Efficient Subgroup Analysis via Optimal Trees with Global Parameter Fusion

TL;DR

The paper tackles heterogeneity in treatment effects within clinical trials by introducing a fused optimal causal tree (FOCT) that uses mixed-integer optimization to obtain globally optimal subgroup partitions while enforcing parameter fusion across related subgroups. This fusion constraint promotes information sharing and improves statistical efficiency, addressing instability and overfitting that plague greedy tree methods, especially with small samples and rare alleles. The authors provide theoretical risk bounds showing near-optimal convergence for the FOCT, compare against CART, and demonstrate superior performance in simulations across varying correlations and sample sizes. The case study on the Health and Aging Brain Study–Health Disparities (HABS-HD) shows FOCT yielding clinically meaningful subgroup insights and interpretable covariate effects, illustrating its potential for precision medicine in AD contexts.

Abstract

Identifying and making statistical inferences on differential treatment effects (commonly known as subgroup analysis in clinical research) is central to precision health. Subgroup analysis allows practitioners to pinpoint populations for whom a treatment is especially beneficial or protective, thereby advancing targeted interventions. Tree based recursive partitioning methods are widely used for subgroup analysis due to their interpretability. Nevertheless, these approaches encounter significant limitations, including suboptimal partitions induced by greedy heuristics and overfitting from locally estimated splits, especially under limited sample sizes. To address these limitations, we propose a fused optimal causal tree method that leverages mixed integer optimization (MIO) to facilitate precise subgroup identification. Our approach ensures globally optimal partitions and introduces a parameter fusion constraint to facilitate information sharing across related subgroups. This design substantially improves subgroup discovery accuracy and enhances statistical efficiency. We provide theoretical guarantees by rigorously establishing out of sample risk bounds and comparing them with those of classical tree based methods. Empirically, our method consistently outperforms popular baselines in simulations. Finally, we demonstrate its practical utility through a case study on the Health and Aging Brain Study Health Disparities (HABS-HD) dataset, where our approach yields clinically meaningful insights.

Figures (9)

• Figure 1: Boxplot of mean square error (MSE) of SATE estimation and out-of-sample risk for different methods with $\rho\in\{0.7,0.8\}$ across 100 Monte Carlo simulations. The purple line is the mean of MSE (or out-of-sample risk) across 100 Monte Carlo simulations for each method. The purple line for CT-CART in Figure \ref{['fig:risk0.8']} is above 2.7.
• Figure 2: Summary of findings by FOCT: HABS-HD case study on black individuals.
• Figure 3: Coefficient estimates in the regression model with parameter fusion pattern learned by FOCT with 90% confidence intervals. Red segments indicate statistically significant coefficients; grey segments indicate non-significant ones.
• Figure 4: Summary of findings by OCT/CT-CART: HABS-HD case study on black individuals.
• Figure 5: Boxplot of mean square error (MSE) of SATE estimation and out-of-sample risk for different methods with $n=100$, $p=0.3$, $\rho=0.5$ and $d=3$ across 50 Monte Carlo simulations.

Theorems & Definitions (11)

• Theorem 4.2: Risk bound for optimal tree
• Theorem 4.3: Risk bound for CART
• Remark A.2
• Definition B.1: Shatter coefficient
• Definition B.2: Vapnik-Chervonenkis dimension
• Definition B.3: $\mathcal{F}^{+}$
• Lemma B.4
• Lemma B.5
• Lemma B.6
• Lemma B.7