Modeling Multilevel Survival Data Using Frailty Models

Often the dependence in multivariate survival data is modeled through an individual level effect called the frailty. Due to its mathematical simplicity, the gamma distribution is often used as the frailty distribution for hazard modeling. However, it is well known that the gamma frailty distribution has many drawbacks. For example, it weakens the effect of covariates. In addition, in the presence of a multilevel model, overall frailty comes from several levels. To overcome such drawbacks, more heavy-tailed distributions are needed to model the frailty distribution in order to incorporate extra variability. In this article, we develop a class of log-skew-t distributions for the frailty. This class includes the log-normal distribution along with many other heavy tailed distributions, e.g., log-Cauchy, log normal, and log-t as special cases.

Conditional on the frailty, the survival times are assumed to be independent with proportional hazard structure. The modeling process is then completed by assuming multilevel frailty-effects. Instead of tuning a strict parameterization of the baseline hazard function, we consider the partial likelihood approach and thus leave the baseline function unspecified. By eliminating the hazard, the pre-specification and computation are simplified considerably.     

Graphical Tests for the Frailty Distribution in the Shared Frailty Model

We extend a diagnostic plot for the frailty distribution in proportional hazards models to the case of shared frailty. The plot is based on a closure property of exponential family failure distributions with canonical statistics z and g(z), namely that the frailty distribution among survivors at time t has the same form, with the same values of the parameters associated with g(z). We extend this property to shared frailty, considering various definitions of a “surviving” cluster at time t. We illustrate the effectiveness of the method in the case where the “death” of the cluster is defined by the first death among its members.     

Extended Weibull type distribution and finite mixture of distributions

An extended form of Weibull distribution is suggested which has two shape parameters (m and δ). Introduction of another shape parameter δ helps to express the extended Weibull distribution not only as an exact form of a mixture of distributions under certain conditions, but also provides extra flexibility to the density function over positive range. The shape of density function of the extended Weibull type distribution for various values of the parameters is shown which may be of some interest to Bayesians. Certain statistical properties such as hazard rate function, mean residual function, rth moment are defined explicitly. The proposed extended Weibull distribution is used to derive an exact form of two, three and k-component mixture of distributions. With the help of a real data set, the usefulness of mixture Weibull type distribution is illustrated by using Markov Chain Monte Carlo (MCMC), Gibbs sampling approach.     

Detection of frailty in weibull lifetime data using outlier tests

Heterogeneity in lifetime data may be modelled by multiplying an individual's hazard by an unobserved frailty. We test for the presence of frailty of this kind in univariate and bivariate data with Weibull distributed lifetimes, using statistics based on the ordered Cox–Snell residuals from the null model of no frailty. The form of the statistics is suggested by outlier testing in the gamma distribution. We find through simulation that the sum of the k largest or k smallest order statistics, for suitably chosen k, provides a powerful test when the frailty distribution is assumed to be gamma or positive stable, respectively. We provide recommended values of k for sample sizes up to 100 and simple formulae for estimated critical values for tests at the 5% level.     

Frailty models of manufacturing effects

The median service lifetime of respirator safety devices produced by different manufacturers is determined using frailty models to account for unobserved differences in manufacturing processes and raw materials. The gamma and positive stable frailty distributions are used to obtain survival distribution estimates when the baseline hazard is assumed to be Weibull. Frailty distributions are compared using laboratory test data of the failure times for 104 respirator cartridges produced by 10 different manufacturers tested with three different challenge agents. Likelihood ratio tests indicate that both frailty models provide a significant improvement over a Weibull model assuming independence. Results are compared to fixed effects approaches for analysis of this data.     

A study of moments and likelihood estimators of the odd Weibull distribution

The odd Weibull distribution is a three-parameter generalization of the Weibull and the inverse Weibull distributions having rich density and hazard shapes for modeling lifetime data. This paper explored the odd Weibull parameter regions having finite moments and examined the relation to some well-known distributions based on skewness and kurtosis functions. The existence of maximum likelihood estimators have shown with complete data for any sample size. The proof for the uniqueness of these estimators is given only when the absolute value of the second shape parameter is between zero and one. Furthermore, elements of the Fisher information matrix are obtained based on complete data using a single integral representation which have shown to exist for any parameter values. The performance of the odd Weibull distribution over various density and hazard shapes is compared with generalized gamma distribution using two different test statistics. Finally, analysis of two data sets has been performed for illustrative purposes.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.     

