Abstract
Causal effects are often characterized with population summaries. These might provide an incomplete picture when there are heterogeneous treatment effects across subgroups. Since the subgroup structure is typically unknown, it is more challenging to identify and evaluate subgroup effects than population effects. We propose a new solution to this problem: Causal k-Means Clustering, which harnesses the widely-used k-means clustering algorithm to uncover the unknown subgroup structure. Our problem differs significantly from the conventional clustering setup since the variables to be clustered are unknown counterfactual functions. We present a plug-in estimator which is simple and readily implementable using off-the-shelf algorithms, and study its rate of convergence. We also develop a new bias-corrected estimator based on nonparametric efficiency theory and double machine learning, and show that this estimator achieves fast root-n rates and asymptotic normality in large nonparametric models. Our proposed methods are especially useful for modern outcome-wide studies with multiple treatment levels. Further, our framework is extensible to clustering with generic pseudo-outcomes, such as partially observed outcomes or otherwise unknown functions. Finally, we explore finite sample properties via simulation, and illustrate the proposed methods in a study of treatment programs for adolescent substance abuse.
Overview
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The paper introduces a method called 'Causal k-Means Clustering', an adaptation of k-means clustering aimed at identifying subgroups with distinct treatment effects in various contexts, including personalized medicine and policy making.
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This method utilizes an outcome regression function and cluster analysis to group units based on homogeneity within and heterogeneity across clusters concerning treatment effects, enhanced by two estimation techniques: a Plug-in Estimator and a Bias-Corrected Estimator.
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Causal k-Means Clustering allows for effective subgroup analysis in fields like medicine and social sciences, improving treatment interventions and strategies through the discovery of precise subgroup responses.
Understanding Subgroup Effects in Treatments through Causal k-Means Clustering
Introduction
When studying the effects of various treatments across different populations, averaging out the effects (known as Average Treatment Effect, or ATE) can sometimes mask underlying variations. These variations or heterogeneous treatment effects are critical, especially in personalized medicine or targeted policy making, where understanding subgroup responses to treatments can lead to more effective interventions.
The paper presents a novel approach called "Causal k-Means Clustering" which is an innovative adaptation of the traditional k-means clustering method used to identify subgroups that exhibit distinct treatment effects. This method particularly shines when the subgroup structure is unknown and needs to be learned from the data.
Core Methodology
The causal k-means clustering proposed leverages the well-known k-means algorithm to identify clusters in the data where the treatment effects are relatively homogenous within each cluster but vary significantly across clusters. Here are the steps and components involved:
- Outcome Regression Function: This function is essential as it represents the expected outcome given covariates and specific treatments. It's unknown and needs to be estimated from the data.
- Cluster Analysis: By focusing on how the treatment effects vary by covariates, the method seeks to group units such that those within each cluster share similar outcomes to treatments. Clustering is done based on the estimated outcome regression functions.
- Estimation Techniques: The paper outlines two main estimation techniques:
- The Plug-in Estimator: This is simpler and directly applies traditional k-means to the estimated outcome regression functions. However, it is generally not $\sqrt{n}$-consistent without specific conditions.
- Bias-Corrected Estimator: An advanced technique that achieves faster convergence rates and asymptotic normality, even under complex model conditions.
Practical Implications
The method finds its use in fields where it is crucial to understand how different subgroups react to treatments, such as medicine (different responses to a drug due to genetic factors), or social sciences (impact of educational programs across various demographic groups). The ability to identify subgroups where interventions are more or less effective can significantly enhance the impact of tailored strategies.
Theory and Speculations
Underpinning the causal k-means algorithm is a solid theoretical foundation that guarantees that as the sample size grows, the cluster discovery process becomes more accurate, assuming certain regularities in the data. The bias-corrected estimator, which is a bit more sophisticated, ensures strong performance even when the baseline estimations of the outcome functions are not perfect.
Looking ahead, the paper suggests potential extensions and refinements, such as improving computational efficiency, handling missing or censored data, and integrating this clustering method with other types of machine learning algorithms for richer insights.
Case Study and Simulations
Illustratively, the method was applied to a real dataset on treatment programs for adolescent substance abuse, helping to uncover that certain programs worked better for specific subgroups than others. The simulation studies reinforce these practical capabilities, showcasing how the method outperforms traditional analysis that ignores subgroup structures.
Conclusion
Causal k-Means Clustering provides a robust tool for uncovering hidden subgroup structures in treatment effect analysis. By integrating this method, researchers and practitioners can gain deeper insights into how different subgroups respond to various treatments, leading to more personalized and effective intervention strategies. The versatility and robust theoretical foundations of this method make it a valuable addition to the toolbox for statisticians and data scientists working in causal inference and personalized medicine.
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