Quantile calculus and censored regression

Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan--Meier method in the $k$-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.Comment: Published in at http://dx.doi.org/10.1214/09-AOS771 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Corrected Score Approach for Proportional Hazards Model with Covariate Measurement Error

In the presence of covariate measurement error with the proportional hazards model, several functional modeling methods have been proposed. These include the conditional score estimator (Tsiatis and Davidian, 2001), the parametric correction estimator (Nakamura, 1992) and the nonparametric correction estimator (Huang and Wang, 2000, 2003) in the order of weaker assumptions on the error. Although they are all consistent, each suffers from potential difficulties with small samples and substantial measurement error. In this article, upon noting that the conditional score and parametric correction estimators are asymptotically equivalent in the case of normal error, we investigate their relative finite sample performance and discover that the former is superior, which may be explained by the unbiasedness of its estimating equation. This finding motivates a general refinement approach to parametric and nonparametric correction methods. The refined correction estimators are asymptotically equivalent to their standard counterparts, but have improved numerical properties. Simulation results and application to an HIV clinical trial are presented

A Corrected Pseudo-score Approach for Additive Hazards Model With Longitudinal Covariates Measured With Error

In medical studies, it is often of interest to characterize the relationship between a time-to-event and covariates, not only time-independent but also time-dependent. Time-dependent covariates are generally measured intermittently and with error. Recent interests focus on the proportional hazards framework, with longitudinal data jointly modeled through a mixed effects model. However, approaches under this framework depend on the normality assumption of the error, and might encounter intractable numerical difficulties in practice. This motivates us to consider an alternative framework, that is, the additive hazards model, under which little has been done when time-dependent covariates are measured with error. We propose a simple corrected pseudo-score approach for the regression parameters with no assumptions on the distribution of the random effects and the error beyond those for the variance structure of the latter. The estimator has an explicit form and is shown to be consistent and asymptotically normal. We illustrate the method via simulations and by application to data from an HIV clinical trial

Marginal Regression of Gaps Between Recurrent Events

Recurrent event data typically exhibit the phenomenon of intra-individual correlation, owing to not only observed covariates but also random effects. In many applications, the population can be reasonably postulated as a heterogeneous mixture of individual renewal processes, and the inference of interest is the effect of individual-level covariates. In this article, we suggest and investigate a marginal proportional hazards model for gaps between recurrent events. A connection is established between observed gap times and clustered survival data, however, with informative cluster size. We then derive a novel and general inference procedure for the latter, based on a functional formulation of standard Cox regression. Large-sample theory is established for the proposed estimators of the regression coefficients and the baseline cumulative hazard function. Numerical studies demonstrate that the procedure performs well under practical sample sizes. Application to the well-known bladder tumor data is given as illustratio

Regression Analysis of Recurrent Gap Times with Time-Dependent Covariates

Individual subjects may experience recurrent events of same type over a relatively long period of time in a longitudinal study. Researchers are often interested in the distributional pattern of gaps between the successive recurrent events and their association with certain concomitant covariates as well. In this article, their probability structure is investigated in presence of censoring. According to the identified structure, we introduce the proportional reverse-time hazards models that allow arbitrary baseline function for every individual in the study, when the time-dependent covariates effect is of main interest. Appropriate inference procedures are proposed and studied to estimate the parameters of interest in the models. The proposed methodology is demonstrated with the Monte-Carlo simulations and applied to a well-known Denmark schizophrenia cohort study data set

Semiparametric Regression Analysis on Longitudinal Pattern of Recurrent Gap Times

In longitudinal studies, individual subjects may experience recurrent events of the same type over a relatively long period of time. The longitudinal pattern of the gaps between the successive recurrent events is often of great research interest. In this article, the probability structure of the recurrent gap times is first explored in the presence of censoring. According to the discovered structure, we introduce the proportional reverse-time hazards models with unspecified baseline functions to accommodate heterogeneous individual underlying distributions, when the ongitudinal pattern parameter is of main interest. Inference procedures are proposed and studied by way of proper riskset construction. The proposed methodology is demonstrated by Monte-Carlo simulations and an application to the well-known Denmark schizophrenia cohort study data se

Covariate Adjustment in Continuous Biomarker Assessment

Continuous biomarkers are common for disease screening and diagnosis. To reach a dichotomous clinical decision, a threshold would be imposed to distinguish subjects with disease from nondiseased individuals. Among various performance metrics, specificity at a controlled sensitivity level (or vice versa) is often desirable because it directly targets the clinical utility of the intended clinical test. Meanwhile, covariates, such as age, race, as well as sample collection conditions, could impact the biomarker distribution and may also confound the association between biomarker and disease status. Therefore, covariate adjustment is important in such biomarker evaluation. Most existing covariate adjustment methods do not specifically target the desired sensitivity/specificity level, but rather do so for the entire biomarker distribution. As such, they might be more prone to model misspecification. In this paper, we suggest a parsimonious quantile regression model for the diseased population, only locally at the controlled sensitivity level, and assess specificity with covariate-specific control of the sensitivity. Variance estimates are obtained from a sample-based approach and bootstrap. Furthermore, our proposed local model extends readily to a global one for covariate adjustment for the receiver operating characteristic (ROC) curve over the sensitivity continuum. We demonstrate computational efficiency of this proposed method and restore the inherent monotonicity in the estimated covariate-adjusted ROC curve. The asymptotic properties of the proposed estimators are established. Simulation studies show favorable performance of the proposal. Finally, we illustrate our method in biomarker evaluation for aggressive prostate cancer

Reliability Analysis Test-Based Interval Estimation Under the Accelerated Failure Time Model

The accelerated failure time (AFT) model is an important regression tool t