Censored Quantile Regression Research Articles

A censored quantile regression approach for the analysis of time to event data.

The commonly used statistical model for studying time to event data, the Cox proportional hazards model, is limited by the assumption of a constant hazard ratio over time (i.e., proportionality), and the fact that it models the hazard rate rather than the survival time directly. The censored quantile regression model, defined on the quantiles of time to event, provides an alternative that is more flexible and interpretable. However, the censored quantile regression model has not been widely adopted in clinical research, due to the complexity involved in interpreting its results properly and consequently the difficulty to appreciate its advantages over the Cox proportional hazards model, as well as the absence of adequate validation procedure. In this paper, we addressed these limitations by (1) using both simulated examples and data from National Wilms' Tumor clinical trials to illustrate proper interpretation of the censored quantile regression model and the differences and the advantages of the model compared to the Cox proportional hazards model; and (2) developing a validation procedure for the predictive censored quantile regression model. The performance of this procedure was examined using simulation studies. Overall, we recommend the use of censored quantile regression model, which permits a more sensitive analysis of time to event data together with the Cox proportional hazards model.

Quantile Autoregression for Censored Data

Quantile autoregression (QAR) is particularly attractive for censored data. However, unlike the standard regression models, the autoregressive models must take account of censoring on both response and regressors. In this article, we show that the existing censored quantile regression methods produce consistent estimators for QAR models when using only the fully observed regressors. A new algorithm is proposed to provide a censored QAR estimator by adopting imputation methods. The algorithm redistributes probability mass of censored points appropriately and iterates towards self‐consistent solutions. Monte Carlo simulations and empirical applications are conducted to demonstrate merits of the proposed method.

Migration and Regional Trade Agreements: A (New) Gravity Estimation

AbstractThis paper investigates the role of regional trade agreements (RTAs) on bilateral international migration. Building on a gravity model for migration, our econometric strategy controls for the multilateral resistance to migration and solves the zero migration flows problem by using a censored quantile regression approach. Further, the endogeneity problem of RTAs in migration settlement is addressed by using instrumental variable censored quantile regression. Our results suggest that the presence of a RTA stimulates the migration stocks among member countries. The pro‐migration effect of RTAs is magnified if the agreement includes also provisions easing bureaucratic procedures for visa and asylum among member countries. Finally, we find an asymmetric effect of RTAs across the quantiles of the distribution of migration settlements.

Smoothed and Corrected Score Approach to Censored Quantile Regression With Measurement Errors

Censored quantile regression is an important alternative to the Cox proportional hazards model in survival analysis. In contrast to the usual central covariate effects, quantile regression can effectively characterize the covariate effects at different quantiles of the survival time. When covariates are measured with errors, it is known that naively treating mismeasured covariates as error-free would result in estimation bias. Under censored quantile regression, we propose smoothed and corrected estimating equations to obtain consistent estimators. We establish consistency and asymptotic normality for the proposed estimators of quantile regression coefficients. Compared with the naive estimator, the proposed method can eliminate the estimation bias under various measurement error distributions and model error distributions. We conduct simulation studies to examine the finite-sample properties of the new method and apply it to a lung cancer study. Supplementary materials for this article are available online.

Outcomes from second-line therapy in long-term responders to first-line tyrosine kinase inhibitor in clear-cell metastatic renal cell carcinoma

Although sequential targeted therapy is standard in patients with metastatic clear-cell renal cell carcinoma (m-ccRCC), the choice of drugs and optimal administration sequence have yet to be established. The objective of this study was to explore whether it is preferable to rechallenge a long-term responder to a first-line tyrosine kinase inhibitor (TKI) with a TKI or whether to switch to a mammalian target of rapamycin inhibitor (mTORi); to determine whether second-line treatment response depends on duration of first-line response (TD1). Retrospective multicenter study (2004-2011) of 241 consecutive mRCC patients (clear-cell histology) who received a first-line TKI for ≥6 months followed by a second-line TKI (n = 118) or mTORi (n = 123). Progression-free survival (PFS) and time-to-treatment failure (TTF) on second-line therapy. Multivariable full-model: second-line drug, TD1, ECOG-PS before first- and second-line, best objective response (first-line), Fuhrman grade, number of metastatic sites, and presence of bone metastases. Adjustment covariable: International mRCC Database Consortium (IMDC) risk score. Multiple propensity score and missing data methods were used. Any correlation between first-line and second-line PFS was investigated using censored quantile regression models (CQRM). Sequence effect in the overall cohort was in favor of the TKI-TKI sequence over the TKI-mTORi sequence on using TD1 as continuous covariable (HR ≈ 0.75 for PFS and TTF). TKI-TKI superiority was attributed in large part to the 11-22 month (TD1) subgroup of patients which displayed significantly better outcomes [HR ≈ 0.5; median PFS (months): 9.4 (5.9-12.2) versus 3.9 (3.0-5.5), P = 0.003; TTF(months): 8.0 (5.5-11.0) versus 3.6 (3.0-4.6), P = 0.009]. Upon full CQRM, long-term second-line responders were more likely to have received a second TKI than an mTORi and to have been long-term responders to first-line TKI. m-ccRCC patients who remained on first-line TKI between 11 and 22 months benefited from a TKI rechallenge rather than from second-line mTORi.