Shared gamma frailty models based on reversed hazard rate for modeling Australian twin data

In this paper, we consider shared gamma frailty model with the reversed hazard rate (RHR) with two different baseline distributions, namely the generalized inverse Rayleigh and the exponentiated Gumbel distributions. With these two baseline distributions we propose two different shared frailty models. We develop the Bayesian estimation procedure using Markov Chain Monte Carlo technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. A search of the literature suggests that currently no work has been done for these two baseline distributions with a shared gamma frailty with the RHR so far. We also apply these two models by using a real life bivariate survival data set of Australian twin data given by Duffy et a1. (1990) and a better model is suggested for the data.     

Fitting parametric frailty and mixture models under biased sampling

Biased sampling from an underlying distribution with p.d.f. f(t), t>0, implies that observations follow the weighted distribution with p.d.f. f ^w (t)=w(t)f(t)/E[w(T)] for a known weight function w. In particular, the function w(t)=t ^α has important applications, including length-biased sampling (α=1) and area-biased sampling (α=2). We first consider here the maximum likelihood estimation of the parameters of a distribution f(t) under biased sampling from a censored population in a proportional hazards frailty model where a baseline distribution (e.g. Weibull) is mixed with a continuous frailty distribution (e.g. Gamma). A right-censored observation contributes a term proportional to w(t)S(t) to the likelihood; this is not the same as S ^w (t), so the problem of fitting the model does not simply reduce to fitting the weighted distribution. We present results on the distribution of frailty in the weighted distribution and develop an EM algorithm for estimating the parameters of the model in the important Weibull–Gamma case. We also give results for the case where f(t) is a finite mixture distribution. Results are presented for uncensored data and for Type I right censoring. Simulation results are presented, and the methods are illustrated on a set of lifetime data.     

Defective models induced by gamma frailty term for survival data with cured fraction

In this paper, we propose a defective model induced by a frailty term for modeling the proportion of cured. Unlike most of the cure rate models, defective models have advantage of modeling the cure rate without adding any extra parameter in model. The introduction of an unobserved heterogeneity among individuals has bring advantages for the estimated model. The influence of unobserved covariates is incorporated using a proportional hazard model. The frailty term assumed to follow a gamma distribution is introduced on the hazard rate to control the unobservable heterogeneity of the patients. We assume that the baseline distribution follows a Gompertz and inverse Gaussian defective distributions. Thus we propose and discuss two defective distributions: the defective gamma-Gompertz and gamma-inverse Gaussian regression models. Simulation studies are performed to verify the asymptotic properties of the maximum likelihood estimator. Lastly, in order to illustrate the proposed model, we present three applications in real data sets, in which one of them we are using for the first time, related to a study about breast cancer in the A.C.Camargo Cancer Center, São Paulo, Brazil.     

Gamma shared frailty model based on reversed hazard rate

Abstract

Frailty models are used in survival analysis to account for unobserved heterogeneity in individual risks to disease and death. To analyze bivariate data on related survival times (e.g., matched pairs experiments, twin, or family data), shared frailty models were suggested. Shared frailty models are frequently used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor(frailty) and baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, we introduce shared gamma frailty models with reversed hazard rate. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. Also, we apply the proposed model to the Australian twin data set.     

Modeling bivariate survival data using shared inverse Gaussian frailty model

ABSTRACT

The shared frailty models are often used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of a random factor (frailty) and the baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and the distribution of frailty. In this paper, we consider inverse Gaussian distribution as frailty distribution and three different baseline distributions, namely the generalized Rayleigh, the weighted exponential, and the extended Weibull distributions. With these three baseline distributions, we propose three different inverse Gaussian shared frailty models. We also compare these models with the models where the above-mentioned distributions are considered without frailty. We develop the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. A search of the literature suggests that currently no work has been done for these three baseline distributions with a shared inverse Gaussian frailty so far. We also apply these three models by using a real-life bivariate survival data set of McGilchrist and Aisbett (1991 McGilchrist, C.A., Aisbett, C.W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to the kidney infection data and a better model is suggested for the data using the Bayesian model selection criteria.     

