Survival Analysis Of Complex Featured Data With Measurement Error

Survival analysis plays an important role in many fields, such as cancer research, clinical trials, epidemiological studies, actuarial science, and so on. A large body of methods on analyzing survival data have been developed. However, many important problems have still not been fully explored. In this thesis, we focus on the analysis of survival data with complex features. In Chapter 1, we review relevant topics including survival analysis, the measurement error model, the graphical model, and variable selection. Graphical models are useful in characterizing the dependence structure of variables. They have been commonly used for analysis of high-dimensional data, including genetic data and data with network structures. Many estimation procedures have been developed under various graphical models with a stringent assumption that the associated variables must be measured precisely. In applications, this assumption, however, is often unrealistic and mismeasurement in variables is usually presented in data. In Chapter 2, we investigate the high-dimensional graphical model with error-prone variables. We propose valid estimation procedures to account for measurement error effects. Theoretical results are established for the proposed methods and numerical studies are reported to assess the performance of our proposed methods. In Chapter 3, we consider survival analysis with network structures and measurement error in covariates. In survival data analysis, the Cox proportional hazards (PH) model is perhaps the most widely used model to feature the dependence of survival times on covariates. While many inference methods have been developed under such a model or its variants, those models are not adequate for handling data with complex structured covariates. High-dimensional survival data often entail several features: (1) many covariates are inactive in explaining the survival information, (2) active covariates are associated in a network structure, and (3) some covariates are error-contaminated. To hand such kinds of survival data, we propose graphical proportional hazards measurement error models, and develop inferential procedures for the parameters of interest. Our proposed models significantly enlarge the scope of the usual Cox PH model and have great flexibility in characterizing survival data. Theoretical results are established to justify the proposed methods. Numerical studies are conducted to assess the performance of the proposed methods. In Chapter 4, we focus on sufficient dimension reduction for high-dimensional survival data with covariate measurement error. Sufficient dimension reduction (SDR) is an important tool in regression analysis which reduces the dimension of covariates without losing predictive information. Several methods have been proposed to handle data with either censoring in the response or measurement error in covariates. However, little research is available to deal with data having these two features simultaneously. Moreover, the analysis becomes more challenging when data contain ultrahigh-dimensional covariates. In Chapter 4, we examine this problem. We start with considering the cumulative distribution function in regular settings and propose a valid SDR method to incorporate the effects of both censored data and covariates measurement error. Next, we extend the proposed method to handle ultrahigh-dimensional data. Theoretical results of the proposed methods are established. Numerical studies are reported to assess the performance of the proposed methods. In Chapter 5, we slightly switch our attention to examine sampling issues concerning survival data. Specifically, we discuss survival analysis for left-truncated and right-censored data with covariate measurement error. Many methods have been developed for analyzing survival data which commonly involve right-censoring. These methods, however, are challenged by complex features pertinent to the data collection as well as the nature of data themselves. Typically, biased samples caused by left-truncation or length-biased sampling and measurement error are often accompanying with survival analysis. While such data frequently arise in practice, little work has been available in the literature. In Chapter 5, we study this important problem and explore valid inference methods for handling left-truncated and right-censored survival data with measurement error under the widely used Cox model. We exploit a flexible estimator for the survival model parameters which does not require specification of the baseline hazard function. To improve the efficiency, we further develop an augmented non-parametric maximum likelihood estimator. We establish asymptotic results for the proposed estimators and examine the efficiency and robustness issues of the proposed estimators. The proposed methods enjoy appealing features that the distributions of the covariates and of the truncation times are left unspecified. Numerical studies are reported to assess the performance of the proposed methods. In Chapter 6, we study outstanding issues on model selection and model averaging for survival data with measurement error. Model selection plays a critical role in statistical inference and a vast literature has been devoted to this topic. Despite extensive research attention on model selection, research gaps still remain. An important but unexplored problem concerns model selection for truncated and censored data with measurement error. Although analysis of left-truncated and right-censored (LTRC) data has received extensive interests in survival analysis, there has been no research on model selection for LTRC data, let alone LTRC data involving with measurement error. In Chapter 6, we take up this important problem and develop inferential procedures to handle model selection for LTRC data with measurement error in covariates. Our development employs the local model misspecification framework and emphasizes the use of the focus information criterion (FIC). We develop valid estimators using the model averaging scheme and establish theoretical results to justify the validity of our methods. Numerical studies are conducted to assess the performance of the proposed methods. Finally, Chapter 7 summarizes the thesis with discussions.