Quantile regression with censoring and endogeneity

In this paper we develop a new censored quantile instrumental variable (CQIV) estimator and describe its properties and computation. The CQIV estimator combines Powell (1986) censored quantile regression (CQR) to deal with censoring, with a control variable approach to incorporate endogenous regressors. The CQIV estimator is obtained in two stages that are nonadditive in the unobservables. The first stage estimates a nonadditive model with infinite dimensional parameters for the control variable, such as a quantile or distribution regression model. The second stage estimates a nonadditive censored quantile regression model for the response variable of interest, including the estimated control variable to deal with endogeneity. For computation, we extend the algorithm for CQR developed by Chernozhukov and Hong (2002) to incorporate the estimation of the control variable. We give generic regularity conditions for asymptotic normality of the CQIV estimator and for the validity of resampling methods to approximate its asymptotic distribution. We verify these conditions for quantile and distribution regression estimation of the control variable. Our analysis covers two-stage (uncensored) quantile regression with nonadditive first stage as an important special case. We illustrate the computation and applicability of the CQIV estimator with a Monte-Carlo numerical example and an empirical application on estimation of Engel curves for alcohol.

Estimation of Censored Quantile Regression for Panel Data With Fixed Effects

This article investigates estimation of censored quantile regression (QR) models with fixed effects. Standard available methods are not appropriate for estimation of a censored QR model with a large number of parameters or with covariates correlated with unobserved individual heterogeneity. Motivated by these limitations, the article proposes estimators that are obtained by applying fixed effects QR to subsets of observations selected either parametrically or nonparametrically. We derive the limiting distribution of the new estimators under joint limits, and conduct Monte Carlo simulations to assess their small sample performance. An empirical application of the method to study the impact of the 1964 Civil Rights Act on the black–white earnings gap is considered. Supplementary materials for this article are available online.

Adjusting VAT rates to promote healthier diets in Norway: A censored quantile regression approach

The effects of health-related taxes and subsidies on food and beverages have mainly been investigated using models that assume identical price responses among high- and low-consuming households. Diet-related health problems are, however, more likely among households with high intakes of unhealthy foods or low intakes of healthy foods than in households with average intakes. In this article, we focus on purchases of healthy and unhealthy foods among low-, median-, and high-purchasing households. The effects of an increase in the Norwegian value-added tax (VAT) on some unhealthy foods and a removal of the VAT on some healthy foods are investigated. Using censored quantile regressions, we reject equality of the own-price elasticities for eight of nine food and beverage groups. We find that a VAT increase is more effective in reducing purchases of unhealthy foods among high-purchasing households than a VAT removal is in increasing the purchases of healthy foods among low-purchasing households.

Quantile Regression for Recurrent Gap Time Data

Evaluating covariate effects on gap times between successive recurrent events is of interest in many medical and public health studies. While most existing methods for recurrent gap time analysis focus on modeling the hazard function of gap times, a direct interpretation of the covariate effects on the gap times is not available through these methods. In this article, we consider quantile regression that can provide direct assessment of covariate effects on the quantiles of the gap time distribution. Following the spirit of the weighted risk-set method by Luo and Huang (2011, Statistics in Medicine 30, 301-311), we extend the martingale-based estimating equation method considered by Peng and Huang (2008, Journal of the American Statistical Association 103, 637-649) for univariate survival data to analyze recurrent gap time data. The proposed estimation procedure can be easily implemented in existing software for univariate censored quantile regression. Uniform consistency and weak convergence of the proposed estimators are established. Monte Carlo studies demonstrate the effectiveness of the proposed method. An application to data from the Danish Psychiatric Central Register is presented to illustrate the methods developed in this article.

Tax Incentives and Charitable Contributions: the Evidence from Censored Quantile Regression

AbstractThis paper investigates the relationship between tax price and charitable contributions using the censored quantile regression (QR) technique, which can provide a complete description of the whole distribution of giving, and data from US Internal Revenue Service individual tax returns. The findings of the present study are as follows. First, the price elasticities of charitable contributions are all negative and consistently decrease in absolute value along the quantiles. Second, donors at lower giving quantiles are price elastic but donors at higher quantiles are price inelastic. Third, the income elasticities are positive and increase along the quantiles. Lastly, the effects of wealth, age, marital status, and the number of dependents vary across quantiles. All the empirical results show that the censored QR offers better explanations on the relationship between tax incentives and charitable contributions.

