Lasso estimation of hierarchical interactions for analyzing heterogeneity of treatment effect

Abstract

Individuals differ in how they respond to a given treatment. In an effort to predict the treatment response and analyze the heterogeneity of treatment effect, we propose a general modeling framework by identifying treatment-covariate interactions honoring a hierarchical condition. We construct a single-step $l_1$ norm penalty procedure that maintains the hierarchical structure of interactions in the sense that a treatment-covariate interaction term is included in the model only when either the covariate or both the covariate and treatment have nonzero main effects. We developed a constrained Lasso approach with two parameterization schemes that enforce the hierarchical interaction restriction differently. We solved the resulting constrained optimization problem using a spectral projected gradient method. We compared our methods to the unstructured Lasso using simulation studies including a scenario that violates the hierarchical condition (misspecified model). The simulations showed that our methods yielded more parsimonious models and outperformed the unstructured Lasso for correctly identifying nonzero treatment-covariate interactions. The superior performance of our methods are also corroborated by an application to a large randomized clinical trial data investigating a drug for treating congestive heart failure (N = 2569). Our methods provide a well-suited approach for doing secondary analysis in clinical trials to analyze heterogeneous treatment effects and to identify predictive biomarkers.

Keywords: Lasso; heterogeneity of treatment effect; hierarchical interaction; treatment-covariate interaction.