Download or read book Semi parametric Regression Analysis of Interval censored Failure Time Data written by Ling Ma and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: By interval-censored data, we mean that the failure time of interest is known only to lie within an interval instead of being observed exactly. Many clinical trials and longitudinal studies may generate interval-censored data. One common example occurs in medical or health studies that entail periodic follow-ups. An important special case of interval-censored data is the so called current status data when each subject is observed only once for the status of the occurrence of the event of interest. That is, instead of observing the survival endpoint directly, we only know the observation time and whether or not the event of interest has occurred at that time. Such data may occur in many fields, for example, cross-sectional studies and tumorigenicity experiments. Sometimes we also refer current status data to as case I interval-censored data and the general case as case II interval-censored data. In the following, for simplicity, we will refer current status data and interval-censored data to case I and case II interval-censored data, respectively. The statistical analysis of both case I and case II interval-censored failure time data has recently attracted a great deal of attention and especially, many procedures have been proposed for their regression analysis under various models. However, due to the strict restrictions of existing regression analysis procedures and practical demands, new methodologies for regression analysis need to be developed. For regression analysis of interval-censored data, many approaches have been proposed and for most of them, the inference is carried out based on the asymptotic normality. It's well known that the symmetric property implied by the normal distribution may not be appropriate sometimes and could underestimate the variance of estimated parameters. In the first part of this dissertation, we adopt the linear transformation models for regression analysis of interval-censored data and propose an empirical likelihood-based procedure to address the underestimating problem from using symmetric property implied by the normal distribution of the parameter estimates. Simulation and analysis of a real data set are conducted to assess the performance of the procedure. The second part of this dissertation discusses regression analysis of current status data under additive hazards models. In this part, we focus on the situation when some covariates could be missing or cannot be measured exactly due to various reasons. Furthermore, for missing covariates, there may exist some related information such as auxiliary covariates (Zhou and Pepe, 1995). We propose an estimated partial likelihood approach for estimation of regression parameters that make use of the available auxiliary information. To assess the finite sample performance of the proposed method, an extensive simulation study is conducted and indicates that the method works well in practical situations. Several semi-parametric and non-parametric methods have been proposed for the analysis of current status data. However, most of these methods deal only with the situation where observation time is independent of the underlying survival time completely or given covariates. The third part of this dissertation discusses regression analysis of current status data when the observation time may be related to survival time. The correlation between observation time and survival time and the covariate effects are described by a copula model and the proportional hazards model, respectively. For estimation, a sieve maximum likelihood procedure with the use of monotone I-spline functions is proposed and the proposed method is examined through a simulation study and illustrated with a real data set. In the fourth part of this dissertation, we discuss the regression analysis of interval- censored data where the censoring mechanism could be related to the failure time. We consider a situation where the failure time depend on the censoring mechanism only through the length of the observed interval. The copula model and monotone I-splines are used and the asymptotic properties of the resulting estimates are established. In particular, the estimated regression parameters are shown to be semiparametrically efficient. An extensive simulation study and an illustrative example is provided. Finally, we will talk about the directions for future research. One topic related the fourth part of this dissertation for future research could be to allow the failure time to depend on both the lower and upper bounds of the observation interval. Another possible future research topic could be to consider a cure rate model for interval-censored data with informative censoring.

Download or read book The Statistical Analysis of Interval censored Failure Time Data written by Jianguo Sun and published by Springer. This book was released on 2007-05-26 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects and unifies statistical models and methods that have been proposed for analyzing interval-censored failure time data. It provides the first comprehensive coverage of the topic of interval-censored data and complements the books on right-censored data. The focus of the book is on nonparametric and semiparametric inferences, but it also describes parametric and imputation approaches. This book provides an up-to-date reference for people who are conducting research on the analysis of interval-censored failure time data as well as for those who need to analyze interval-censored data to answer substantive questions.

Download or read book Nonparametric and Semiparametric Methods for Interval censored Failure Time Data written by Chao Zhu and published by . This book was released on 2006 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interval-censored failure time data commonly arise in follow-up studies such as clinical trials and epidemiology studies. For their analysis, what interests researcher most includes comparisons of survival functions for different groups and regression analysis. This dissertation, which consists of three parts, consider these problems on two types of interval-censored data by using nonparametric and semiparametric methods.