A two-state analysis of estate taxes and charitable bequests from the most generous decedents

This study uses Censored Quantile Regression (CQR) and probate records for the years 2000–2005 to analyze charitable bequests of the most generous decedents from two states, Virginia and Louisiana. We focus on the most generous decedents because they account for a large portion of total charitable bequests. The most generous decedents' charitable bequest tax price (i.e., one minus the marginal tax rate) elasticity is not significantly greater than one in both our full sample and the subsample of the decedents filing federal estate tax returns. Our finding suggests that allowing charitable bequests to be deductible is not a treasury efficient policy for the most generous decedents. We also find that the tax price and wealth effects for the most generous decedents are overestimated by the Tobit model that has been traditionally utilized by prior studies. Furthermore, filers' charitable bequests are predicted to decrease if the estate tax is repealed.

A censored quantile regression analysis of employee stock options substitution for debt and the impact of SFAS 123R

PurposeThis paper aims to examine whether the substitution of employee stock options (ESOs) for debt occurs for firms with different tax status classifications throughout the conditional distribution of interest expense before and after the implementation of Statement of Financial Accounting Standard 123R (SFAS 123R).Design/methodology/approachThis study uses Censored Quantile Regression (CQR) to assess whether the substitution effect is dependent on firms' position in the conditional distribution of interest expense. Our sample firms are categorized into two groups: one group (tax‐sensitive) that is sensitive to additional deductions due to a moderate income level and the other group (tax‐insatiable) that is not sensitive because of very high income level.FindingsThe substitution effect is not present for firms with below medium level of interest expense. Only tax‐sensitive firms substitute at medium levels of interest expense while both tax‐sensitive and tax‐insatiable firms substitute at high levels of interest expense. Tax‐insatiable firms with very high levels of interest expense also substitute; however, tax‐sensitive firms with very high levels of interest expense only substitute after SFAS 123R required firms to report ESO expense in financial statements. We attribute the substitution patterns revealed by the CQR analysis to a positive relationship between interest expense and cost of debt.Originality/valueTo the authors' knowledge, this is the first paper to analyze firms' tax status classification impact on the substitution of ESO expense for interest expense across different levels of interest expense. Our application of CQR should benefit researchers who are interested in examining explanatory variables' impact at various points in the conditional distribution of the dependent variable. This study also refines the conjecture that ESOs are substitutes for debt by demonstrating that such relationship is dependent on the level of interest expense and tax status. Furthermore, the finding of firms substituting ESOs for debt provides accounting standard setters a reason to begin requiring firms to re‐measure the value of ESOs after the grant date until the exercise date.

R&D Strategy of Small and Medium Enterprises in India

Liberalisation of economic policies in the last two decades and intensifying market competition tend to be a cause of policy concern for the survival of SMEs in emerging economies like India. These SMEs account for the largest chunk of industrial units and employment in the national economy. Yet, most of them are competing with deeply inadequate resources, especially by means of weak technological capabilities. The present study has provided not only preliminary estimates on SME R&D investments in Indian manufacturing and their broad trends and patterns, but also contributed to the understanding of factors driving the SME in-house R&D activities. It shows that Indian SMEs continue to be vulnerable among all firmas they have the lowest incidence of doing in-house R&D and their R&D intensitieshave fallen in the last decade. Based on the results from three-step Censored Quantile Regression, this study has suggested a set of useful policy implications for enhancing SME R&D.

Charitable giving in the German welfare state: fiscal incentives and crowding out

There are two ways that government activities influence private charitable giving: (1) government spending on the provision of public goods may cause crowding out of private charitable contributions; and (2) tax incentives may boost private charitable giving. From a sample of German income tax returns, we estimate the elasticity of charitable giving relative to tax incentives, income, and government spending. Using censored quantile regression analysis, we derive results for different points of the underlying distribution of charitable giving. Evaluating overall treasury efficiency, the tax deductibility of charitable donations fosters enough private giving to offset foregone tax revenues.

Heterogeneity in the intergenerational transmission of alcohol consumption: A quantile regression approach

This paper addresses the question of whether the effect of parental drinking on children's later consumption of alcohol – which is frequently found to be of positive sign – exhibits a certain pattern of heterogeneity. In particular, if this effect is more prominent in the upper tail than elsewhere in the distribution of children's alcohol consumption, conventional regression analyses that focus on the mean effect may substantially underrate parental drinking as a risk factor for children's later alcohol abuse. In our empirical application, we address this issue by applying censored quantile regression methods to German survey data. The supposed pattern of heterogeneity is indeed found in the data, at least for daily parental drinking. In addition, the intergenerational transmission of alcohol consumption exhibits gender-specific heterogeneity.

