformation of the discrete optimization problem, and provides effective solvers implemented in a user friendly R software package. We exemplify the use of oscar-package in the context of prostate cancer prognostic prediction using both real-world hospital registry and clinical cohort data. By benchmarking the methodology against existing regularization methods, we illustrate the advantages of the L0-pseudonorm for better clinical applicability, selection of grouped features, and demonstrate its applicability in high-dimensional transcriptomics datasets.