Download or read book Interval Censored Time to Event Data written by Ding-Geng (Din) Chen and published by CRC Press. This book was released on 2012-07-19 with total page 435 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interval-Censored Time-to-Event Data: Methods and Applications collects the most recent techniques, models, and computational tools for interval-censored time-to-event data. Top biostatisticians from academia, biopharmaceutical industries, and government agencies discuss how these advances are impacting clinical trials and biomedical research. Divided into three parts, the book begins with an overview of interval-censored data modeling, including nonparametric estimation, survival functions, regression analysis, multivariate data analysis, competing risks analysis, and other models for interval-censored data. The next part presents interval-censored methods for current status data, Bayesian semiparametric regression analysis of interval-censored data with monotone splines, Bayesian inferential models for interval-censored data, an estimator for identifying causal effect of treatment, and consistent variance estimation for interval-censored data. In the final part, the contributors use Monte Carlo simulation to assess biases in progression-free survival analysis as well as correct bias in interval-censored time-to-event applications. They also present adaptive decision making methods to optimize the rapid treatment of stroke, explore practical issues in using weighted logrank tests, and describe how to use two R packages. A practical guide for biomedical researchers, clinicians, biostatisticians, and graduate students in biostatistics, this volume covers the latest developments in the analysis and modeling of interval-censored time-to-event data. It shows how up-to-date statistical methods are used in biopharmaceutical and public health applications.

Download or read book Regression Analysis of Interval censored Failure Time Data with Non Proportional Hazards Models written by Han Zhang (Graduate of University of Missouri) and published by . This book was released on 2018 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interval-censored failure time data arises when the failure time of interest is known only to lie within an interval or window instead of being observed exactly. Many clinical trials and longitudinal studies may generate interval-censored data. One common area that often produces such data is medical or health studies with periodic follow-ups, in which the medical condition of interest such as the onset of a disease is only known to occur between two adjacent examination times. An important special case of interval-censored data is the so-called current status data when each study subject is observed only once for the status of the event of interest. That is, instead of observing the survival endpoint directly, we will only know the observation time and whether or not the event of interest has occurred by that time. Such data may occur in many fields as cross-sectional studies and tumorigenicity experiments. Sometimes we also refer current status data as case I interval-censored data and the general case as case II interval-censored data. Recently the semi-parametric statistical analysis of both case I and case II intervalcensored failure time data has attracted a great deal of attention. Many procedures have been proposed for their regression analysis under various models. We will describe the structure of interval-censored data in Chapter 1 and provides two specific examples. Also some special situations like informative censoring and failure time data with missing covariates are discussed. Besides, a brief review of the literature on some important topics, including nonparametric estimation and regression analysis are performed. However, there are still a number of problems that remain unsolved or lack approaches that are simpler, more efficient and could apply to more general situations compared to the existing ones. For regression analysis of interval-censored data, many approaches have been proposed and more specifically most of them are developed for the widely used proportional hazards model. The research in this dissertation focuses on the statistical analysis on non-proportional hazards models. In Chapter 2 we will discuss the regression analysis of interval-censored failure time data with possibly crossing hazards. For the problem of crossing hazards, people assume that the hazard functions with two samples considered may cross each other where most of the existing approaches cannot deal with such situation. Many authors has provided some efficient methods on right-censored failure time data, but little articles could be found on interval-censored data. By applying the short-term and long-term hazard ratio model, we develop a spline-based maximum likelihood estimation procedure to deal with this specific situation. In the method, a splined-based sieve estimation are used to approximate the underlying unknown function. The proposed estimators are shown to be strongly consistent and the asymptotic normality of the estimators of regression parameters are also shown to be true. In addition, we also provided a Cramer-Raw type of criterion to do the model validation. Simulation study was conducted for the assessment of the finite sample properties of the presented procedure and suggests that the method seems to work well for practical situations. Also an illustrative example using a data set from a tumor study is provided. As we discussed in Chapter 1, several semi-parametric and non-parametric methods have been proposed for the analysis of current status data. However, most of them only deal with the situation where observation time is independent of the underlying survival time. In Chapter 3, we consider regression analysis of current status data with informative observation times in additive hazards model. In many studies, the observation time may be correlated to the underlying failure time of interest, which is often referred to as informative censoring. Several authors have discussed the problem and in particular, an estimating equation-based approach for fitting current status data to additive hazards model has been proposed previously when informative censoring occurs. However, it is well known that such procedure may not be efficient and to address this, we propose a sieve maximum likelihood procedure. In particular, an EM algorithm is developed and the resulting estimators of regression parameters are shown to be consistent and asymptotically normal. An extensive simulation study was conducted for the assessment of the finite sample properties of the presented procedure and suggests that it seems to work well for practical situations. An application to a tumorigenicity experiment is also provided. In Chapter 4, we considered another special case under the additive hazards model, case II interval-censored data with possibly missing covariates. In many areas like demographical, epidemiological, medical and sociological studies, a number of nonparametric or semi-parametric methods have been developed for interval-censored data when the covariates are complete. However, it is well-known that in reality some covariates may suffer missingness due to various reasons, data with missing covariates could be very common in these areas. In the case of missing covariates, a naive method is clearly the complete-case analysis, which deletes the cases or subjects with missing covariates. However, it's apparent that such analysis could result in loss of efficiency and furthermore may lead to biased estimation. To address this, we propose the inverse probability weighted method and reweighting approach to estimate the regression parameters under the additive hazards model when some of the covariates are missing at random. The resulting estimators of regression parameters are shown to be consistent and asymptotically normal. Several numerical results suggest that the proposed method works well in practical situations. Also an application to a health survey is provided. Several directions for future research are discussed in Chapter 5.

