From Conditional Quantile Regression to Marginal Quantile Estimation with Applications to Missing Data and Causal Inference

Abstract

It is well known that information on the conditional distribution of an outcome variable given covariates can be used to obtain an enhanced estimate of the marginal outcome distribution. This can be done easily by integrating out the marginal covariate distribution from the conditional outcome distribution. However, to date, no analogy has been established between marginal quantile and conditional quantile regression. This article provides a link between them. We propose two novel marginal quantile and marginal mean estimation approaches through conditional quantile regression when some of the outcomes are missing at random. The first of these approaches is free from the need to choose a propensity score. The second is double robust to model misspecification: it is consistent if either the conditional quantile regression model is correctly specified or the missing mechanism of outcome is correctly specified. Consistency and asymptotic normality of the two estimators are established, and the second double robust estimator achieves the semiparametric efficiency bound. Extensive simulation studies are performed to demonstrate the utility of the proposed approaches. An application to causal inference is introduced. For illustration, we apply the proposed methods to a job training program dataset.

Keyword:

• Augmented inverse probability weighting
• Causal inference
• Marginal mean
• Marginal quantile
• Missing at random
• Quantile regression

Supplementary Materials

The supplementary materials include the justification of transforming the solution finding problems to minimization, proofs of Theorems 4–6, additional simulation studies, simulation studies for QTT and ATT, and an additional real-data example.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The authors thank the Editor, Professor Jianqing Fan, an Associate Editor, and two reviewers for their insightful comments and suggestions that greatly improved the article.

Additional information

Funding

This work is partly supported by the National Natural Science Foundation of China (71931004, 11901200, 71971083, and 11971170) and the National Key R&D Program of China (2021YFA1000100 and 2021YFA1000101).