Lasso estimation of hierarchical interactions for analyzing heterogeneity of treatment effect

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‐stepl 1norm 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.