Download or read book Maximum Penalized Likelihood Estimation for Semi parametric Regression Models with Partly Interval censored Failure Time Data written by Jinqing Li and published by . This book was released on 2015 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interval-censored failure time data arise in many areas including demographical, financial, actuarial, medical and sociological studies. By interval censoring we mean that the failure time is not always exactly observed and we can only observe an interval within which the failure event has occurred. The goal of this dissertation is to develop maximum penalized likelihood (MPL) methods for ptoportional hazard (PH), additive hazard (AH) and accelerated failure time (AFT) models with partly interval-censored failure time data, which contains exactly observed, left-censored, finite interval-censored and right-censored data.

Download or read book Semiparametric Analysis of Failure Time Data with Complex Structures written by Yeqian Liu and published by . This book was released on 2016 with total page 119 pages. Available in PDF, EPUB and Kindle. Book excerpt: Failure time data arise in many fields including biomedical studies and industrial life testing. Right-censored failure time data are often observed from a cohort of prevalent cases that are subject to length-biased sampling, which are termed as length-biased and right-censored data. Interval-censored failure time data arise when the failure time of interest in a survival study is not exactly observed but known only to fall within some interval or window. One area that often produces such data is medical studies with periodic follow-ups, in which the medical condition of interest such as the onset of a disease is only known to occur between two adjacent examination times. An important special case of interval-censored data is current status data which arise when each study subject is observed only once and the only information available is whether the failure event of interest has occurred or not by the observation time. Sometimes we also refer current status data as case I interval-censored data and the general case as case II interval-censored data. Semiparametric regression analysis of both right-censored and interval-censored failure time data has recently attracted a great deal of attention. Many procedures have been proposed for their regression analysis under various models. However, in many settings, the population include a cured (nonsusceptible) subpopulation, where only individuals in the susceptible subpopulation will go on to experience the event. Since classical survival models implicitly assume that all individuals will eventually experience the event of interest, they cannot be used in such contexts. They would in fact lead to incorrect results such as, among others, an overestimation of the survival of the non-cured subjects. The research in this dissertation focuses on the statistical analysis for right-censored data with length-biased sampling, interval-censored data with a cured subgroup in the presence of potential dependent censoring and measurement errors. Chapter 1 describes specific examples of right-censored and interval-censored failure time data and reviews the literature on some important topics, including nonparametric and semiparametric estimation, regression analysis in the presence of length-biased sampling and a cured subgroup respectively. Chapter 2 discusses regression analysis of length-biased and right-censored data with with partially linear varying effects. For this problem, we consider quantile regression analysis of right-censored and length-biased data and present a semiparametric varying coefficient partially linear model. For estimation of regression parameters, a three-stage procedure that makes use of the inverse probability weighted technique is developed, and the asymptotic properties of the resulting estimators are established. In addition, the approach allows the dependence of the censoring variable on covariates, while most of the existing methods assume the independence between censoring variables and covariates. A simulation study is conducted and suggests that the proposed approach works well in practical situations. Also an illustrative example is provided. Chapter 3 considers regression analysis of current status data in the presence of a cured subgroup and dependent censoring. For the problem, we develop a sieve maximum likelihood estimation approach with the use of latent variables and Bernstein polynomials. For the determination of the proposed estimators, an EM algorithm and the asymptotic properties of the estimators are established. An extensive simulation study conducted to asses the finite sample performance of the method indicates that it performs well for practical situations. An illustrative example using a data set from a tumor toxicological study is provided. Chapter 4 considers regression analysis of interval-censored data in the presence of a cured subgroup and the case where one or more explanatory variables in the model are subject to measurement errors. These errors should be taken into account in the estimation of the model, to avoid biased estimations. A general approach that exists in the literature is the SIMEX algorithm, a method based on simulations which allows one to estimate the effect of measurement error on the bias of the estimators and to reduce this bias. We extend the SIMEX approach to the mixture cure model with interval-censored data. Comprehensive simulations study as well as a real data application are provided. Several directions for future research are discussed in Chapter 5.

