A Mixture Shared Inverse Gaussian Frailty Model under Modified Weibull Baseline Distribution

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyse the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data), the shared frailty models were suggested. Shared frailty models are used despite their limitations. To overcome their disadvantages correlated frailty models may be used. In this paper, we introduce the gamma correlated frailty models based on reversed hazard rate (RHR) with three different baseline distributions namely, the generalised log-logistic type I, the generalised log-logistic type II and the modified inverse Weibull. We introduce 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. We also apply the proposed models to the Australian twin dataset and a better model is suggested.

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.

ABSTRACTThe unknown or unobservable risk factors in the survival analysis cause heterogeneity between individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the shared frailty models were suggested. The most common shared frailty model is a model in which frailty act multiplicatively on the hazard function. In this paper, we introduce the shared gamma frailty model and the inverse Gaussian frailty model with the reversed hazard rate. We introduce the 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. We also apply the proposed models to the Australian twin data set and a better model is suggested.

In this article, we propose additive shared gamma frailty model and additive shared inverse Gaussian frailty model with Lindley as baseline distribution to analyze the infectious disease data set of McGilchrist and Aisbett. The Bayesian approach of Markov Chain Monte Carlo technique was employed to estimate the parameters involved in the models. A simulation study was also carried out to compare the true values and estimated values of the parameters. Comparison of proposed models and the existing models was also done by using Bayesian information criteria such as BIC, WBIC and Bayes factor. A better model was also recommended for infectious disease data.

Due to the unavailability of complete data in various circumstances in biological, epidemiological, and medical studies, the analysis of censored data is very common among practitioners. But the analysis of bivariate censored data is not a regular mechanism because it is not necessary to always have independent data. Observed and unobserved covariates affect the variables under study. So, heterogeneity is present in the data. Ignoring observed and unobserved covariates may have objectionable consequences. But it is not easy to find that whether there is any effect of the unobserved covariate or not. Shared frailty models are the viable choice to counter such scenarios. However, due to certain restrictions such as the identifiability condition and the requirement that their Laplace transform exists, finding a frailty distribution can be difficult. As a result, in this paper, we introduce a new frailty distribution generalized Lindley (GL) for reversed hazard rate (RHR) setup that outperforms the gamma frailty distribution. So, our main motive is to establish a new frailty distribution under the RHR setup. By assuming exponential Gumbel (EG) and generalized inverted exponential (GIE) baseline distributions, we propose a new class of shared frailty models based on RHR. We estimate the parameters in these frailty models and use the Bayesian paradigm of the Markov Chain Monte Carlo (MCMC) technique. Model selection criteria have been performed for the comparison of models. We analyze Australian twin data and suggest a better model.

Many analyses in the epidemiological and the prognostic studies and in the studies of event history data require methods that allow for unobserved covariates or “frailties”. We consider the shared frailty model in the framework of parametric proportional hazard model. There are certain assumptions about the distribution of frailty and baseline distribution. The exponential distribution is the commonly used distribution for analyzing lifetime data. In this paper, we consider shared gamma frailty model with bivariate exponential of Marshall and Olkin (1967) distribution as baseline hazard for bivariate survival times. We solve the inferential problem in a Bayesian framework with the help of a comprehensive simulation study and real data example. We fit the model to the real-life bivariate survival data set of diabetic retinopathy data. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the proposed model and then compare the true values of the parameters with the estimated values for different sample sizes.

Frailty Models in Public Health

In the environmental sciences, a large knowledge base is typically available on an investigated system or at least on similar systems. This makes the application of Bayesian inference techniques in environmental modeling very promising. However, environmental systems are often described by complex, computationally demanding simulation models. This strongly limits the application of Bayesian inference techniques, because numerical implementation of these techniques requires a very large number of simulation runs. The development of efficient sampling techniques that attempt to approximate the posterior distribution with a relatively small parameter sample can extend the range of applicability of Bayesian inference techniques to such models. In this article a sampling technique is presented that tries to achieve this goal. The proposed technique combines numerical techniques typically applied in Bayesian inference, including posterior maximization, local normal approximation, and importance sampling, with copula techniques for the construction of a multivariate distribution with given marginals and correlation structure and with low-discrepancy sampling. This combination improves the approximation of the posterior distribution by the sampling distribution and improves the accuracy of results for small sample sizes. The usefulness of the proposed technique is demonstrated for a simple model that contains the major elements of models used in the environmental sciences. The results indicate that the proposed technique outperforms conventional techniques (random sampling from simpler distributions or Markov chain Monte Carlo techniques) in cases in which the analysis can be limited to a relatively small number of parameters.

In this article, estimation problems for the Burr X distribution under a progressive-stress accelerated life test with progressive type-II censoring are studied. The stress is assumed to be a linearly increasing function of time. The inverse power law and the cumulative exposure model are considered. The classical and Bayesian estimations for the model parameters are obtained by using maximum likelihood method and Markov chain Monte Carlo technique,respectively. The asymptotic confidence intervals are constructed and highest posterior density intervals are also established. A simulation study is conducted to investigate the performance of the proposed point and interval estimations. Finally, a real data set is analysed for illustration.

