Technical Summary: RuleSHAP for Nonlinear Epidemiological Inference

Problem Statement

Machine Learning (ML) is increasingly utilized in epidemiology for "hypothesis-free discovery" of risk and protective factors because it can capture complex nonlinearities and interactions that rigid linear models miss. However, a critical limitation persists: standard ML models lack reliable statistical inference. While feature importance measures like SHAP (Shapley Additive Explanations) provide local effect estimates, they typically lack valid uncertainty quantification (UQ). Existing attempts to quantify SHAP uncertainty often focus only on aleatoric uncertainty (noise in the data) while neglecting epistemic uncertainty (instability in model parameters). Furthermore, methods that combine ML feature selection with subsequent linear regression often yield overly optimistic uncertainty estimates and fail to capture nonlinear interactions consistently. The challenge is to develop a framework that offers the flexibility of ML for detecting complex patterns while providing rigorous, local-level statistical inference with credible intervals.

Methodology: The RuleSHAP Framework

The authors propose RuleSHAP, a framework that integrates Bayesian sparse regression, tree ensembles, and Shapley values to address the limitations of existing RuleFit family models. The methodology consists of three core innovations:

1. Improved Linear Fit with Structured Horseshoe Prior

RuleSHAP employs a Bayesian linear regression model enhanced with dichotomous decision rules. Unlike the standard HorseRule model, which uses a single global shrinkage parameter that can cause excessive shrinkage of linear terms (leading to the loss of linear signals), RuleSHAP introduces a hierarchical structured horseshoe prior.

• Separate Global Shrinkages: The model utilizes distinct global shrinkage parameters ($\tau_L$ for linear terms and $\tau_R$ for rule terms). This ensures that uninformative rules do not inflate the global shrinkage to the detriment of linear terms.
• Inference Focus: This separation allows for more accurate uncertainty quantification for linear effects, which is crucial because linear terms are not conditional on the data in the same way rules are, making them more amenable to inference.

2. Robust Rule Generation (Smoothing Random Forest)

To address the issue that rule generation is random and dependent on the specific dataset (leading to overfitting and underestimated uncertainty), RuleSHAP introduces a "Smoothing Random Forest" approach:

• Parametric Bootstrap Integration: Instead of using the raw observed outcomes for rule generation, the method fits a Random Forest, estimates the parameters of the outcome distribution (e.g., mean and variance) using out-of-bag predictions, and then generates rules by fitting a new Random Forest on synthetic outcomes sampled from these estimated distributions.
• Residual-Based Fitting: For the initial linear fit, rules are generated from the residuals of a Bayesian Horseshoe regression on linear terms. This ensures rules model beyond-linear trends rather than re-capturing linear signals.
• Rule Disaggregation: To enhance interpretability and prevent overfitting to complex interactions, the method fully disaggregates tree paths. Conditions in a tree path are recombined in all possible ways, allowing the model to utilize simpler, necessary conditions without being forced to use unnecessarily complex interaction rules.

3. Efficient Shapley Value Computation

The framework derives explicit formulas for computing marginal Shapley values and interaction Shapley values directly from the rule terms.

• Direct Estimation: Unlike TreeSHAP, which requires converting rules back into full trees (increasing computational cost), RuleSHAP provides an unbiased estimator for Shapley values as sample means over the dataset.
• Uncertainty Propagation: By computing Shapley values for every posterior sample of the Bayesian model, the framework generates full posterior distributions for feature effects, enabling the construction of credible intervals for local inference.

Key Results

Simulation Studies

The authors validated RuleSHAP on simulated data (Friedman's benchmark) with varying sample sizes ($n$) and dimensions ($p$), comparing it against OLS, LASSO, RuleFit, HorseRule, Random Forest, and BART.

• Signal Reconstruction: RuleSHAP successfully identified linear signals ($x_4, x_5$) that RuleFit and HorseRule often missed due to excessive shrinkage. It reconstructed interaction effects ($x_1, x_2$) and purely nonlinear effects ($x_3$) with accuracy comparable to or better than competitors at larger sample sizes.
• Noise Detection: RuleSHAP demonstrated superior ability to distinguish signal from noise. While BART and HorseRule frequently produced significant local effects for noise features (due to underestimated uncertainty), RuleSHAP's credible intervals for noise features consistently contained zero.
• Uncertainty Quantification: RuleSHAP achieved the lowest rejection rates for noise features and the highest for signal features, indicating a better balance between Type I and Type II errors in local inference.

Epidemiological Application (HELIUS Study)

The framework was applied to the HELIUS cohort ($n=21,570$) to investigate risk factors for systolic blood pressure (SBP) and cholesterol levels.

• Predictive Performance: RuleSHAP matched linear models for SBP (a largely linear relationship) but significantly outperformed linear models for cholesterol, capturing necessary nonlinearities.
• Inference Findings:
  • SBP: Confirmed positive associations with age, BMI, and glucose. It revealed heterogeneous effects of age across ethnicities and sexes.
  • Cholesterol: Detected a positive association with smoking and a negative association with glucose. Crucially, it uncovered a nonlinear interaction between age and sex, showing that the difference in cholesterol levels between sexes becomes most pronounced after age 52, aligning with menopausal age ranges.
  • Noise Control: All four added noise features were correctly identified as having non-significant effects.

Significance and Claims

The paper claims that RuleSHAP offers a valid framework for hypothesis-free discovery with local statistical inference. Its primary significance lies in bridging the gap between the flexibility of ML and the rigor of statistical inference:

1. Local Inference: Unlike Generalized Additive Models (GAMs), which are limited to global inference or suboptimal local intervals, RuleSHAP provides valid credible intervals for local feature effects.
2. Epistemic Uncertainty: By integrating Bayesian regression, the method accounts for model instability (epistemic uncertainty), which is often neglected in other SHAP-based inference attempts.
3. Linear vs. Nonlinear Balance: The structured prior prevents the "overshrinking" of linear terms, a common failure mode in previous RuleFit variants, ensuring that linear signals are not lost in the pursuit of complex interactions.
4. Computational Efficiency: The derived explicit formulas for Shapley values allow for efficient computation that scales better than converting rules to full trees for TreeSHAP, particularly when analyzing individual rules.

The authors conclude that while computational cost remains a limitation (scaling quadratically with sample size for the Bayesian fit), RuleSHAP provides a robust tool for transparent ML in healthcare, enabling the detection of complex, nonlinear risk factors with statistically valid uncertainty quantification.