Towards R-learner of conditional average treatment effects with a continuous treatment: T-identification, estimation, and inference

The R-learner has been popular in causal inference as a flexible and efficient meta-learning approach for heterogeneous treatment effect estimation. In this article, we show the identifiability transition of the generalized R-learning framework from a binary treatment to continuous treatment. To resolve the non-identification issue with continuous treatment, we propose a novel identification strategy named T-identification, acknowledging the use of Tikhonov regularization rooted in the nonlinear functional analysis. Following the new identification strategy, we introduce an $\ell_2$-penalized R-learner framework to estimate the conditional average treatment effect with continuous treatment. The new R-learner framework accommodates modern, flexible machine learning algorithms for both nuisance function and target estimand estimation. Asymptotic properties are studied when the target estimand is approximated by sieve approximation, including general error bounds, asymptotic normality, and inference. Simulations illustrate the superior performance of our proposed estimator. An application of the new method to the medical information mart for intensive care data reveals the heterogeneous treatment effect of oxygen saturation on survival in sepsis patients.