A General Framework for Quantile Estimation with Incomplete Data

Abstract

Summary Quantile estimation has attracted significant research interest in recent years. However, there has been only a limited literature on quantile estimation in the presence of incomplete data. We propose a general framework to address this problem. Our framework combines the two widely adopted approaches for missing data analysis, the imputation approach and the inverse probability weighting approach, via the empirical likelihood method. The method proposed is capable of dealing with many different missingness settings. We mainly study three of them: estimating the marginal quantile of a response that is subject to missingness while there are fully observed covariates; estimating the conditional quantile of a fully observed response while the covariates are partially available; estimating the conditional quantile of a response that is subject to missingness with fully observed covariates and extra auxiliary variables. The method proposed allows multiple models for both the missingness probability and the data distribution. The resulting estimators are multiply robust in the sense that they are consistent if any one of these models is correctly specified. The asymptotic distributions are established by using empirical process theory.

Censored quantile regression based on multiply robust propensity scores

“…The new method is inspired by the multiply robust PS estimators of Han 23 and Han et al 24 in terms of dealing with missing data, 25–29 which provides new insights into PS estimation. In particular, we propose an ISUB-QR estimator based on multiply robust PS.…”

“…Secondly, instead of being used to deal with missing data, PS in the current study is adopted as a criterion for selecting the informative subset in censored QR. Moreover, in contrast with the estimators of Han 23 and Han et al, 24 where the PS is estimated for fully observed subjects only, we estimate PS for both fully observed and censored subjects. The theoretical investigation of the proposed ISUB-QR estimator also differs from that for missing data, where the exact representation of the PS estimator is needed, while in our setting, only the uniform consistency at a certain rate is required.…”

Augmented inverse probability weighted fractional imputation in quantile regression

“…It may be satisfactory with small amounts of missing data. However, it can lead to biased and inefficient estimators especially when drawing inferences for subpopulations (Little and Rubin 2002) and the rate of missingness is high (Han et al 2019) 4 . Another popular and simple method is inverse probability weighting (IPW) (Horvitz and Thompson 1952) 5 .…”

Covariate Distribution Balance via Propensity Scores

“…Most recently, Han et al (2019) propose a hybrid framework that combines PS models and calibration weights and is also suitable for estimating quantile treatment effects. Despite achieving improved robustness properties, we note that Han et al (2019)'s method uses the ML estimate of the PS, does not fully exploit the covariate balancing of the PS and requires one to compute different calibration weights for each analyzed quantile. Given that the IPS does not consult the outcome data, it can be used in conjunction with many existing PS methods including Han et al (2019).…”

“…Despite achieving improved robustness properties, we note that Han et al (2019)'s method uses the ML estimate of the PS, does not fully exploit the covariate balancing of the PS and requires one to compute different calibration weights for each analyzed quantile. Given that the IPS does not consult the outcome data, it can be used in conjunction with many existing PS methods including Han et al (2019). We view this flexibility as one of the main attractive features of our proposal.…”