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Joint Score-Threshold Optimization for Interpretable Risk Assessment Under Partial Supervision

Fardin Gankhanloo, Emmett Springer, Erik H. Hoyer, Daniel L. Young, Kimia Ghobadi

TL;DR

This work tackles risk scoring in healthcare by addressing partial supervision from intervention-censored outcomes and asymmetric misclassification costs in ordinal risk categories. It introduces a joint optimization framework that learns scoring weights $\beta$ and thresholds $\tau$ under per-instance feasible label sets $\mathcal{S}_i$, using an exact mixed-integer program (MIP) and a convex constrained score optimization (CSO) relaxation. The authors add governance constraints for interpretability, including sign, sparsity, and minimal modification restrictions, and employ a two-phase CSO–MIP strategy to enable scalable, deployable solutions. Empirical evaluation on the Johns Hopkins Fall Risk Assessment Tool demonstrates improved separation and predictive performance over the baseline, with tunable trade-offs between false positives and false negatives that support safe, practical deployment across different clinical settings.

Abstract

Risk assessment tools in healthcare commonly employ point-based scoring systems that map patients to ordinal risk categories via thresholds. While electronic health record (EHR) data presents opportunities for data-driven optimization of these tools, two fundamental challenges impede standard supervised learning: (1) partial supervision arising from intervention-censored outcomes, where only extreme categories can be reliably labeled, and (2) asymmetric misclassification costs that increase with ordinal distance. We propose a mixed-integer programming (MIP) framework that jointly optimizes scoring weights and category thresholds under these constraints. Our approach handles partial supervision through per-instance feasible label sets, incorporates asymmetric distance-aware objectives, and prevents middle-category collapse via minimum threshold gaps. We further develop a CSO relaxation using softplus losses that preserves the ordinal structure while enabling efficient optimization. The framework supports governance constraints including sign restrictions, sparsity, and minimal modifications to incumbent tools, ensuring practical deployability in clinical workflows.

Joint Score-Threshold Optimization for Interpretable Risk Assessment Under Partial Supervision

TL;DR

This work tackles risk scoring in healthcare by addressing partial supervision from intervention-censored outcomes and asymmetric misclassification costs in ordinal risk categories. It introduces a joint optimization framework that learns scoring weights and thresholds under per-instance feasible label sets , using an exact mixed-integer program (MIP) and a convex constrained score optimization (CSO) relaxation. The authors add governance constraints for interpretability, including sign, sparsity, and minimal modification restrictions, and employ a two-phase CSO–MIP strategy to enable scalable, deployable solutions. Empirical evaluation on the Johns Hopkins Fall Risk Assessment Tool demonstrates improved separation and predictive performance over the baseline, with tunable trade-offs between false positives and false negatives that support safe, practical deployment across different clinical settings.

Abstract

Risk assessment tools in healthcare commonly employ point-based scoring systems that map patients to ordinal risk categories via thresholds. While electronic health record (EHR) data presents opportunities for data-driven optimization of these tools, two fundamental challenges impede standard supervised learning: (1) partial supervision arising from intervention-censored outcomes, where only extreme categories can be reliably labeled, and (2) asymmetric misclassification costs that increase with ordinal distance. We propose a mixed-integer programming (MIP) framework that jointly optimizes scoring weights and category thresholds under these constraints. Our approach handles partial supervision through per-instance feasible label sets, incorporates asymmetric distance-aware objectives, and prevents middle-category collapse via minimum threshold gaps. We further develop a CSO relaxation using softplus losses that preserves the ordinal structure while enabling efficient optimization. The framework supports governance constraints including sign restrictions, sparsity, and minimal modifications to incumbent tools, ensuring practical deployability in clinical workflows.
Paper Structure (22 sections, 2 theorems, 17 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 22 sections, 2 theorems, 17 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Problem Formulation and Notation
  4. Mixed-Integer Programming Formulation
  5. Non-Degenerate Categories
  6. Governance and Interpretability Constraints
  7. Constrained Score Optimization (CSO) Relaxation
  8. Two-Phase Optimization: CSO Warm-Start with MIP Refinement
  9. Experiments
  10. Dataset Construction and Partial Labels
  11. MIP Models
  12. Experimental Setup
  13. Baselines and Evaluation Metrics
  14. Results
  15. Optimized Risk Factor Coefficients
  16. ...and 7 more sections

Key Result

Proposition 1

The CSO objective eq:cso-program-detailed is convex in $(\beta, \tau)$ jointly.

Figures (4)

  • Figure 1: Stratification of inpatient admissions based on fall events and targeted interventions. The intervention cohort was excluded from primary optimization analyses but retained for post-optimization validation.
  • Figure 2: Score distributions for all labeled data (test and train data together)
  • Figure 3: Score differentials (MIP score - JHFRAT score) for all patients in dataset
  • Figure 4: Receiver Operating Characteristic (ROC) and Precision-Recall performance curves across JHFRAT and five optimization models on training dataset (above) and testing dataset (below).

Theorems & Definitions (4)

  • Definition 1: Ordinal Risk Assessment Problem
  • Proposition 1: Convexity
  • proof : Proof sketch
  • Proposition 2: Approximation Quality