Download or read book Regression Analysis of Clustered Interval censored Failure Time Data with Informative Cluster Size written by Xinyan Zhang and published by . This book was released on 2010 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt: Correlated or clustered failure time data often occur in many research fields including epidemiological, geographical, sociological and medical studies. Sometimes such data arise together with interval censoring and the failure time of interest may be related to the cluster size. Various approaches have been proposed to analyze failure time data with interval censoring. However, these approaches ignore the informativeness of the cluster size. Due to the lack of proper inference procedures for direct analysis, these methods merely simplified or converted interval-censored data into right-censored data, which inevitably resulted in biased parameter estimates. In this dissertation, both parametric and semiparametric approaches are presented for regression analyses of clustered failure time data that allow both interval-censoring and informative cluster size. We further validate these approaches by conducting various simulation studies and apply them to a lymphatic filariasis example.

Download or read book Semiparametric Regression Under Left truncated and Interval censored Competing Risks Data and Missing Cause of Failure written by Jun Park and published by . This book was released on 2020 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Observational studies and clinical trials with time-to-event data frequently involve multiple event types, known as competing risks. The cumulative incidence function (CIF) is a particularly useful parameter as it explicitly quantifies clinical prognosis. Common issues in competing risks data analysis on the CIF include interval censoring, missing event types, and left truncation. Interval censoring occurs when the event time is not observed but is only known to lie between two observation times, such as clinic visits. Left truncation, also known as delayed entry, is the phenomenon where certain participants enter the study after the onset of disease under study. These individuals with an event prior to their potential study entry time are not included in the analysis and this can induce selection bias. In order to address unmet needs in appropriate methods and software for competing risks data analysis, this thesis focuses the following development of application and methods. First, we develop a convenient and exible tool, the R package intccr, that performs semiparametric regression analysis on the CIF for interval-censored competing risks data. Second, we adopt the augmented inverse probability weighting method to deal with both interval censoring and missing event types. We show that the resulting estimates are consistent and double robust. We illustrate this method using data from the East-African International Epidemiology Databases to Evaluate AIDS (IeDEA EA) where a significant portion of the event types is missing. Last, we develop an estimation method for semiparametric analysis on the CIF for competing risks data subject to both interval censoring and left truncation. This method is applied to the Indianapolis-Ibadan Dementia Project to identify prognostic factors of dementia in elder adults. Overall, the methods developed here are incorporated in the R package intccr.

Download or read book Emerging Topics in Modeling Interval Censored Survival Data written by Jianguo Sun and published by Springer Nature. This book was released on 2022-11-29 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book primarily aims to discuss emerging topics in statistical methods and to booster research, education, and training to advance statistical modeling on interval-censored survival data. Commonly collected from public health and biomedical research, among other sources, interval-censored survival data can easily be mistaken for typical right-censored survival data, which can result in erroneous statistical inference due to the complexity of this type of data. The book invites a group of internationally leading researchers to systematically discuss and explore the historical development of the associated methods and their computational implementations, as well as emerging topics related to interval-censored data. It covers a variety of topics, including univariate interval-censored data, multivariate interval-censored data, clustered interval-censored data, competing risk interval-censored data, data with interval-censored covariates, interval-censored data from electric medical records, and misclassified interval-censored data. Researchers, students, and practitioners can directly make use of the state-of-the-art methods covered in the book to tackle their problems in research, education, training and consultation.

Download or read book Multiple Imputation Approaches to Regression Analysis of Interval censored Failure Time Data written by Ling Chen and published by . This book was released on 2009 with total page 109 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation discusses regression analysis of interval-censored failure time data, which occur in many fields including demographical, epidemiological, financial, medical, and sociological studies (Sun, 2006). It consists of three parts. The first part considers regression analysis of current status data under the additive hazards model and in particular, we considered the situation where the observation times depend on covariates. The second part considers regression analysis of interval-censored failure time data under the additive hazards model and time-dependent covariates. The third part considers regression analysis of interval-censored failure time data under the linear transformation model. For these situations, we proposed a general semiparametric method based on multiple imputation for inference under the regression models. This multiple imputation converts the analysis of interval-censored failure time data to that of right-censored failure time data. A major advantage of the approach is its simplicity and it can be easily implemented by using the existing software packages for right-censored failure time data. Extensive simulation studies are conducted and indicate that the approaches perform well for practical situations and are comparable to the existing methods. Real data applications are provided and model checking is discussed.