In survival data analysis, covariates are often subject to measurement error. A naive analysis with measurement error ignored commonly leads to biased estimation of parameters of survival models. Measurement error also causes efficiency loss for detecting possible association between risk factors and time to event. Furthermore, it induces difficulty on model building and model checking, because the presence of measurement error frequently masks true underlying patterns of data. Although there has been a large body of literature to handle error-prone survival data since the paper by Prentice (1982), many important issues still remain unexplored in this area. This thesis focuses on several important issues of survival analysis with covariate measurement error. One problem that has received little attention is on misspecification of measurement error models. In this thesis, we investigate this important problem with the attention particularly paid to error-contaminated survival data under the Cox model. In particular, we conduct bias analysis which offers a way to unify many existing methods of survival data with measurement error, and study the impact of misspecifying the error models in survival data analysis. A simple expression is obtained to quantify the bias of "working" estimators derived under misspecified error models. Consistent estimators under general error models are derived based on this simple expression. Furthermore, we study hypothesis testing with both model misspecification and measurement error present. A second problem of our interest is about the validity of survival model assumptions when measurement error is involved. In the literature, a large number of methods have been developed to correct for measurement error effects, and these methods basically assume the survival model to be the Cox model. When the Cox model or the error model assumptions fail to hold, existing methods would break down. In this thesis, we address the issue of checking the Cox model assumptions with measurement error. We propose valid goodness of fit tests for survival data with covariate measurement error. This research offers us an important addition to the literature of survival data with measurement error. Our third topic concerns survival data analysis under additive hazards models with covariate measurement error. The additive hazards model is a useful and important alternative to the Cox model. However, this model is relatively less studied for situations where covariates are measured with error. In this thesis, we make important contributions to this topic. Specifically, we explore asymptotic bias induced from ignoring measurement error. A number of inference methods are developed to correct for error effects. The validity of the proposed methods is justified both theoretically and empirically. We investigate issues of model checking and model misspecification as well. In many studies, collection of data often includes a large number of variables in which many of them are unimportant in explaining survival of an individual. An important task is thus to identify relevant risk factors which are linked to the hazards of subjects. Although there is work on variable selection for survival data analysis, the available methods typically require all variables be precisely measured. This requirement is, however, often infeasible. More challengingly, in some studies, the dimension of the risk factors can be quite large or even much larger than the size of subjects. Our fourth topic concerns about estimation and variable selection for survival data with high dimensional mismeasured covariates. We propose corrected penalized methods. Our methods can adjust for measurement error effects, and perform estimation and variable selection simultaneously. Our research on this topic closes multiple gaps among the areas of survival analysis, measurement error and variable selection.

Making complex methods more accessible to applied researchers without an advanced mathematical background, the authors present the essence of new techniques available, as well as classical techniques, and apply them to data. Practical suggestions for implementing the various methods are set off in a series of practical notes at the end of each section, while technical details of the derivation of the techniques are sketched in the technical notes. This book will thus be useful for investigators who need to analyse censored or truncated life time data, and as a textbook for a graduate course in survival analysis, the only prerequisite being a standard course in statistical methodology.

Survival Analysis Using S: Analysis of Time-to-Event Data is designed as a text for a one-semester or one-quarter course in survival analysis for upper-level or graduate students in statistics, biostatistics, and epidemiology. Prerequisites are a standard pre-calculus first course in probability and statistics, and a course in applied linear regression models. No prior knowledge of S or R is assumed. A wide choice of exercises is included, some intended for more advanced students with a first course in mathematical statistics. The authors emphasize parametric log-linear models, while also detailing nonparametric procedures along with model building and data diagnostics. Medical and public health researchers will find the discussion of cut point analysis with bootstrap validation, competing risks and the cumulative incidence estimator, and the analysis of left-truncated and right-censored data invaluable. The bootstrap procedure checks robustness of cut point analysis and determines cut point(s). In a chapter written by Stephen Portnoy, censored regression quantiles - a new nonparametric regression methodology (2003) - is developed to identify important forms of population heterogeneity and to detect departures from traditional Cox models. By generalizing the Kaplan-Meier estimator to regression models for conditional quantiles, this methods provides a valuable complement to traditional Cox proportional hazards approaches.

