Censored Quantile Regression Research Articles

Interval-Censored Linear Quantile Regression

Censored quantile regression has emerged as a prominent alternative to classical Cox’s proportional hazards model or accelerated failure time model in both theoretical and applied statistics. While quantile regression has been extensively studied for right-censored survival data, methodologies for analyzing interval-censored data remain limited in the survival analysis literature. This article introduces a novel local weighting approach for estimating linear censored quantile regression, specifically tailored to handle diverse forms of interval-censored survival data. The estimation equation and the corresponding convex objective function for the regression parameter can be constructed as a weighted average of quantile loss contributions at two interval endpoints. The weighting components are nonparametrically estimated using local kernel smoothing or ensemble machine learning techniques. To estimate the nonparametric distribution mass for interval-censored data, a modified EM algorithm for nonparametric maximum likelihood estimation is employed by introducing subject-specific latent Poisson variables. The proposed method’s empirical performance is demonstrated through extensive simulation studies and real data analyses of two HIV/AIDS datasets. Supplementary materials for this article are available online.

A censored quantile transformation model for Alzheimer’s Disease data with multiple functional covariates

Abstract Alzheimer’s disease (AD) is a progressive disease that starts from mild cognitive impairment and may eventually lead to irreversible memory loss. It is imperative to explore the risk factors associated with the conversion time to AD that is usually right-censored. Classical statistical models like mean regression and Cox models fail to quantify the impact of risk factors across different quantiles of a response distribution, and previous research has primarily focused on modelling a single functional covariate, possibly overlooking the interdependence among multiple functional covariates and other crucial features of the distribution. To address these issues, this paper proposes a multivariate functional censored quantile regression model based on dynamic power transformations, which relaxes the global linear assumption and provides more robustness and flexibility. Uniform consistency and weak convergence of the quantile process are established. Simulation studies suggest that the proposed method outperforms the existing approaches. Real data analysis shows the importance of both left and right hippocampal radial distance curves for predicting the conversion time to AD at different quantile levels.

Distributed Censored Quantile Regression

This article discusses an extension of censored quantile regression to a distributed setting. With the growing availability of massive datasets, it is oftentimes an arduous task to analyze all the data with limited computational facilities efficiently. Our proposed method, which attempts to overcome this challenge, is comprised of two key steps, namely: (i) estimation of both Kaplan-Meier estimator and model coefficients in a parallel computing environment; (ii) aggregation of coefficient estimations from individual machines. We study the upper limit of the order of the number of machines for this computing environment, which, if fulfilled, guarantees that the proposed estimator converges at a comparable rate to that of the oracle estimator. In addition, we also provide two further modifications for distributed systems including (i) a communication-facilitated adaptation in the sense of Chen, Liu, and Zhang and (ii) a nonparametric counterpart along the direction of Kong and Xia for censored quantile regression. Numerical experiments are conducted to compare the proposed and the existing estimators. The promising results demonstrate the computation efficiency of the proposed methods. Finally, for practical concerns, a cross-validation procedure is also developed which can better select the hyperparameters for the proposed methodologies. Supplementary materials for this article are available online.

Identifying important risk factors for survival of HIV-infected patients using censored quantile regression

ENWEndNote BIBJabRef, Mendeley RISPapers, Reference Manager, RefWorks, Zotero AMA Hamidi O, Maroufizadeh S, Poorolajal J, Azimi T, Amini P. Identifying important risk factors for survival of HIV-infected patients using censored quantile regression. HIV & AIDS Review. International Journal of HIV-Related Problems. 2023. doi:10.5114/hivar.2023.125016. APA Hamidi, O., Maroufizadeh, S., Poorolajal, J., Azimi, T., & Amini, P. (2023). Identifying important risk factors for survival of HIV-infected patients using censored quantile regression. HIV & AIDS Review. International Journal of HIV-Related Problems. https://doi.org/10.5114/hivar.2023.125016 Chicago Hamidi, Omid, Saman Maroufizadeh, Jalal Poorolajal, Tara Azimi, and Payam Amini. 2023. "Identifying important risk factors for survival of HIV-infected patients using censored quantile regression". HIV & AIDS Review. International Journal of HIV-Related Problems. doi:10.5114/hivar.2023.125016. Harvard Hamidi, O., Maroufizadeh, S., Poorolajal, J., Azimi, T., and Amini, P. (2023). Identifying important risk factors for survival of HIV-infected patients using censored quantile regression. HIV & AIDS Review. International Journal of HIV-Related Problems. https://doi.org/10.5114/hivar.2023.125016 MLA Hamidi, Omid et al. "Identifying important risk factors for survival of HIV-infected patients using censored quantile regression." HIV & AIDS Review. International Journal of HIV-Related Problems, 2023. doi:10.5114/hivar.2023.125016. Vancouver Hamidi O, Maroufizadeh S, Poorolajal J, Azimi T, Amini P. Identifying important risk factors for survival of HIV-infected patients using censored quantile regression. HIV & AIDS Review. International Journal of HIV-Related Problems. 2023. doi:10.5114/hivar.2023.125016.