Download or read book Statistical Analysis of Panel Count Data written by Jianguo Sun and published by Springer Science & Business Media. This book was released on 2013-10-09 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Panel count data occur in studies that concern recurrent events, or event history studies, when study subjects are observed only at discrete time points. By recurrent events, we mean the event that can occur or happen multiple times or repeatedly. Examples of recurrent events include disease infections, hospitalizations in medical studies, warranty claims of automobiles or system break-downs in reliability studies. In fact, many other fields yield event history data too such as demographic studies, economic studies and social sciences. For the cases where the study subjects are observed continuously, the resulting data are usually referred to as recurrent event data. This book collects and unifies statistical models and methods that have been developed for analyzing panel count data. It provides the first comprehensive coverage of the topic. The main focus is on methodology, but for the benefit of the reader, the applications of the methods to real data are also discussed along with numerical calculations. There exists a great deal of literature on the analysis of recurrent event data. This book fills the void in the literature on the analysis of panel count data. This book provides an up-to-date reference for scientists who are conducting research on the analysis of panel count data. It will also be instructional for those who need to analyze panel count data to answer substantive research questions. In addition, it can be used as a text for a graduate course in statistics or biostatistics that assumes a basic knowledge of probability and statistics.

Download or read book Statistical Analysis of Bivariate Interval censored Failure Time Data written by Qingning Zhou and published by . This book was released on 2015 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation deals with various issues in the statistical analysis of bivariate interval-censored failure time data, including regression analysis, model selection and estimation of the association between failure times. In particular, it includes three projects. The first project discusses regression analysis of bivariate current status data under the marginal proportional hazards model. For the problem, by using Bernstein polynomials and an unspecified copula model, we develop a sieve maximum likelihood estimation approach that applies to very general situations. In particular, it allows one to estimate the underlying copula model and can be easily implemented. The strong consistency, asymptotic normality and efficiency of the estimators of regression parameters are established. In the second project, we consider regression analysis of bivariate case II interval-censored data. For this problem, we present a class of semiparametric transformation models which is very flexible and in particular includes the commonly used proportional hazards model as a special case. Also, for inference, we develop a sieve maximum likelihood approach based on Bernstein polynomials. The strong consistency, asymptotic normality and efficiency of the resulting estimators of the regression parameters are established. In the third project, we consider the class of semiparametric copula-based models, in which multivariate survival functions are characterized by parametric copulas and nonparametric marginal survival functions. One important issue in applying this class of models to a given data set is how to choose an appropriate parametric copula. We propose two model selection procedures for Archimedean copulas with bivariate interval-censored data. The first procedure is based on a comparison of the nonparametric and model-based estimators of the probability integral transformation K, while the second procedure is based on a pseudo-likelihood function.

Download or read book Introduction to Empirical Processes and Semiparametric Inference written by Michael R. Kosorok and published by Springer Science & Business Media. This book was released on 2007-12-29 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: Kosorok’s brilliant text provides a self-contained introduction to empirical processes and semiparametric inference. These powerful research techniques are surprisingly useful for developing methods of statistical inference for complex models and in understanding the properties of such methods. This is an authoritative text that covers all the bases, and also a friendly and gradual introduction to the area. The book can be used as research reference and textbook.

Download or read book The Frailty Model written by Luc Duchateau and published by Springer Science & Business Media. This book was released on 2007-10-23 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: Readers will find in the pages of this book a treatment of the statistical analysis of clustered survival data. Such data are encountered in many scientific disciplines including human and veterinary medicine, biology, epidemiology, public health and demography. A typical example is the time to death in cancer patients, with patients clustered in hospitals. Frailty models provide a powerful tool to analyze clustered survival data. In this book different methods based on the frailty model are described and it is demonstrated how they can be used to analyze clustered survival data. All programs used for these examples are available on the Springer website.

Download or read book Survival Analysis written by John P. Klein and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Mixed Effects Models for Complex Data written by Lang Wu and published by CRC Press. This book was released on 2009-11-11 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.