Quantile estimation of time-to-event data plays a key role in many medical applications, especially conditional on covariates of interest. In such settings, bias due to model misspecification is an important concern. As such, Empirical Likelihood (EL) is a particularly attractive estimation approach, making minimal parametric modeling assumptions without unduly compromising statistical efficiency. However, observed survival times are typically subject to right-censoring, in which case most EL approaches cannot be applied directly. In this thesis, we revisit a widely-applicable Expectation-Maximization (EM) algorithm for right-censored EL. As the covariate-free EL function becomes discontinuous in the conditional setting, we propose a continuity correction for which the computational properties of EM are retained. Several approaches to obtaining confidence intervals are explored. We provide an implementation of our method and related algorithms in the R package flexEL. The source code is written in C++ for high computational performance, and a straightforward interface allows users to fit arbitrary EL models with little programming effort.

Empirical Likelihood Method in Survival Analysis explains how to use the empirical likelihood method for right censored survival data. The author uses R for calculating empirical likelihood and includes many worked out examples with the associated R code. The datasets and code are available for download on his website and CRAN. The book focuses on all the standard survival analysis topics treated with empirical likelihood, including hazard functions, cumulative distribution functions, analysis of the Cox model, and computation of empirical likelihood for censored data. It also covers semi-parametric accelerated failure time models, the optimality of confidence regions derived from empirical likelihood or plug-in empirical likelihood ratio tests, and several empirical likelihood confidence band results. While survival analysis is a classic area of statistical study, the empirical likelihood methodology has only recently been developed. Until now, just one book was available on empirical likelihood and most statistical software did not include empirical likelihood procedures. Addressing this shortfall, this book provides the functions to calculate the empirical likelihood ratio in survival analysis as well as functions related to the empirical likelihood analysis of the Cox regression model and other hazard regression models.

In Survival analysis, it is vital to understand the effect of the covariates on the survival time. Commonly studied models are the Cox [1972] proportional hazards model and the accelerated failure time model. These methods mainly focus on one characteristic of the survival time. In reality, the association between the response and risk factors is not homogeneous always. This leads to the use of quantile regression [Koenker and Basset, 1978] models, which provide a global description of the association. In quantile regression modeling of the survival data, the problem of estimating the regression coefficients for extreme quantiles can be affected by severe censoring [Portnoy, 2003], especially when the sample size is small. In epidemiological studies, however, there are often times when only a subset of the whole study cohort is accurately observed. The rest of the cohort has only some auxiliary covariate available. The naive use of the auxiliary covariate in the model without the accurately measured covariate could lead to biased estimates. To deal with this problem in censored quantile regression, we propose a regression calibration based method when there is a linear relationship between the auxiliary covariate and the accurately measured covariate. When the relationship is non-linear, we propose a non-parametric kernel smoothing technique. We also propose an empirical likelihood [Owen, 1998, 2001] based weighted censored quantile regression to improve the efficiency of the censored quantile regression estimation by utilizing the auxiliary information about the target population parameters available through scientific facts/previous studies. The proposed estimators are consistent and have asymptotically Gaussian distributions. The efficiency gain compared to the existing methods is remarkable. These methods provide the possibilities of looking into extreme quantiles of the survival distribution. We also applied our proposed methods in real case examples.

Empirical likelihood provides inferences whose validity does not depend on specifying a parametric model for the data. Because it uses a likelihood, the method has certain inherent advantages over resampling methods: it uses the data to determine the shape of the confidence regions, and it makes it easy to combined data from multiple sources. It al

Quantile regression constitutes an ensemble of statistical techniques intended to estimate and draw inferences about conditional quantile functions. Median regression, as introduced in the 18th century by Boscovich and Laplace, is a special case. In contrast to conventional mean regression that minimizes sums of squared residuals, median regression minimizes sums of absolute residuals; quantile regression simply replaces symmetric absolute loss by asymmetric linear loss. Since its introduction in the 1970's by Koenker and Bassett, quantile regression has been gradually extended to a wide variety of data analytic settings including time series, survival analysis, and longitudinal data. By focusing attention on local slices of the conditional distribution of response variables it is capable of providing a more complete, more nuanced view of heterogeneous covariate effects. Applications of quantile regression can now be found throughout the sciences, including astrophysics, chemistry, ecology, economics, finance, genomics, medicine, and meteorology. Software for quantile regression is now widely available in all the major statistical computing environments. The objective of this volume is to provide a comprehensive review of recent developments of quantile regression methodology illustrating its applicability in a wide range of scientific settings. The intended audience of the volume is researchers and graduate students across a diverse set of disciplines.

A guide to the implementation and interpretation of Quantile Regression models This book explores the theory and numerous applications of quantile regression, offering empirical data analysis as well as the software tools to implement the methods. The main focus of this book is to provide the reader with a comprehensive description of the main issues concerning quantile regression; these include basic modeling, geometrical interpretation, estimation and inference for quantile regression, as well as issues on validity of the model, diagnostic tools. Each methodological aspect is explored and followed by applications using real data. Quantile Regression: Presents a complete treatment of quantile regression methods, including, estimation, inference issues and application of methods. Delivers a balance between methodolgy and application Offers an overview of the recent developments in the quantile regression framework and why to use quantile regression in a variety of areas such as economics, finance and computing. Features a supporting website (www.wiley.com/go/quantile_regression) hosting datasets along with R, Stata and SAS software code. Researchers and PhD students in the field of statistics, economics, econometrics, social and environmental science and chemistry will benefit from this book.