Modeling Censored Mobility Demand Through Censored Quantile Regression Neural Networks

Shared mobility services require accurate demand models for effective service planning. On the one hand, modeling the full probability distribution of demand is advantageous because the entire uncertainty structure preserves valuable information for decision-making. On the other hand, demand is often observed through the usage of the service itself, so that the observations are censored, as they are inherently limited by available supply. Since the 1980s, various works on Censored Quantile Regression models have performed well under such conditions. Further, in the last two decades, several papers have proposed to implement these models flexibly through Neural Networks. However, the models in current works estimate the quantiles individually, thus incurring a computational overhead and ignoring valuable relationships between the quantiles. We address this gap by extending current Censored Quantile Regression models to learn multiple quantiles at once and apply these to synthetic baseline datasets and datasets from two shared mobility providers in the Copenhagen metropolitan area in Denmark. The results show that our extended models yield fewer quantile crossings and less computational overhead without compromising model performance.

Scalable estimation and inference for censored quantile regression process

Censored quantile regression (CQR) has become a valuable tool to study the heterogeneous association between a possibly censored outcome and a set of covariates, yet computation and statistical inference for CQR have remained a challenge for large-scale data with many covariates. In this paper, we focus on a smoothed martingale-based sequential estimating equations approach, to which scalable gradient-based algorithms can be applied. Theoretically, we provide a unified analysis of the smoothed sequential estimator and its penalized counterpart in increasing dimensions. When the covariate dimension grows with the sample size at a sublinear rate, we establish the uniform convergence rate (over a range of quantile indexes) and provide a rigorous justification for the validity of a multiplier bootstrap procedure for inference. In high-dimensional sparse settings, our results considerably improve the existing work on CQR by relaxing an exponential term of sparsity. We also demonstrate the advantage of the smoothed CQR over existing methods with both simulated experiments and data applications.

Efficient Estimation for Censored Quantile Regression

ABSTRACT Censored quantile regression (CQR) has received growing attention in survival analysis because of its flexibility in modeling heterogeneous effect of covariates. Advances have been made in developing various inferential procedures under different assumptions and settings. Under the conditional independence assumption, many existing CQR methods can be characterized either by stochastic integral-based estimating equations (see, e.g., Peng and Huang) or by locally weighted approaches to adjust for the censored observations (see, for instance, Wang and Wang). While there have been proposals of different apparently dissimilar strategies in terms of formulations and the techniques applied for CQR, the inter-relationships amongst these methods are rarely discussed in the literature. In addition, given the complicated structure of the asymptotic variance, there has been limited investigation on improving the estimation efficiency for censored quantile regression models. This article addresses these open questions by proposing a unified framework under which many conventional approaches for CQR are covered as special cases. The new formulation also facilitates the construction of the most efficient estimator for the parameters of interest amongst a general class of estimating functions. Asymptotic properties including consistency and weak convergence of the proposed estimator are established via the martingale-based argument. Numerical studies are presented to illustrate the promising performance of the proposed estimator as compared to existing contenders under various settings. Supplementary materials for this article are available online.

Inference for High-Dimensional Censored Quantile Regression

With the availability of high-dimensional genetic biomarkers, it is of interest to identify heterogeneous effects of these predictors on patients’ survival, along with proper statistical inference. Censored quantile regression has emerged as a powerful tool for detecting heterogeneous effects of covariates on survival outcomes. To our knowledge, there is little work available to draw inferences on the effects of high-dimensional predictors for censored quantile regression (CQR). This article proposes a novel procedure to draw inference on all predictors within the framework of global CQR, which investigates covariate-response associations over an interval of quantile levels, instead of a few discrete values. The proposed estimator combines a sequence of low-dimensional model estimates that are based on multi-sample splittings and variable selection. We show that, under some regularity conditions, the estimator is consistent and asymptotically follows a Gaussian process indexed by the quantile level. Simulation studies indicate that our procedure can properly quantify the uncertainty of the estimates in high-dimensional settings. We apply our method to analyze the heterogeneous effects of SNPs residing in lung cancer pathways on patients’ survival, using the Boston Lung Cancer Survival Cohort, a cancer epidemiology study on the molecular mechanism of lung cancer.

