Learning an Interpretable Risk Scoring System for Maximizing Decision Net Benefit

Many clinical risk scores are deployed as additive rules with nonnegative integer points assigned to relevant binary predictive features. These integer weights not only make the score easier to use in practice but also promote sparsity in the resulting prediction model. Such risk scores are often derived by first fitting a regression model and then rounding the estimated coefficients to the nearest integer after appropriate scaling. This approach is computationally fast but does not guarantee optimality of the resulting score. Alternatively, one may search over all possible integer weights to directly optimize a value function by posing the problem as an integer programming task. However, the associated computational burden can be substantial, especially when the value function is nonconcave or even discontinuous. In this paper, we develop new machine learning algorithms that employ a flexible greedy optimization strategy to learn such additive scoring directly under explicit and sensible optimality objectives. We apply the proposed method to a large electronic health record (EHR) cohort in Epic Cosmos to construct an integer-weighted comorbidity score for measuring the risk of post-discharge mortality. We also conduct a simulation study to examine the finite-sample operating characteristics.

Risk scores are an interpretable and actionable class of machine learning models with applications in medicine, insurance, and risk management. Unlike most computational methods, risk scores are designed to be computed by a human by attributing points to a data sample based on a limited set of criteria. The most common approaches for generating risk scores use linear regressions to estimate the effect of selected variables. We propose a simple and effective approach towards building compact and predictive risk scores. We provide an algorithm based on gradient boosting that is capable of modeling nonlinear effects, along with a C++ implementation with Python and R bindings. Through extensive empirical evaluation on twelve tabular datasets spanning regression, classification, and time-to-event tasks, we show that our method achieves competitive predictive performance while producing substantially more compact scores than regression-based alternatives, with 60% fewer rules for classification tasks and 16% fewer rules for time-to-event tasks on average, compared to AutoScore.

Objective Prediction models are commonly trained using objectives such as Bernoulli negative log-likelihood (NLL), although downstream clinical decisions may depend on specific risk thresholds. We introduce Smooth Net Benefit ($\sigma$NB), a differentiable approximation of Net Benefit designed to align model training with threshold-specific clinical utility. Materials and Methods We evaluated $\sigma$NB as a training objective for logistic regression, generalized additive models (GAMs), and XGBoost with three Hessian implementations. Experiments used the Framingham cardiovascular risk dataset and 44 TabZilla datasets comprising 72 dataset-threshold combinations. Results $\sigma$NB training did not consistently improve Net Benefit in Framingham. Across the TabZilla benchmark, mean standardized Net Benefit for logistic regression increased from 0.5669 with NLL to 0.5765 with $\sigma$NB (mean difference 0.0096, 95% CI -0.0001 to 0.0193). For GAMs, mean standardized Net Benefit decreased from 0.5921 to 0.5625 (mean difference -0.0296, 95% CI -0.0721 to 0.0129). For XGBoost, NLL achieved 0.6745 compared with 0.6723--0.6735 across $\sigma$NB implementations. In logistic regression, $\sigma$NB gains were positively associated with the performance advantage of XGBoost over NLL-trained logistic regression. Discussion The effect of $\sigma$NB was context dependent, with modest gains concentrated in logistic regression and little benefit for more flexible model classes. This suggests that decision-focused optimization may be most useful when limited model flexibility leaves greater scope for improvement. Conclusion Our results do not support $\sigma$NB as a general replacement for NLL training, but support further investigation of decision-focused objectives in settings where conventional likelihood-based training may not adequately capture decision-relevant structure.