Nazir Lone

Usher Institute, University of Edinburgh, Edinburgh, UK Abstract Finding patient subgroups with similar characteristics is crucial for personalized decision-making in various disciplines such as healthcare and policy evaluation. While most existing approaches rely on unsupervised clustering methods, there is a growing trend toward using supervised clustering methods that identify operationalizable subgroups in the context of a specific outcome of interest. We propose Bayesian Supervised Causal Clustering (bscc), with treatment effect as outcome to guide the clustering process. bscc identifies homogenous subgroups of individuals who are similar in their covariate profiles as well as their treatment effects. We evaluate bscc on simulated datasets as well as real-world dataset from the third International Stroke Trial to assess the practical usefulness of the framework. 1 Introduction Patient stratification, i.e., the identification of subgroups sharing clinical, molecular, or genetic characteristics, is a cornerstone of precision medicine and data-driven decision-making [Abdelnour et al., 2022]. By uncovering structured variation across high-dimensional population data, stratification enables clinicians to tailor treatments to specific groups, maximizing efficacy while minimizing adverse effects [Dubois et al., 2015]. Crucially, this approach addresses the challenge of Heterogeneity of Treatment Effects (HTE), where identical interventions yield divergent responses across subpopulations due to complex interactions between patient characteristics and treatment mechanisms [Thelen and Hennessy, 2025]. While Randomized Controlled Trials (RCTs) generally report Average Treatment Effects (ATE), such aggregate summaries often mask substantial variability in treatment response [Shehabi et al., 2021]. Ignoring this heterogeneity can lead to ineffective or even harmful clinical decisions: a modest overall benefit might obscure a large effect in a small, identifiable subgroup, or conversely, significant harm in another [Kravitz et al., 2004, Kent et al., 2018]. Consequently, identifying clinically operationalizable subgroups that are distinct not only in their observable features but also in their response to treatment is essential for guiding personalized therapeutic strategies. Traditionally, unsupervised clustering methods such as Gaussian Mixture Models (gmm) [Fraley and Raftery, 2002] and Latent Class Analysis (lca) [Collins and Lanza, 2010] have been the primary tools for data-driven patient stratification. These methods excel at uncovering hidden population structures by minimizing within-group variance in the covariate space [Murphy, 2012]. Approaches like kk-means and lca have been successfully applied to characterize disease phenotypes in conditions such as sepsis [Seymour et al., 2019], acute respiratory distress syndrome (ARDS) [Sinha et al., 2018], and hypertension [Guo et al., 2017]. However, a key limitation of unsupervised clustering is that the resulting subgroups are defined solely by covariate similarity, ignoring the outcome or treatment response. As a consequence, the identified clusters may be phenotypically distinct but heterogeneous with respect to treatment effects, limiting their utility for prescriptive decision-making (Figure 1(a, left)). Figure 1: (a) Illustrating the conceptual difference between (left) unsupervised clustering with Gaussian Mixture Model (gmm) (right) supervised clustering with bscc. gmm fails to recover the subgroups relevant to treatment while bscc can. (b) Difference between bscc clustering samples based on both covariates and treatment effect in a supervised clustering set-up, as opposed to causal clustering or supervised learning based approaches clustering based on potential outcomes or treatment effect only. To address this limitation, recent methodological research has shifted toward supervised clustering, which incorporates outcome information into the clustering process to ensure that the discovered groups are predictive of the target variable [Rouanet et al., 2023]. However, most existing supervised clustering methods focus on predicting an outcome rather than on predicting the difference between two potential outcomes. In the context of precision medicine, it might also be beneficial to identify groups with different responses to an intervention (i.e., predictive enrichment) alongside those with different outcomes (i.e., prognostic enrichment) [Temple, 2010]. This motivates the development of supervised causal clustering, where subgroup discovery is guided jointly by covariate similarity and treatment-effect