A single-index model with multiple-links

In a regression model for treatment outcome in a randomized clinical trial, a treatment effect modifier is a covariate that has an interaction with the treatment variable, implying that the treatment efficacies vary across values of such a covariate. In this paper, we present a method for determining a composite variable from a set of baseline covariates, that can have a nonlinear association with the treatment outcome, and acts as a composite treatment effect modifier. We introduce a parsimonious generalization of the single-index models that targets the effect of the interaction between the treatment conditions and the vector of covariates on the outcome, a single-index model with multiple-links (SIMML) that estimates a single linear combination of the covariates (i.e., a single-index), with treatment-specific nonparametric link functions. The approach emphasizes a focus on the treatment-by-covariates interaction effects on the treatment outcome that are relevant for making optimal treatment decisions. Asymptotic results for estimator are obtained under possible model misspecification. A treatment decision rule based on the derived single-index is defined, and it is compared to other methods for estimating optimal treatment decision rules. An application to a clinical trial for the treatment of depression is presented.