Another Proof of Borel's Strong Law of Large Numbers

Let X be a continuous nonnegative random variable with finite first and second moments and a continuous pdf that is positive on the interior of its support. A nonzero limiting density at the origin and a coefficient of variation (CV) greater than 1 are shown to be sufficient conditions for the distribution truncated below at t > 0 to have a variance greater than the variance of the full distribution. Distributions that satisfy these conditions include those with decreasing hazard rates (e.g., the gamma and Weibull distributions with shape parameters less than 1) and the beta distribution with parameter values p and q for which q > p(p + q + 1). The bound T for which truncation at 0 < t < T increases the variance relative to the full distribution is shown to be greater than the (1 — 1/CV)th percentile of the full distribution.     

Gamma frailty models for bivariate survival data

In this paper, we introduce the shared gamma frailty models with two different baseline distributions namely, the generalized log-logistic and the generalized Weibull. We introduce the Bayesian estimation procedure to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. We apply these models to a real-life bivariate survival data set of McGilchrist and Aisbett related to the kidney infection data and a better model is suggested for the data.     

Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

Objective Bayesian Analysis of Skew‐t Distributions

Abstract. We study the Jeffreys prior and its properties for the shape parameter of univariate skew‐t distributions with linear and nonlinear Student's t skewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student's t approximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum a posteriori estimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location‐scale models under scale mixtures of skew‐normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew‐t distributions.     

Modeling of Inverse Gaussian Frailty Model for Bivariate Survival Data

Shared frailty models are often used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of random factor (frailty) and baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and distribution of frailty. In this article, we consider inverse Gaussian distribution as frailty distribution and three different baseline distributions namely, Weibull, generalized exponential, and exponential power distribution. With these three baseline distributions, we propose three different inverse Gaussian shared frailty models. To estimate the parameters involved in these models we adopt Markov Chain Monte Carlo (MCMC) approach. We present a simulation study to compare the true values of the parameters with the estimated values. Also, we apply these three models to a real life bivariate survival data set of McGilchrist and Aisbett (1991 McGilchrist , C. A. , Aisbett , C. W. ( 1991 ). Regression with frailty in survival analysis . Biometrics 47 : 461 – 466 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to kidney infection and a better model is suggested for the data.     

On extreme order statistics from heterogeneous Weibull variables

In this paper, we focus on stochastic comparisons of extreme order statistics from heterogeneous independent/interdependent Weibull samples. Specifically, we study extreme order statistics from Weibull distributions with (i) common shape parameter but different scale parameters, and (ii) common scale parameter but different shape parameters. Several new comparison results in terms of the likelihood ratio order, reversed hazard rate order and usual stochastic order are studied in those scenarios. The results established here strengthen and generalize some of the results known in the literature including Khaledi and Kochar [Weibull distribution: some stochastic comparisons. J Statist Plann Inference. 2006;136:3121–3129], Fang and Zhang [Stochastic comparisons of series systems with heterogeneous Weibull components. Statist Probab Lett. 2013;83:1649–1653], Torrado [Comparisons of smallest order statistics from Weibull distributions with different scale and shape parameters. J Korean Statist Soc. 2015;44:68–76] and Torrado and Kochar [Stochastic order relations among parallel systems from Weibull distributions. J Appl Probab. 2015;52:102–116]. Some numerical examples are also provided for illustration.     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

A class of regression models for parallel and series systems with a random number of components

We extend the Weibull power series (WPS) class of distributions to the new class of extended Weibull power series (EWPS) class of distributions. The EWPS distributions are related to series and parallel systems with a random number of components, whereas the WPS distributions [Morais AL, Barreto-Souza W. A compound class of Weibull and power series distributions. Computational Statistics and Data Analysis. 2011;55:1410–1425] are related to series systems only. Unlike the WPS distributions, for which the Weibull is a limiting special case, the Weibull law is a particular case of the EWPS distributions. We prove that the distributions in this class are identifiable under a simple assumption. We also prove stochastic and hazard rate order results and highlight that the shapes of the EWPS distributions are markedly more flexible than the shapes of the WPS distributions. We define a regression model for the EWPS response random variable to model a scale parameter and its quantiles. We present the maximum likelihood estimator and prove its consistency and asymptotic normal distribution. Although series and parallel systems motivated the construction of this class, the EWPS distributions are suitable for modelling a wide range of positive data sets. To illustrate potential uses of this model, we apply it to a real data set on the tensile strength of coconut fibres and present a simple device for diagnostic purposes.