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Estimation and variable selection via frailty models with penalized likelihood

The penalized likelihood methodology has been consistently demonstrated to be an attractive shrinkage and selection method. It does not only automatically and consistently select the important variables but also produces estimators that are as efficient as the oracle estimator. In this paper, we apply this approach to a general likelihood function for data organized in clusters, which corresponds to a class of frailty models, which includes the Cox model and the Gamma frailty model as special cases. Our aim was to provide practitioners in the medical or reliability field with options other than the Gamma frailty model, which has been extensively studied because of its mathematical convenience. We illustrate the penalized likelihood methodology for frailty models through simulations and real data.

Estimation Of Density For Arbitrarily Censored And Truncated Data

We consider survival data that are both interval censored and truncated. Turnbull [Tur76] proposed in 1976 a nice method for nonparametric maximum likelihood estimation of the distribution function in this case, which has been used since by many authors. But, to our knowledge, the consistency of the resulting estimate was never proved. We prove here the consistency of Turnbull’s NPMLE under appropriate conditions on the involved distributions: the censoring, truncation and survival distributions.

Tuning Parameter Selection in Penalized Frailty Models

The penalized likelihood approach of Fan and Li (2001, 2002) differs from the traditional variable selection procedures in that it deletes the non-significant variables by estimating their coefficients as zero. Nevertheless, the desirable performance of this shrinkage methodology relies heavily on an appropriate selection of the tuning parameter which is involved in the penalty functions. In this work, new estimates of the norm of the error are firstly proposed through the use of Kantorovich inequalities and, subsequently, applied to the frailty models framework. These estimates are used in order to derive a tuning parameter selection procedure for penalized frailty models and clustered data. In contrast with the standard methods, the proposed approach does not depend on resampling and therefore results in a considerable gain in computational time. Moreover, it produces improved results. Simulation studies are presented to support theoretical findings and two real medical data sets are analyzed.

Frailty or Transformation Models in Survival Analysis and Reliability

Frailty models are generalizations of the well-known Cox model (Cox, J Roy Stat Soc B 34:187–202, 1972), introduced by Vaupel et al. (Demography 16:439–454, 1979) which are included in a bigger class of models called transformation models. They have received considerable attention over the past couple of decades, especially for the analysis of medical and reliability data that display heterogeneity, which cannot be sufficiently explained by the Cox model. More specifically, the frailty parameter is a random effect term that acts multiplicatively on the hazard intensity function of the Cox model. In this paper we present older and recent results on frailty and transformation models in the parametric and semiparametric setting and for various observational schemes. We deal with efficient estimation of parameters in the uncensored case, right censored case and interval censored and truncated data case.

A real survival analysis application via variable selection methods for Cox's proportional hazards model

Variable selection is fundamental to high-dimensional statistical modeling in diverse fields of sciences. In our health study, different statistical methods are applied to analyze trauma annual data, collected by 30 General Hospitals in Greece. The dataset consists of 6334 observations and 111 factors that include demographic, transport, and clinical data. The statistical methods employed in this work are the nonconcave penalized likelihood methods, Smoothly Clipped Absolute Deviation, Least Absolute Shrinkage and Selection Operator, and Hard, the maximum partial likelihood estimation method, and the best subset variable selection, adjusted to Coxs proportional hazards model and used to detect possible risk factors, which affect the length of stay in a hospital. A variety of different statistical models are considered, with respect to the combinations of factors while censored observations are present. A comparative survey reveals several differences between results and execution times of each method. Finally, we provide useful biological justification of our results.

Semiparametric Transformation Models for Arbitrarily Censored and Truncated Data

A regression analysis for arbitrarily censored and truncated data was proposed by (1996), using Cox’s proportional hazards model, and based on (1976) for nonparametric estimation of a distribution function. We propose here a generalization of their method to the case where there is an unobserved heterogeneity in the data taken into account by a frailty model. Our methodology is applied to a set of real data on transfusion-related AIDS that has been used among others by (1989).

Estimation in Two — Sample Nonproportional Hazards Models in Clinical Trials by an Algorithmic Method

A regression nonproportional hazards model in which the structural parameter is the vector of regression coefficients is considered. Jointly (implicitly) defined estimators of the structural and nuisance parameters are proposed and for the special case of the two — sample problem, an algorithmic procedure that provides these estimators is designed. The behavior of the algorithm is illustrated through extensive simulation of survival data.

A new method for the analysis of supersaturated designs with discrete data

Supersaturated designs are factorial designs in which the number of potential effects is greater than the run size. They are commonly used in screening experiments, with the aim of identifying the dominant active factors with low cost. However, an important research field, which is poorly developed, is the analysis of such designs with non-normal response. In this article, we develop a variable selection strategy, through the modification of the PageRank algorithm, which is commonly used in the Google search engine for ranking Webpages. The proposed method incorporates an appropriate information theoretical measure into this algorithm and as a result, it can be efficiently used for factor screening. A noteworthy advantage of this procedure is that it allows the use of supersaturated designs for analyzing discrete data and therefore a generalized linear model is assumed. As it is depicted via a thorough simulation study, in which the Type I and Type II error rates are computed for a wide range of underlying models and designs, the presented approach can be considered quite advantageous and effective.

Information Measures in Biostatistics and Reliability Engineering

In this paper, we discuss the basic tools for modelling in Biomedicine and Reliability. In particular, we present the divergence measures and the tests of fit while optimal modelling issues are also addressed. The last section is devoted to various applications in Reliability, Biomedicine, Hydrology, and Insurance and Actuarial Science.

The frailty model

Survival analysis is a collection of statistical procedures for the analysis of survival data, namely data for which the outcome variable of interest is the time until an event occurs. Proportional hazards models and accelerated failure time models are classic survival models. In recent years frailty models and copula models have been introduced for handling clustered survival data. This book focuses on frailty models and provides an in-depth discussion of the basics of frailty model methodology using numerous real data sets. Through a frailty parameter, these models capture and describe either the dependence within a cluster or the heterogeneity between clusters or both. Chapter 1 consists of two parts. The first part is dedicated to the presentation of 11 data sets that are extensively used in subsequent chapters. The second part provides the basics of survival analysis with references to survival likelihood, proportional hazards models and accelerated failure time models. The simplest frailty model, the parametric proportional hazards model with a 1-parameter gamma frailty, is thoroughly covered in Chapter 2. The model is fitted based on both the frequentist and the Bayesian approach based on the Metropolis algorithm. The discussion covers both rightcensored and interval-censored data. Chapter 3 deals with alternatives to the frailty model. More specifically, the fixed effects, copula and marginal models are discussed. The chapter includes short discussions about the differences between each of the alternative models and the frailty model. The Clayton copula and its differences with the frailty model are also covered. Different distributions for the frailty factor are presented in Chapter 4. Some general characteristics including means of dependence such as Kendall’s τ and the cross ratio function are initially provided. The chapter covers the gamma, the inverse Gaussian, the positive stable and the compound Poisson distribution. A section is devoted to the general power variance function family that includes all of the above distributions. The lognormal, despite not being a member of the above family, is also discussed. Chapter 5 focuses on the extension of the semiparametric proportional hazards model to the semiparametric frailty model. The EM algorithm and the penalised partial likelihood maximisation for semiparametric frailty models with gamma frailty are presented. A section of this chapter is devoted to the Bayesian approach to fit such models based on Gibbs sampling. Multifrailty models are presented in Chapter 6. The one-cluster level with two frailties within a cluster is discussed. The frequentist and the Bayesian approach are considered, both of which are based on the Laplacian integration for fitting such a model. Hierarchical models with two nested