Certain Survival Models And Their Semiparametric Estimation Methods

Parametric and semiparametric models are tools with a wide range of applications to reliability, survival analysis, and quality of life. This self-contained volume examines these tools in survey articles written by experts currently working on the development and evaluation of models and methods. While a number of chapters deal with general theory, several explore more specific connections and recent results in "real-world" reliability theory, survival analysis, and related fields. Specific topics covered include: * cancer prognosis using survival forests * short-term health problems related to air pollution: analysis using semiparametric generalized additive models * semiparametric models in the studies of aging and longevity This book will be of use as a reference text for general statisticians, theoreticians, graduate students, reliability engineers, health researchers, and biostatisticians working in applied probability and statistics.

Survival analysis deals with the distribution of life times, essentially the times from an initiating event such as birth or the start of a job to some terminal event such as death or pension. This book, originally published in 1980, surveys and analyzes methods that use survival measurements and concepts, and helps readers apply the appropriate method for a given situation. Four broad sections cover introductions to data, univariate survival function, multiple-failure data, and advanced topics.

Handbook of Survival Analysis presents modern techniques and research problems in lifetime data analysis. This area of statistics deals with time-to-event data that is complicated by censoring and the dynamic nature of events occurring in time. With chapters written by leading researchers in the field, the handbook focuses on advances in survival analysis techniques, covering classical and Bayesian approaches. It gives a complete overview of the current status of survival analysis and should inspire further research in the field. Accessible to a wide range of readers, the book provides: An introduction to various areas in survival analysis for graduate students and novices A reference to modern investigations into survival analysis for more established researchers A text or supplement for a second or advanced course in survival analysis A useful guide to statistical methods for analyzing survival data experiments for practicing statisticians

Many conventional survival analysis methods, such as the Kaplan-Meier method for survival function estimation and the partial likelihood method for Cox model regression coefficients estimation, were developed under the assumption that survival times are subject to right censoring only. However, in practice, survival time observations may include interval-censored data, especially when the exact time of the event of interest cannot be observed. When interval-censored observations are present in a survival dataset, one generally needs to consider likelihood-based methods for inference. If the survival model under consideration is fully parametric, then likelihood-based methods impose neither theoretical nor computational challenges. However, if the model is semi-parametric, there will be difficulties in both theoretical and computational aspects. Likelihood Methods in Survival Analysis: With R Examples explores these challenges and provides practical solutions. It not only covers conventional Cox models where survival times are subject to interval censoring, but also extends to more complicated models, such as stratified Cox models, extended Cox models where time-varying covariates are present, mixture cure Cox models, and Cox models with dependent right censoring. The book also discusses non-Cox models, particularly the additive hazards model and parametric log-linear models for bivariate survival times where there is dependence among competing outcomes. Features Provides a broad and accessible overview of likelihood methods in survival analysis Covers a wide range of data types and models, from the semi-parametric Cox model with interval censoring through to parametric survival models for competing risks Includes many examples using real data to illustrate the methods Includes integrated R code for implementation of the methods Supplemented by a GitHub repository with datasets and R code The book will make an ideal reference for researchers and graduate students of biostatistics, statistics, and data science, whose interest in survival analysis extend beyond applications. It offers useful and solid training to those who wish to enhance their knowledge in the methodology and computational aspects of biostatistics.

ABSTRACT: First, we present two novel semiparametric survival models with log-linear median regression functions for right censored survival data. These models are useful alternatives to the popular Cox (1972) model and linear transformation models (Cheng et al., 1995). Compared to existing semiparametric models, our models have many important practical advantages, including interpretation of the regression parameters via the median and the ability to address heteroscedasticity. We demonstrate that our modeling techniques facilitate the ease of prior elicitation and computation for both parametric and semiparametric Bayesian analysis of survival data. We illustrate the advantages of our modeling, as well as model diagnostics, via reanalysis of a small-cell lung cancer study. Results of our simulation study provide further guidance regarding appropriate modelling in practice. Our second goal is to develop the methods of analysis and associated theoretical properties for interval censored and current status survival data. These new regression models use log-linear regression function for the median. We present frequentist and Bayesian procedures for estimation of the regression parameters. Our model is a useful and practical alternative to the popular semiparametric models which focus on modeling the hazard function. We illustrate the advantages and properties of our proposed methods via reanalyzing a breast cancer study. Our other aim is to develop a model which is able to account for the heteroscedasticity of response, together with robust parameter estimation and outlier detection using sparsity penalization. Some preliminary simulation studies have been conducted to compare the performance of proposed model and existing median lasso regression model. Considering the estimation bias, mean squared error and other identication benchmark measures, our proposed model performs better than the competing frequentist estimator.