Moment estimation for censored quantile regression

In influential articles Powell (Journal of Econometrics 25(3):303–325, 1984; Journal of Econometrics 32(1):143–155, 1986) proposed optimization-based censored least absolute deviations estimator (CLAD) and general censored quantile regression estimator (CQR). It has been recognized, however, that this optimization-based estimator may perform poorly in finite samples (e.g., Khan and Powell, Journal of Econometrics 103(1–2):73–110, 2001; Fitzenberger, Handbook of Statistics. Elsevier, 1996; Fitzenberger and Winker, Computational Statistics & Data Analysis 52(1):88–108, 2007; Koenker, Journal of Statistical Software 27(6), 2008). In this paper we propose a moment-based censored quantile regression estimator (MCQR). While both the CQR and MCQR estimators have the same large sample properties, our simulation results suggest certain advantage of our moment-based estimator (MCQR). In addition, the empirical likelihood methods for the uncensored model (e.g., Whang 2006; Otsu, Journal of Econometrics 142(1):508–538, 2008) can readily be adapted to the censored model within our method of moment estimation framework. When both censoring and endogeneity are present, we develop an instrumental variable censored quantile regression estimator (IVCQR) by combining the insights of Chernozhukov and Hansen’s (Journal of Econometrics 132(2):491–525, 2006; Journal of Econometrics 142(1):379–398, 2008) instrumental variables quantile regression estimator (IVQR) and the MCQR. Simulation results indicate that the IVCQR estimator performs well.

Estimation of causal quantile effects with a binary instrumental variable and censored data.

The causal effect of a treatment is of fundamental interest in the social, biological, and health sciences. Instrumental variable (IV) methods are commonly used to determine causal treatment effects in the presence of unmeasured confounding. In this work, we study a new binary IV framework with randomly censored outcomes where we propose to quantify the causal treatment effect by the concept of complier quantile causal effect (CQCE). The CQCE is identifiable under weaker conditions than the complier average causal effect when outcomes are subject to censoring, and it can provide useful insight into the dynamics of the causal treatment effect. Employing the special characteristic of the binary IV and adapting the principle of conditional score, we uncover a simple weighting scheme that can be incorporated into the standard censored quantile regression procedure to estimate CQCE. We develop robust nonparametric estimation of the derived weights in the first stage, which permits stable implementation of the second stage estimation based on existing software. We establish rigorous asymptotic properties for the proposed estimator, and confirm its validity and satisfactory finite-sample performance via extensive simulations. The proposed method is applied to a bone marrow transplant dataset to evaluate the causal effect of rituximab in diffuse large B-cell lymphoma patients.

Exploring the determinants of real estate liquidity from an alternative perspective: censored quantile regression in real estate research

In this paper, the liquidity (inverse of time on market) of rental dwellings and its determinants for different liquidity quantiles are examined for the seven largest German cities. The determinants are estimated using censored quantile regressions in order to investigate the impact on very liquid to very illiquid dwellings. As market heterogeneity is not only observed between cities but also within a city, each of the seven cities is considered individually. Micro data for almost 500,000 observations from 2013 to 2017 is used to examine the time on market. Substantial differences in the magnitude and direction of the regression coefficients for the different liquidity quantiles are found. Furthermore, both the magnitude and direction of the impact of an explanatory variable on the liquidity, differ between the cities. To the best of the authors’ knowledge this is the first paper, to apply censored quantile regressions to liquidity analysis of the real estate rental market. The model reveals that the proportionality assumption underlying the Cox proportional hazards model cannot be confirmed for all variables across all cities, but for most of them.

P1519IL6 trans-signalling affects risk of cardiovascular events pre-eminently in men