Although standard mixed effects models are useful in a range of studies, other approaches must often be used in correlation with them when studying complex or incomplete data. Mixed Effects Models for Complex Data discusses commonly used mixed effects models and presents appropriate approaches to address dropouts, missing data, measurement errors, censoring, and outliers. For each class of mixed effects model, the author reviews the corresponding class of regression model for cross-sectional data. An overview of general models and methods, along with motivating examples After presenting real data examples and outlining general approaches to the analysis of longitudinal/clustered data and incomplete data, the book introduces linear mixed effects (LME) models, generalized linear mixed models (GLMMs), nonlinear mixed effects (NLME) models, and semiparametric and nonparametric mixed effects models. It also includes general approaches for the analysis of complex data with missing values, measurement errors, censoring, and outliers. Self-contained coverage of specific topics Subsequent chapters delve more deeply into missing data problems, covariate measurement errors, and censored responses in mixed effects models. Focusing on incomplete data, the book also covers survival and frailty models, joint models of survival and longitudinal data, robust methods for mixed effects models, marginal generalized estimating equation (GEE) models for longitudinal or clustered data, and Bayesian methods for mixed effects models. Background material In the appendix, the author provides background information, such as likelihood theory, the Gibbs sampler, rejection and importance sampling methods, numerical integration methods, optimization methods, bootstrap, and matrix algebra. Failure to properly address missing data, measurement errors, and other issues in statistical analyses can lead to severely biased or misleading results. This book explores the biases that arise when naïve methods are used and shows which approaches should be used to achieve accurate results in longitudinal data analysis.

Measurement error models describe functional relationships among variables observed, subject to random errors of measurement. This book treats general aspects of the measurement problem and features a discussion of the history of measurement error models.

Multiple Imputation of Missing Data in Practice: Basic Theory and Analysis Strategies provides a comprehensive introduction to the multiple imputation approach to missing data problems that are often encountered in data analysis. Over the past 40 years or so, multiple imputation has gone through rapid development in both theories and applications. It is nowadays the most versatile, popular, and effective missing-data strategy that is used by researchers and practitioners across different fields. There is a strong need to better understand and learn about multiple imputation in the research and practical community. Accessible to a broad audience, this book explains statistical concepts of missing data problems and the associated terminology. It focuses on how to address missing data problems using multiple imputation. It describes the basic theory behind multiple imputation and many commonly-used models and methods. These ideas are illustrated by examples from a wide variety of missing data problems. Real data from studies with different designs and features (e.g., cross-sectional data, longitudinal data, complex surveys, survival data, studies subject to measurement error, etc.) are used to demonstrate the methods. In order for readers not only to know how to use the methods, but understand why multiple imputation works and how to choose appropriate methods, simulation studies are used to assess the performance of the multiple imputation methods. Example datasets and sample programming code are either included in the book or available at a github site (https://github.com/he-zhang-hsu/multiple_imputation_book). Key Features Provides an overview of statistical concepts that are useful for better understanding missing data problems and multiple imputation analysis Provides a detailed discussion on multiple imputation models and methods targeted to different types of missing data problems (e.g., univariate and multivariate missing data problems, missing data in survival analysis, longitudinal data, complex surveys, etc.) Explores measurement error problems with multiple imputation Discusses analysis strategies for multiple imputation diagnostics Discusses data production issues when the goal of multiple imputation is to release datasets for public use, as done by organizations that process and manage large-scale surveys with nonresponse problems For some examples, illustrative datasets and sample programming code from popular statistical packages (e.g., SAS, R, WinBUGS) are included in the book. For others, they are available at a github site (https://github.com/he-zhang-hsu/multiple_imputation_book)

Well received in its first edition, Survival Analysis: A Practical Approach is completely revised to provide an accessible and practical guide to survival analysis techniques in diverse environments. Illustrated with many authentic examples, the book introduces basic statistical concepts and methods to construct survival curves, later developing them to encompass more specialised and complex models. During the years since the first edition there have been several new topics that have come to the fore and many new applications. Parallel developments in computer software programmes, used to implement these methodologies, are relied upon throughout the text to bring it up to date.

Critically acclaimed and resoundingly popular in its first edition, Modelling Survival Data in Medical Research has been thoroughly revised and updated to reflect the many developments and advances--particularly in software--made in the field over the last 10 years. Now, more than ever, it provides an outstanding text for upper-level and graduate courses in survival analysis, biostatistics, and time-to-event analysis.The treatment begins with an introduction to survival analysis and a description of four studies that lead to survival data. Subsequent chapters then use those data sets and others to illustrate the various analytical techniques applicable to such data, including the Cox regression model, the Weibull proportional hazards model, and others. This edition features a more detailed treatment of topics such as parametric models, accelerated failure time models, and analysis of interval-censored data. The author also focuses the software section on the use of SAS, summarising the methods used by the software to generate its output and examining that output in detail. Profusely illustrated with examples and written in the author's trademark, easy-to-follow style, Modelling Survival Data in Medical Research, Second Edition is a thorough, practical guide to survival analysis that reflects current statistical practices.