Joint modeling of survival and longitudinal data has been studied extensively in recent literature. The likelihood approach is one of the most popular estimation methods employed within the joint modeling framework. Typically the parameters are estimated using maximum likelihood, with computation performed by the EM algorithm. However, one drawback of this approach is that standard error (SE) estimates are not automatically produced when using the EM algorithm. Many different procedures have been proposed to obtain the asymptotic covariance matrix for the parameters when the number of parameters is typically small. In the joint modeling context, however, there may be an infinite-dimensional parameter, the baseline hazard function, which greatly complicates the problem so that the existing methods cannot be readily applied. The profile likelihood (PL) and the bootstrap methods overcome the difficulty to some extent, however, they can be computationally intensive. In the first chapter, we propose two new methods for SE estimation using the EM algorithm that allow for more efficient computation of the SE of a subset of parametric components in a semiparametric or high-dimensional parametric model. The precision and computation time are evaluated through a thorough simulation study. We conclude with an application of our SE estimation method to analyze an HIV clinical trial dataset. In the second chapter, our interest is in doubly-censored data, which are subject to both left- and right-censoring and are not uncommon in lifetime data. The censoring mechanism is different from the case with right-censoring only and the situation is even more complicated when the data are clustered. In this chapter, we consider a class of linear transformation models, which includes the proportional hazards and the proportional odds models as special cases, to model this kind of data. While the transformation models expand the horizon of survival models, they pose considerable computational challenges to the likelihood approach, especially for doubly-censored clustered data. An effective EM algorithm is developed to overcome the computational difficulties and it leads to stable nonparametric maximum likelihood estimates (NPMLEs). The NPMLEs are shown to be consistent and the estimates of the finite-dimensional parameters are semiparametric efficient. A computationally efficient method proposed in the first chapter is extended and applied to estimate the standard errors (SE) of the parameter estimates. Simulation studies demonstrate that the proposed EM algorithm and SE estimates perform well, both are then applied to a dataset from a Hepatitis B clinical study.

Statistical models and methods for lifetime and other time-to-event data are widely used in many fields, including medicine, the environmental sciences, actuarial science, engineering, economics, management, and the social sciences. For example, closely related statistical methods have been applied to the study of the incubation period of diseases such as AIDS, the remission time of cancers, life tables, the time-to-failure of engineering systems, employment duration, and the length of marriages. This volume contains a selection of papers based on the 1994 International Research Conference on Lifetime Data Models in Reliability and Survival Analysis, held at Harvard University. The conference brought together a varied group of researchers and practitioners to advance and promote statistical science in the many fields that deal with lifetime and other time-to-event-data. The volume illustrates the depth and diversity of the field. A few of the authors have published their conference presentations in the new journal Lifetime Data Analysis (Kluwer Academic Publishers).

Survival analysis is a highly active area of research with applications spanning the physical, engineering, biological, and social sciences. In addition to statisticians and biostatisticians, researchers in this area include epidemiologists, reliability engineers, demographers and economists. The economists survival analysis by the name of duration analysis and the analysis of transition data. We attempted to bring together leading researchers, with a common interest in developing methodology in survival analysis, at the NATO Advanced Research Workshop. The research works collected in this volume are based on the presentations at the Workshop. Analysis of survival experiments is complicated by issues of censoring, where only partial observation of an individual's life length is available and left truncation, where individuals enter the study group if their life lengths exceed a given threshold time. Application of the theory of counting processes to survival analysis, as developed by the Scandinavian School, has allowed for substantial advances in the procedures for analyzing such experiments. The increased use of computer intensive solutions to inference problems in survival analysis~ in both the classical and Bayesian settings, is also evident throughout the volume. Several areas of research have received special attention in the volume.

We provide ways to test the fit of a parametric copula family for bivariate censored data with or without covariates. The proposed copula family is tested by embedding it in an expanded parametric family of copulas. When parameters in the proposed and the expanded copula models are estimated by maximum likelihood, a likelihood ratio test can be used. However, when they are estimated by two-stage pseudolikelihood estimation, the corresponding test is a pseudolikelihood ratio test. The two-stage procedures offer less computation, which is especially attractive when the marginal lifetime distributions are specified nonparametrically or semiparametrically. It is shown that the likelihood ratio test is consistent even when the expanded model is misspecified. Power comparisons of the likelihood ratio and the pseudolikelihood ratio tests with some other goodness-of-fit tests are performed both when the expanded family is correct and when it is misspecified. They indicate that model expansion provides a convenient, powerful and robust approach. We introduce a semiparametric maximum likelihood estimation method in which the copula parameter is estimated without assumptions on the marginal distributions. This method and the two-stage semiparametric estimation method suggested by Shih and Louis (1995) are generalized to regression models with Cox proportional hazards margins. The two-stage semiparametric estimator of the copula parameter is found to be about as good as the semiparametric maximum likelihood estimator. Semiparametric likelihood ratio and pseudolikelihood ratio tests are considered to provide goodness of fit tests for a copula model without making parametric assumptions for the marginal distributions. Both when the expanded family is correct and when it is misspecified, the semiparametric pseudolikelihood ratio test is almost as powerful as the parametric likelihood ratio and pseudolikelihood ratio tests while achieving robustness to the form of the marginal distributions. The methods are illustrated on applications in medicine and insurance. Sequentially observed survival times are of interest in many studies but there are difficulties in modeling and analyzing such data. First, when the duration of followup is limited and the times for a given individual are not independent, the problem of induced dependent censoring arises for the second and subsequent survival times. Non-identifiability of the marginal survival distributions for second and later times is another issue, since they are observable only if preceding survival times for an individual are uncensored. In addition, in some studies, a significant proportion of individuals may never have the first event. Fully parametric models can deal with these features, but lack of robustness is a concern, and methods of assessing fit are lacking. We introduce an approach to address these issues. We model the joint distribution of the successive survival times by using copula functions, and provide semiparametric estimation procedures in which copula parameters are estimated without parametric assumptions on the marginal distributions. The performance of semiparametric estimation methods is compared with some other estimation methods in simulation studies and shown to be good. The methodology is applied to a motivating example involving relapse and survival following colon cancer treatment.