Towards symbolic regression for interpretable clinical decision scores

TL;DR

Brush introduces a split-aware symbolic regression algorithm that integrates decision-tree-like splitting with nonlinear parameter optimization within a genetic programming framework. It achieves Pareto-optimal performance on the SRBench benchmark and successfully reproduces classic clinical scores (CART, MEWS) with compact, interpretable models. Applied to MIMIC-IV-ED data, Brush performs competitively against traditional and SR baselines in both regression and deterioration classification tasks while preserving interpretability. The work highlights the practical promise of combining rule-based logic with data-driven SR for interpretable clinical decision support.

Abstract

Medical decision-making makes frequent use of algorithms that combine risk equations with rules, providing clear and standardized treatment pathways. Symbolic regression (SR) traditionally limits its search space to continuous function forms and their parameters, making it difficult to model this decision-making. However, due to its ability to derive data-driven, interpretable models, SR holds promise for developing data-driven clinical risk scores. To that end we introduce Brush, an SR algorithm that combines decision-tree-like splitting algorithms with non-linear constant optimization, allowing for seamless integration of rule-based logic into symbolic regression and classification models. Brush achieves Pareto-optimal performance on SRBench, and was applied to recapitulate two widely used clinical scoring systems, achieving high accuracy and interpretable models. Compared to decision trees, random forests, and other SR methods, Brush achieves comparable or superior predictive performance while producing simpler models.

Figures (5)

• Figure 1: Brush Overview. Nodes are sampled from the search space to build mathematical expression trees, including those with split operations. A randomly generated population of expressions goes into an evolutionary loop. The set of nodes are used to create variations on the population. After several generations, the final solution is selected from the evolved individuals.
• Figure 2: Brush evaluation, split, and optimization. (A) Evaluation starts at the root, recursively propagating input $\mathbf{x}$ and parameters $\theta$, applying each node's symbol to its children. (B) Splits can occur anywhere, with the conditional subtree directing data flow based on the condition. (C) Optimization with splits in three steps: optimize $\text{child}_1$; find the $\tau$ minimizing $y$ variance across $\text{child}_2$ and $\text{child}_3$; then fit the rest of the tree, ignoring already fitted parameters.
• Figure 3: Brush's SRBench results. Bars denotes the bootstrapped confidence intervals (CI).
• Figure 4: Exemplar Brush models edited as Python code for catastrophic deterioration prediction using two different scoring systems CART and simplified MEWS.
• Figure 5: Metrics and sizes for the clinical decision experiment. The size of the original decision model is denoted as a dashed line. Statistical comparisons are conducted using Mann-Whitney-Wilcoxon two-sided test with Holm-Bonferroni corrections. *: 1.00e-02 $<$ p $\leq$ 5.00e-02; **: 1.00e-03 $<$ p $\leq$ 1.00e-02 ***: 1.00e-04 $<$ p $\leq$ 1.00e-03 ****: p $\leq$ 1.00e-04. Non-significant comparisons were omitted.