skip to main content

Title: Quantile regression for survival data with covariates subject to detection limits

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.

more » « less
Award ID(s):
Author(s) / Creator(s):
 ;  ;  
Publisher / Repository:
Oxford University Press
Date Published:
Journal Name:
Medium: X Size: p. 610-621
["p. 610-621"]
Sponsoring Org:
National Science Foundation
More Like this
  1. Summary

    In many observational longitudinal studies, the outcome of interest presents a skewed distribution, is subject to censoring due to detection limit or other reasons, and is observed at irregular times that may follow a outcome-dependent pattern. In this work, we consider quantile regression modeling of such longitudinal data, because quantile regression is generally robust in handling skewed and censored outcomes and is flexible to accommodate dynamic covariate-outcome relationships. Specifically, we study a longitudinal quantile regression model that specifies covariate effects on the marginal quantiles of the longitudinal outcome. Such a model is easy to interpret and can accommodate dynamic outcome profile changes over time. We propose estimation and inference procedures that can appropriately account for censoring and irregular outcome-dependent follow-up. Our proposals can be readily implemented based on existing software for quantile regression. We establish the asymptotic properties of the proposed estimator, including uniform consistency and weak convergence. Extensive simulations suggest good finite-sample performance of the new method. We also present an analysis of data from a long-term study of a population exposed to polybrominated biphenyls (PBB), which uncovers an inhomogeneous PBB elimination pattern that would not be detected by traditional longitudinal data analysis.

    more » « less
  2. Abstract

    Statistical analysis of longitudinal data often involves modeling treatment effects on clinically relevant longitudinal biomarkers since an initial event (the time origin). In some studies including preventive HIV vaccine efficacy trials, some participants have biomarkers measured starting at the time origin, whereas others have biomarkers measured starting later with the time origin unknown. The semiparametric additive time-varying coefficient model is investigated where the effects of some covariates vary nonparametrically with time while the effects of others remain constant. Weighted profile least squares estimators coupled with kernel smoothing are developed. The method uses the expectation maximization approach to deal with the censored time origin. The Kaplan–Meier estimator and other failure time regression models such as the Cox model can be utilized to estimate the distribution and the conditional distribution of left censored event time related to the censored time origin. Asymptotic properties of the parametric and nonparametric estimators and consistent asymptotic variance estimators are derived. A two-stage estimation procedure for choosing weight is proposed to improve estimation efficiency. Numerical simulations are conducted to examine finite sample properties of the proposed estimators. The simulation results show that the theory and methods work well. The efficiency gain of the two-stage estimation procedure depends on the distribution of the longitudinal error processes. The method is applied to analyze data from the Merck 023/HVTN 502 Step HIV vaccine study.

    more » « less
  3. Abstract

    Popular parametric and semiparametric hazards regression models for clustered survival data are inappropriate and inadequate when the unknown effects of different covariates and clustering are complex. This calls for a flexible modeling framework to yield efficient survival prediction. Moreover, for some survival studies involving time to occurrence of some asymptomatic events, survival times are typically interval censored between consecutive clinical inspections. In this article, we propose a robust semiparametric model for clustered interval‐censored survival data under a paradigm of Bayesian ensemble learning, called soft Bayesian additive regression trees or SBART (Linero and Yang, 2018), which combines multiple sparse (soft) decision trees to attain excellent predictive accuracy. We develop a novel semiparametric hazards regression model by modeling the hazard function as a product of a parametric baseline hazard function and a nonparametric component that uses SBART to incorporate clustering, unknown functional forms of the main effects, and interaction effects of various covariates. In addition to being applicable for left‐censored, right‐censored, and interval‐censored survival data, our methodology is implemented using a data augmentation scheme which allows for existing Bayesian backfitting algorithms to be used. We illustrate the practical implementation and advantages of our method via simulation studies and an analysis of a prostate cancer surgery study where dependence on the experience and skill level of the physicians leads to clustering of survival times. We conclude by discussing our method's applicability in studies involving high‐dimensional data with complex underlying associations.

    more » « less
  4. 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. 
    more » « less
  5. When analyzing bivariate outcome data, it is often of scientific interest to measure and estimate the association between the bivariate outcomes. In the presence of influential covariates for one or both of the outcomes, conditional association measures can quantify the strength of association without the disturbance of the marginal covariate effects, to provide cleaner and less‐confounded insights into the bivariate association. In this work, we propose estimation and inferential procedures for assessing the conditional Kendall's tau coefficient given the covariates, by adopting the quantile regression and quantile copula framework to handle marginal covariate effects. The proposed method can flexibly accommodate right censoring and be readily applied to bivariate survival data. It also facilitates an estimator of the conditional concordance measure, namely, a conditionalindex, where the unconditionalindex is commonly used to assess the predictive capacity for survival outcomes. The proposed method is flexible and robust and can be easily implemented using standard software. The method performed satisfactorily in extensive simulation studies with and without censoring. Application of our methods to two real‐life data examples demonstrates their desirable practical utility.

    more » « less