Heterogeneity in the Intergenerational Transmission of Alcohol Consumption – A Quantile Regression Approach

This paper addresses the question of whether the effect of parental drinking on children’s later consumption of alcohol – which is frequently found to be of positive sign – exhibits a certain pattern of heterogeneity. In particular, if this effect is more prominent in the upper tail than elsewhere in the distribution of children’s alcohol consumption, conventional regression analyses that focus on the mean effect may substantially underrate parental drinking as a risk factor for children’s later alcohol abuse. In our empirical application, we address this issue by applying censored quantile regression methods to German survey data. The supposed pattern of heterogeneity is indeed found in the data, at least for daily parental drinking. In addition, the intergenerational transmission of alcohol consumption exhibits gender-specific heterogeneity.

Locally Weighted Censored Quantile Regression

Censored quantile regression offers a valuable supplement to Cox proportional hazards model for survival analysis. Existing work in the literature often requires stringent assumptions, such as unconditional independence of the survival time and the censoring variable or global linearity at all quantile levels. Moreover, some of the work uses recursive algorithms, making it challenging to derive asymptotic normality. To overcome these drawbacks, we propose a new locally weighted censored quantile regression approach that adopts the redistribution-of-mass idea and employs a local reweighting scheme. Its validity only requires conditional independence of the survival time and the censoring variable given the covariates, and linearity at the particular quantile level of interest. Our method leads to a simple algorithm that can be conveniently implemented with R software. Applying recent theory of M-estimation with infinite dimensional parameters, we establish the consistency and asymptotic normality of the proposed estimator. The proposed method is studied via simulations and is illustrated with the analysis of an acute myocardial infarction dataset.

Censored quantile regression and purchases of ice cream

Using Norwegian household data, we estimate the joint effects of increasing the value added tax (VAT) for less healthy food and removing the VAT for healthy food on purchases of ice cream and thereby weight outcomes. We estimate these effects with a censored quantile regression (CQR) model because about half the sample does not purchase ice cream and also because the probability of obesity is greater for heavy ice cream eaters. The CQR model allows for differing responses for high and low-purchasing households. We use an algorithm for the CQR which is simple, robust to a high degree of censoring, and performs well near the censoring point. In response to the VAT changes, we find that high-purchasing households would reduce their annual per capita purchases of ice cream by 1.8 kilograms, which corresponds ceteris paribus to an annual weight reduction of more than half a kilogram of body weight.

Statistical Downscaling of Extreme Precipitation Events Using Censored Quantile Regression

Abstract A statistical downscaling approach for extremes using censored quantile regression is presented. Conditional quantiles of station data (e.g., daily precipitation sums) in Germany are estimated by means of the large-scale circulation as represented by the NCEP reanalysis data. It is shown that a mixed discrete–continuous response variable, such as a daily precipitation sum, can be statistically modeled by a censored variable. Furthermore, a conditional quantile skill score is formulated to assess the relative gain of a quantile forecast compared with a reference forecast. Just like multiple regression for expectation values, quantile regression provides a tool to formulate a model output statistics system for extremal quantiles.

A Censored Quantile Regression Analysis of Vegetable Demand: The Effects of Changes in Prices and Total Expenditure

Many diseases are linked to dietary behavior. One major diet‐related risk factor is a low consumption of vegetables. The consumption may be increased through public policies. The effects on vegetable purchases of either removing the value added tax on vegetables or a general income support are investigated. Adverse health effects are most serious in households consuming low quantities of vegetables. Therefore, the effects on high‐ and low‐consuming households are estimated by using quantile regressions (QRs). Since many households did not purchase any vegetable during each survey period, censored as well as ordinary QRs are used. Our results suggest that the effects of the policy variables differ in different parts of the conditional distribution of vegetable purchases. None of the proposed policy options is likely to substantially increase vegetable purchases among low‐consuming households. Bon nombre de maladies découlent des habitudes alimentaires. La faible consommation de légumes constitue un important facteur de risque liéà l'alimentation. Cette consommation pourrait être accrue par l'instauration de politiques gouvernementales. Nous avons examiné les effets de l'abolition de la taxe sur la valeur ajoutée ou d'un soutien du revenu sur les achats de légumes. Les effets néfastes sur la santé sont plus graves chez les ménages qui consomment de faibles quantités de légumes. Nous avons donc estimé les effets chez les ménages à forte et à faible consommation de légumes à l'aide de régressions par quantile. Comme de nombreux ménages n'ont pas acheté de légumes au cours des périodes sondées, nous avons utilisé des régressions par quantile censurées et des régressions par quantile ordinaires. Nos résultats ont indiqué que les effets des variables concernant les politiques diffèrent dans différentes parties de la distribution conditionnelle des achats de légumes. Aucune des options politiques proposées ne semble susceptible d'accroître substantiellement les achats de légumes chez les ménages qui en consomment peu.