The study deals with the analysis of Type-II hybrid censored data from the modified Weibull distribution. We provide maximum-likelihood estimates of the parameters, reliability, and hazard rate functions along with their standard errors. The confidence intervals along with their widths have also been obtained. Assuming gamma and Jeffrey’s invariant priors for the unknown parameters, Bayes estimates along with its posterior errors and highest posterior density credible intervals are obtained. The Markov Chain Monte Carlo technique has been used to simulate draws from the complicated posterior densities of the parameters. A simulation study is conducted to compare the performances of classical and Bayesian methods of estimation. Finally, a real data analysis is performed for illustrative purpose.

The paper presents a novel statistical approach for analyzing the daily coronavirus case and fatality statistics. The survival discretization method was used to generate a two-parameter discrete distribution. The resulting distribution is referred to as the "Discrete Marshall–Olkin Length Biased Exponential (DMOLBE) distribution". Because of the varied forms of its probability mass and failure rate functions, the DMOLBE distribution is adaptable. We calculated the mean and variance, skewness, kurtosis, dispersion index, hazard and survival functions, and second failure rate function for the suggested distribution. The DI index demonstrates that the proposed model can represent both over-dispersed and under-dispersed data sets. We estimated the parameters of the DMOLBE distribution. The behavior of ML estimates is checked via a comprehensive simulation study. The behavior of Bayesian estimates is checked by generating 10,000 iterations of Markov chain Monte Carlo techniques, plotting the trace, and checking the proposed distribution. From simulation studies, it was observed that the bias and mean square error decreased with an increase in sample size. To show the importance and flexibility of DMOLBE distribution using two data sets about deaths due to coronavirus in China and Pakistan are analyzed. The DMOLBE distribution provides a better fit than some important discrete models namely the discrete Burr-XII, discrete Bilal, discrete Burr-Hatke, discrete Rayleigh distribution, and Poisson distributions. We conclude that the new proposed distribution works well in analyzing these data sets. The data sets used in the paper was collected from 2020 year.

Bayesian inference and Markov chain Monte Carlo techniques have enjoyed enormous popularity since they were introduced into phylogenetics about a decade ago. We provide an overview of the field, with emphasis on recent developments of importance to empirical systematists. In particular, we describe a number of recent advances in the stochastic modeling of evolution that address major deficiencies in current models in a computationally efficient way. These include models of process heterogeneity across sites and lineages, as well as alignment-free models and model averaging approaches. Many of these methods should find their way into standard analyses in the near future. We also summarize the influence of Bayesian methods on insect systematics, with particular focus on current practices and how they could be improved using existing and emerging techniques.

We present a Bayesian Markov chain Monte Carlo technique for estimating the astrophysical parameters of gravitational radiation signals from a neutron star in laser interferometer data. This computational algorithm can estimate up to six unknown parameters of the target, including the rotation frequency and frequency derivative, using reparametrization, delayed rejection and simulated annealing. We highlight how a simple extension of the method, distributed over multiple computer processors, will allow for a search over a narrow frequency band. The ultimate goal of this research is to search for sources at known locations, but uncertain spin parameters; an example would be SN1987A.

Demand for organic and conventional vegetables is investigated using data from A.C. Nielsen’s 2006 Homescan panel. We use a Bayesian Markov chain Monte Carlo technique, along with data augmentation, to estimate a large linear approximate Almost Ideal Demand System with censored dependent variables. Demands are price elastic, and expenditure elasticities are very high for organic vegetables, whilst demands for conventional vegetables are primarily inelastic. We find a mix of gross substitution and complementarity among the vegetable products, but net substitution is the dominant pattern. Socio‐demographic characteristics also play important roles in demands. These findings can inform deliberations about marketing campaigns, nutrition education and policy interventions.

Nanoparticle dispersion plays a crucial role in the mechanical properties of polymer nanocomposites. Transmission Electron Microscope/Scanning Electron Microscope (TEM/SEM) images are commonly used to represent nanoparticle dispersion without further quantifications on its properties. Therefore, there is a strong need to develop a quantitative measure to effectively describe nanoparticle dispersion from a TEM/SEM image. This article reports an effective modeling strategy to characterize nanoparticle dispersion states among different locations of a nanocomposite surface. An engineering-driven inhomogenous Poisson random field is proposed to represent the nanoparticle dispersion at the nanoscale. The model parameters are estimated through the Bayesian Markov Chain Monte Carlo technique to overcome the challenge of the limited amount of accessible data due to the time-consuming sample collection process. The TEM images taken from nano-silica/epoxy composites are used to support the proposed methodology. The research strategy and framework are generally applicable to other nanocomposite materials.