Abstract Background Interleukin 6 (IL6) is a known cardiovascular risk marker. The pro-atherogenic effects of IL6 are mediated by the IL6 trans-signalling pathway via a binary complex of IL6 and the soluble IL6 receptor (IL6:sIL6R). The binary complex is however neutralised by the natural inhibitor, sgp130 when forming the ternary complex (IL6:sIL6R:sgp130). To assess the risk of cardiovascular events (CVE) with IL6 trans-signalling, a ratio between the active binary complex and the neutralised ternary complex was calculated. We recently demonstrated that high levels of the binary/ternary complex ratio (b/t ratio > the median) representing an excess of the active binary complex, was independently associated with a 44% increased risk of future CVE in subjects free of prevalent cardiovascular disease. Purpose In this study we aimed to analyse the risk of CVE and time to event associated with the b/t ratio in men and women separately. Methods In a cohort of 60 year old men and women from Stockholm, the molar concentrations of the binary and ternary IL6 complex were estimated at baseline. Subjects free of prevalent cardiovascular disease were followed through national registers to assess future CVE (myocardial infarction, hospitalised angina pectoris and ischemic stroke). During a 16 year follow-up, 525 first time CVEs were registered. The risk for CVE and time to CVE was calculated for men and women, separately. To evaluate the risk associated with IL6 trans-signalling, the b/t ratio dichotomised at the median was modelled in a Cox regression model and risk was expressed as hazard ratios (HR) with 95% confidence intervals (CI). In addition, analysis was performed using censored quantile regression that allows measuring the effect of covariates on different quantiles of the time to CVE. Results Approximately half of the population were men and 64% of the CVE occurred in men. The risk of CVE during follow-up was significantly higher in men with b/t ratio > median (HR 1.70; 95% CI 1.35–2.15), while no significant difference was found in women (HR 1.12; 95% CI 0.84–1.50). Consistently, quantile regression showed that, men with a b/t ratio > median suffered their CVEs at an earlier time point. The time at which 15% of the male population was observed to have experienced CVE was 5.6 years shorter (95% CI: 4.0–7.2) in the high b/t ratio group. In women there was no significant difference in time to CVE. Conclusion The risk of CVE and early events associated with IL6 trans-signalling estimated by a b/t ratio >median is significantly increased in men. Acknowledgement/Funding The Stockholm County Council ALF project, Strategic research in Epidemiology

An Adapted Loss Function for Censored Quantile Regression

ABSTRACTIn this article, we study a novel approach for the estimation of quantiles when facing potential right censoring of the responses. Contrary to the existing literature on the subject, the adopted strategy of this article is to tackle censoring at the very level of the loss function usually employed for the computation of quantiles, the so-called “check” function. For interpretation purposes, a simple comparison with the latter reveals how censoring is accounted for in the newly proposed loss function. Subsequently, when considering the inclusion of covariates for conditional quantile estimation, by defining a new general loss function the proposed methodology opens the gate to numerous parametric, semiparametric, and nonparametric modeling techniques. To illustrate this statement, we consider the well-studied linear regression under the usual assumption of conditional independence between the true response and the censoring variable. For practical minimization of the studied loss function, we also provide a simple algorithmic procedure shown to yield satisfactory results for the proposed estimator with respect to the existing literature in an extensive simulation study. From a more theoretical prospect, consistency and asymptotic normality of the estimator for linear regression are obtained using several recent results on nonsmooth semiparametric estimation equations with an infinite-dimensional nuisance parameter, while numerical examples illustrate the adequateness of a simple bootstrap procedure for inferential purposes. Lastly, an application to a real dataset is used to further illustrate the validity and finite sample performance of the proposed estimator. Supplementary materials for this article are available online.

Variable screening for ultrahigh dimensional censored quantile regression

ABSTRACTQuantile regression is a flexible approach to assessing covariate effects on failure time, which has attracted considerable interest in survival analysis. When the dimension of covariates is much larger than the sample size, feature screening and variable selection become extremely important and indispensable. In this article, we introduce a new feature screening method for ultrahigh dimensional censored quantile regression. The proposed method can work for a general class of survival models, allow for heterogeneity of data and enjoy desirable properties including the sure screening property and the ranking consistency property. Moreover, an iterative version of screening algorithm has also been proposed to accommodate more complex situations. Monte Carlo simulation studies are designed to evaluate the finite sample performance under different model settings. We also illustrate the proposed methods through an empirical analysis.

Determining Overall Survival and Risk Factors in Esophageal Cancer Using Censored Quantile Regression

Background:Esophageal cancer is one of the leading causes of death worldwide. The global increasing rate of this type of cancer requires more attention. The purpose of this study was to determine the overall survival probability of esophageal cancer after diagnosis and to assess the potential risk factors in a population of Iranian patients.Materials and Methods:This retrospective cohort study was conducted on 127 cases with esophageal cancer in the Azarbaijan province, East of Iran. Participants in the study were diagnosed during 2009-2010 and were followed up for 5 years. The event was considered death due to esophageal cancer and those who survived until the end of the study were assumed as right censored. Censored quntile regression was fitted to find the overall survival of the patients using adjusted effects of variables and was compared with Cox regression model.Results:Patients’ mean and median survival time were 16.99 and 10.06 months respectively and 89% off cases died by the end of the study. The 1, 3, 6, 12 and 36-month survival probabilities were 0.95, 0.76, 0.60, 0.43, and 0.18. The median survival time for females and males without surgery were 21.79 and 14.76 month respectively. The accuracy of predictions were 0.99 and 0.74 for the censored quantile regression and Cox, respectively.Conclusion:We concluded that being male, not having surgery, longer wait time between having symptoms and being diagnosed, low socioeconomic status and old age to be significant risk factors in reducing the probability of survival from esophageal cancer.

