Direct Doubly Robust Estimation of Conditional Quantile Contrasts
Josh Givens, Song Liu, Henry W J Reeve, Katarzyna Reluga
TL;DR
This work introduces the first direct estimator for the conditional quantile comparator (CQC), a distributional heterogeneous treatment effect that maps untreated responses to treated quantiles via a transport $g^{*}(y_0|\mathbf x)$. By explicitly parameterising the CQC as $g_\theta(y_0|\mathbf x)$ and minimising a carefully constructed loss $\bar{\ell}$ with a doubly robust gradient $\hat{\zeta}_{\text{dr}}$, the method enables direct modelling, interpretability, and constraint imposition while preserving double robustness to nuisance estimation errors. The authors derive finite-sample convergence guarantees under convex and strongly convex conditions, show robustness to nuisance estimation, and demonstrate through simulations that the direct estimator outperforms the prior inversion-based CQC approach across varying CQC slopes, nuisance errors, and sample sizes. Real-world applications to employment program data illustrate the method's interpretability, revealing how treatment effects depend on baseline earnings and age, with additional neural network-based CQC modelling providing flexibility for nonlinear relationships. Overall, the direct CQC estimator advances HTE analysis by offering a transparent, versatile, and accurate tool for quantile-level treatment effect assessment.
Abstract
Within heterogeneous treatment effect (HTE) analysis, various estimands have been proposed to capture the effect of a treatment conditional on covariates. Recently, the conditional quantile comparator (CQC) has emerged as a promising estimand, offering quantile-level summaries akin to the conditional quantile treatment effect (CQTE) while preserving some interpretability of the conditional average treatment effect (CATE). It achieves this by summarising the treated response conditional on both the covariates and the untreated response. Despite these desirable properties, the CQC's current estimation is limited by the need to first estimate the difference in conditional cumulative distribution functions and then invert it. This inversion obscures the CQC estimate, hampering our ability to both model and interpret it. To address this, we propose the first direct estimator of the CQC, allowing for explicit modelling and parameterisation. This explicit parameterisation enables better interpretation of our estimate while also providing a means to constrain and inform the model. We show, both theoretically and empirically, that our estimation error depends directly on the complexity of the CQC itself, improving upon the existing estimation procedure. Furthermore, it retains the desirable double robustness property with respect to nuisance parameter estimation. We further show our method to outperform existing procedures in estimation accuracy across multiple data scenarios while varying sample size and nuisance error. Finally, we apply it to real-world data from an employment scheme, uncovering a reduced range of potential earnings improvement as participant age increases.