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1. Censored Quantile Regression Forest

   Li, Alexander Hanbo; Bradic, Jelena (January 2020, Proceedings of Machine Learning Research)

   Chiappa, Silvia; Calandra, Roberto (Ed.)

   Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases. Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models. Regression adjustment is based on a new estimating equation that adapts to censoring and leads to quantile score whenever the data do not exhibit censoring. The proposed procedure named censored quantile regression forest, allows us to estimate quantiles of time-to-event without any parametric modeling assumption. We establish its consistency under mild model specifications. Numerical studies showcase a clear advantage of the proposed procedure. 

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2. Envelopes for censored quantile regression

   https://doi.org/10.1111/sjos.12602  

   Zhao, Yue; Van Keilegom, Ingrid; Ding, Shanshan (June 2022, Scandinavian Journal of Statistics)

   Abstract We propose an efficient estimator for the coefficients in censored quantile regression using the envelope model. The envelope model uses dimension reduction techniques to identify material and immaterial components in the data, and forms the estimator based only on the material component, thus reducing the variability of estimation. We will demonstrate the guaranteed asymptotic efficiency gain of our proposed envelope estimator over the traditional estimator for censored quantile regression. Our analysis begins with the local weighing approach that traditionally relies on semiparametric ‐estimation involving the conditional Kaplan–Meier estimator. We will instead invoke the independent identically distributed (i.i.d.) representation of the Kaplan–Meier estimator, which eliminates this infinite‐dimensional nuisance and transforms our objective function in ‐estimation into a ‐process indexed by only an Euclidean parameter. The modified ‐estimation problem becomes entirely parametric and hence more amenable to analysis. We will also reconsider the i.i.d. representation of the conditional Kaplan–Meier estimator. 

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3. Quantile regression for survival data with covariates subject to detection limits

   https://doi.org/10.1111/biom.13309  

   Yu, Tonghui; Xiang, Liming; Wang, Huixia Judy (June 2020, Biometrics)

   Abstract With advances in biomedical research, biomarkers are becoming increasingly important prognostic factors for predicting overall survival, while the measurement of biomarkers is often censored due to instruments' lower limits of detection. This leads to two types of censoring: random censoring in overall survival outcomes and fixed censoring in biomarker covariates, posing new challenges in statistical modeling and inference. Existing methods for analyzing such data focus primarily on linear regression ignoring censored responses or semiparametric accelerated failure time models with covariates under detection limits (DL). In this paper, we propose a quantile regression for survival data with covariates subject to DL. Comparing to existing methods, the proposed approach provides a more versatile tool for modeling the distribution of survival outcomes by allowing covariate effects to vary across conditional quantiles of the survival time and requiring no parametric distribution assumptions for outcome data. To estimate the quantile process of regression coefficients, we develop a novel multiple imputation approach based on another quantile regression for covariates under DL, avoiding stringent parametric restrictions on censored covariates as often assumed in the literature. Under regularity conditions, we show that the estimation procedure yields uniformly consistent and asymptotically normal estimators. Simulation results demonstrate the satisfactory finite‐sample performance of the method. We also apply our method to the motivating data from a study of genetic and inflammatory markers of Sepsis. 

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4. Debiased inference on heterogeneous quantile treatment effects with regression rank scores

   https://doi.org/10.1093/jrsssb/qkad075  

   Giessing, Alexander; Wang, Jingshen (August 2023, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Understanding treatment effect heterogeneity is vital to many scientific fields because the same treatment may affect different individuals differently. Quantile regression provides a natural framework for modelling such heterogeneity. We propose a new method for inference on heterogeneous quantile treatment effects (HQTE) in the presence of high-dimensional covariates. Our estimator combines an ℓ1-penalised regression adjustment with a quantile-specific bias correction scheme based on rank scores. We study the theoretical properties of this estimator, including weak convergence and semi-parametric efficiency of the estimated HQTE process. We illustrate the finite-sample performance of our approach through simulations and an empirical example, dealing with the differential effect of statin usage for lowering low-density lipoprotein cholesterol levels for the Alzheimer’s disease patients who participated in the UK Biobank study. 

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5. Joint Quantile Regression for Spatial Data

   https://doi.org/10.1111/rssb.12467  

   Chen, Xu; Tokdar, Surya_T (August 2021, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. For analysing spatially indexed data, we address this difficulty by generalizing the joint quantile regression model of Yang and Tokdar (Journal of the American Statistical Association, 2017, 112(519), 1107–1120) and characterizing spatial dependence via a Gaussian or t-copula process on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out spatial quantile smoothing. An effective model comparison criteria is provided, particularly for selecting between different model specifications of tail heaviness and tail dependence. Extensive simulation studies and two real applications to particulate matter concentration and wildfire risk are presented to illustrate substantial gains in inference quality, prediction accuracy and uncertainty quantification over existing alternatives. 

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