variation. Such a framework offers a promising path toward discovering subgroups that are both interpretable in terms of patient characteristics and actionable in terms of distinct treatment responses. Related Work We summarize existing approaches for subgroup discovery in Table 1. Unsupervised clustering (UC) methods cluster patients based on covariate similarity alone, ignoring treatment response [Fraley and Raftery, 2002, Collins and Lanza, 2010]. While useful for exploratory analysis, they do not guarantee that the resulting clusters exhibit homogeneous treatment effects. Supervised clustering with outcome (SC) as side information, including Bayesian profile regression [Molitor et al., 2010] and supervised GMM (sgmm) [Shou et al., 2019, Lyu et al., 2024], jointly model covariates for clustering while predicting outcomes via fixed effects. Extensions accommodate various outcome types, e.g., categorical and count data [Liverani et al., 2015] and longitudinal outcomes [Rouanet et al., 2023]. However, treatment effect has not been the primary focus of these studies, and they typically model the conditional expectation of the outcome rather than the contrast between potential outcomes. Tree-based subgroup analysis (SA) methods, including interaction trees (it) [Xiaogang et al., 2008] and model-based recursive partitioning (mob) [Zeileis et al., 2008, Seibold et al., 2016], partition the covariate space recursively based on treatment effect heterogeneity. While directly supervised by treatment effect, these greedy and deterministic approaches may miss subgroups defined by complex variable interactions or soft boundaries, and can be unstable [Thelen and Hennessy, 2025]. Effect modeling (EM) methods estimate individual treatment effects (ITE) using meta-learners [Kennedy, 2023]: doubly robust learner (DR-learner) [Kennedy, 2020] and residual-learner (R-learner)) [Nie and Wager, 2020]; causal forests (cf) [Athey et al., 2019], Bayesian additive regression trees (bart) [Sparapani et al., 2021], or penalized regressions like Finding Heterogeneous Treatment Effects (FindIt) [Imai and Ratkovic, 2013] and Outcome Weighted Estimation (owe) [Chen et al., 2017]. While these methods provide flexible effect estimates at an individual level, they lack interpretable subgroup structures. Recovering subgroups from ITE estimates typically requires a post-hoc clustering step, [Bonvini et al., 2023]. Causal clustering (cc) [Kim et al., 2024] clusters individuals based on their potential outcomes estimated via counterfactual regression. This approach disentangles the potential outcome structure but ignores the covariate structure, potentially conflating structurally distinct subpopulations that happen to share similar potential outcomes (Figure 1(b)). Bayesian methods (BM) have been used to capture HTE without focusing on finding subgroups in the covariate space. Xu et al. [2016] use mixture modelling to group longitudinal treatment effects, and Shahn and Madigan [2017] captures treatment heterogeneity as a mixture model. Table 1: Summary of approaches. TE: treatment-effect-based. Category Supervised Probabilistic Covariate-based TE Example UC ×\times ✓\checkmark ✓\checkmark ×\times gmm SC ✓\checkmark ✓\checkmark ✓\checkmark ×\times sgmm SA ✓\checkmark ×\times ✓\checkmark ✓\checkmark it, mob ✓\checkmark ✓\checkmark ✓\checkmark ✓\checkmark bscc (ours) EM ✓\checkmark ×\times ×\times ✓\checkmark cf, bart, DR-l, R-l, FindIt, owe, cc BM ✓\checkmark ✓\checkmark ×\times ✓\checkmark Xu et al. [2016], Shahn and Madigan [2017] Contributions We propose Bayesian Supervised Causal Clustering, or bscc (“basic”), as a probabilistic framework for operationalizable patient stratification. bscc jointly clusters individuals based on both their covariate structure and their response to treatment. Unlike traditional unsupervised clustering methods that only consider covariate similarity, or traditional effect modelling methods that focus on ITE estimation without finding subgroups, or causal clustering methods that do not consider covariate structure, bscc explicitly integrates causal effect as a supervisory signal into the clustering process. Similar to tree-based subgroup analysis, this allows for the identification of subgroups that are homogeneous in covariates and treatment effects, but with more flexible subgroups through Bayesian mixture modelling. We explore the effectiveness of bscc in both simulated and real datasets extensively, demonstrating that it can identify meaningful and consistent subgroups of individuals. 2 Methods Standard mixture model with KK components is described as p​(𝐱n)=∑k=