Efficiency Driver in Nigerian Airports: A Bootstrap DEA–Censored Quantile Regression Approach

This paper reports on the use of a novel two-stage approach for assessing the efficiency of 30 major Nigerian airports from 2003 to 2013 based on bootstrapped data envelopment analysis (DEA) and censored quantile regression. In the first stage, bootstrapped efficiency estimates are computed. They enable bias correction and testing for significant differences in efficiency levels among airports. Subsequently, bootstrapped DEA results are combined with censored quantile regression to assess the impact of contextual variables—related to the airports’ ownership, location, and network connectivity—on different efficiency percentiles. The results reveal that the intensity of significant impacts regarding airports’ contextual variables may vary between high-/low-efficiency airports. Policy implications are derived accordingly.

HIGH DIMENSIONAL CENSORED QUANTILE REGRESSION.

Censored quantile regression (CQR) has emerged as a useful regression tool for survival analysis. Some commonly used CQR methods can be characterized by stochastic integral-based estimating equations in a sequential manner across quantile levels. In this paper, we analyze CQR in a high dimensional setting where the regression functions over a continuum of quantile levels are of interest. We propose a two-step penalization procedure, which accommodates stochastic integral based estimating equations and address the challenges due to the recursive nature of the procedure. We establish the uniform convergence rates for the proposed estimators, and investigate the properties on weak convergence and variable selection. We conduct numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposals.

A multiple imputation method based on weighted quantile regression models for longitudinal censored biomarker data with missing values at early visits

BackgroundIn patient-based studies, biomarker data are often subject to left censoring due to the detection limits, or to incomplete sample or data collection. In the context of longitudinal regression analysis, inappropriate handling of these issues could lead to biased parameter estimates. We developed a specific multiple imputation (MI) strategy based on weighted censored quantile regression (CQR) that not only accounts for censoring, but also missing data at early visits when longitudinal biomarker data are modeled as a covariate.MethodsWe assessed through simulation studies the performances of developed imputation approach by considering various scenarios of covariance structures of longitudinal data and levels of censoring. We also illustrated the application of the proposed method to the Prospective Study of Outcomes in Ankylosing spondylitis (AS) (PSOAS) data to address the issues of censored or missing C-reactive protein (CRP) level at early visits for a group of patients.ResultsOur findings from simulation studies indicated that the proposed method performs better than other MI methods by having a higher relative efficiency. We also found that our approach is not sensitive to the choice of covariance structure as compared to other methods that assume normality of biomarker data. The analysis results of PSOAS data from the imputed CRP levels based on our method suggested that higher CRP is significantly associated with radiographic damage, while those from other methods did not result in a significant association.ConclusionThe MI based on weighted CQR offers a more valid statistical approach to evaluate a biomarker of disease in the presence of both issues with censoring and missing data in early visits.

Heterogeneity in demand for performances and seats in the theatre

Studying the heterogeneity of consumers allows to price the product differently for consumer segments or groups of a product. In this paper, we estimate a model of aggregate demand for Perm Opera and Ballet Theatre focusing on the heterogeneity in price effect on demand for tickets on different performances and seats. We estimate the parameters of demand function using censored quantile regression that accounts for the limited capacity of the theatre house. We reveal the price effect variation across different types of theatrical productions and seats with lower elastic demand on ballets and for seats of higher quality.

A Stochastic EM Algorithm for Quantile and Censored Quantile Regression Models

We proposed a stochastic EM algorithm for quantile and censored quantile regression models in order to circumvent some limitations of the EM algorithm and Gibbs sampler. We conducted several simulation studies to illustrate the performance of the algorithm and found that the procedure performs as better as the Gibbs sampler, and outperforms the EM algorithm in uncensored situation. Finally we applied the methodology to the classical Engel food expenditure data and the labour supply data with left censoring, finding that the SEM algorithm behaves more satisfying than the Gibbs sampler does.