Estimating Frailty Models via Poisson Hierarchical Generalized Linear Models

A general class of small area estimation using calibrated hierarchical likelihood approach with applications to COVID-19 data

The direct estimation techniques in small area estimation (SAE) models require sufficiently large sample sizes to provide accurate estimates. Hence, indirect model-based methodologies are developed to incorporate auxiliary information. The most commonly used SAE models, including the Fay-Herriot (FH) model and its extended models, are estimated using marginal likelihood estimation and the Bayesian methods, which rely heavily on the computationally intensive integration of likelihood function. In this article, we propose a Calibrated Hierarchical (CH) likelihood approach to obtain SAE through hierarchical estimation of fixed effects and random effects with the regression calibration method for bias correction. The latent random variables at the domain level are treated as 'parameters' and estimated jointly with other parameters of interest. Then the dispersion parameters are estimated iteratively based on the Laplace approximation of the profile likelihood. The proposed method avoids the intractable integration to estimate the marginal distribution. Hence, it can be applied to a wide class of distributions, including generalized linear mixed models, survival analysis, and joint modeling with distinct distributions. We demonstrate our method using an area-level analysis of publicly available count data from the novel coronavirus (COVID-19) positive cases.

Multi-parameter regression survival modelling with random effects

We consider a parametric modelling approach for survival data where covariates are allowed to enter the model through multiple distributional parameters (i.e., scale and shape). This is in contrast with the standard convention of having a single covariate-dependent parameter, typically the scale. Taking what is referred to as a multi-parameter regression (MPR) approach to modelling has been shown to produce flexible and robust models with relatively low model complexity cost. However, it is very common to have clustered data arising from survival analysis studies, and this is something that is under developed in the MPR context. The purpose of this article is to extend MPR models to handle multivariate survival data by introducing random effects in both the scale and the shape regression components. We consider a variety of possible dependence structures for these random effects (independent, shared and correlated), and estimation proceeds using a h-likelihood approach. The performance of our estimation procedure is investigated by a way of an extensive simulation study, and the merits of our modelling approach are illustrated through applications to two real data examples, a lung cancer dataset and a bladder cancer dataset.

Big data ordination towards intensive care event count cases using fast computing GLLVMS

BACKGROUND

In heart data mining and machine learning, dimension reduction is needed to remove multicollinearity. Meanwhile, it has been proven to improve the interpretation of the parameter model. In addition, dimension reduction can also increase the time of computing in high dimensional data.

METHODS

In this paper, we perform high dimensional ordination towards event counts in intensive care hospital for Emergency Department (ED 1), First Intensive Care Unit (ICU1), Second Intensive Care Unit (ICU2), Respiratory Care Intensive Care Unit (RICU), Surgical Intensive Care Unit (SICU), Subacute Respiratory Care Unit (RCC), Trauma and Neurosurgery Intensive Care Unit (TNCU), Neonatal Intensive Care Unit (NICU) which use the Generalized Linear Latent Variable Models (GLLVM's).

RESULTS

During the analysis, we measure the performance and calculate the time computing of GLLVM by employing variational approximation and Laplace approximation, and compare the different distributions, including Negative Binomial, Poisson, Gaussian, ZIP, and Tweedie, respectively. GLLVMs (Generalized Linear Latent Variable Models), an extended version of GLMs (Generalized Linear Models) with latent variables, have fast computing time. The major challenge in latent variable modelling is that the function [Formula: see text] is not trivial to solve since the marginal likelihood involves integration over the latent variable u.

CONCLUSIONS

In a nutshell, GLLVMs lead as the best performance reaching the variance of 98% comparing other methods. We get the best model negative binomial and Variational approximation, which provides the best accuracy by accuracy value of AIC, AICc, and BIC. In a nutshell, our best model is GLLVM-VA Negative Binomial with AIC 7144.07 and GLLVM-LA Negative Binomial with AIC 6955.922.

Penalized variable selection for cause-specific hazard frailty models with clustered competing-risks data

Competing risks data usually arise when an occurrence of an event precludes other types of events from being observed. Such data are often encountered in a clustered clinical study such as a multi-center clinical trial. For the clustered competing-risks data which are correlated within a cluster, competing-risks models allowing for frailty terms have been recently studied. To the best of our knowledge, however, there is no literature on variable selection methods for cause-specific hazard frailty models. In this article, we propose a variable selection procedure for fixed effects in cause-specific competing risks frailty models using a penalized h-likelihood (HL). Here, we study three penalty functions, LASSO, SCAD, and HL. Simulation studies demonstrate that the proposed procedure using the HL penalty works well, providing a higher probability of choosing the true model than LASSO and SCAD methods without losing prediction accuracy. The proposed method is illustrated by using two kinds of clustered competing-risks cancer data sets.

Semiparametric estimation for nonparametric frailty models using nonparametric maximum likelihood approach

A consequence of using a parametric frailty model with nonparametric baseline hazard for analyzing clustered time-to-event data is that its regression coefficient estimates could be sensitive to the underlying frailty distribution. Recently, there has been a proposal for specifying both the baseline hazard and the frailty distribution nonparametrically, and estimating the unknown parameters by the maximum penalized likelihood method. Instead, in this paper, we propose the nonparametric maximum likelihood method for a general class of nonparametric frailty models, i.e. models where the frailty distribution is completely unspecified but the baseline hazard can be either parametric or nonparametric. The implementation of the estimation procedure can be based on a combination of either the Broyden-Fletcher-Goldfarb-Shanno or expectation-maximization algorithm and the constrained Newton algorithm with multiple support point inclusion. Simulation studies to investigate the performance of estimation of a regression coefficient by several different model-fitting methods were conducted. The simulation results show that our proposed regression coefficient estimator generally gives a reasonable bias reduction when the number of clusters is increased under various frailty distributions. Our proposed method is also illustrated with two data examples.

A Case Study on Modeling the Time to Recurrence of Gastric Cancer Patients

BACKGROUND

Gastric cancer is a malignant tumor of the stomach and it is one of the leading causes of death worldwide. The study aimed to model the time to first recurrence of gastric cancer patients at the Tikur Anbesa specialized hospital.

METHODS

The data for this study were gastric cancer patients followed up from January 1, 2013 to February 29, 2020 at Tikur Anbesa Specialized Hospital, Oncology Center, Addis Ababa. We used Weibull, log-logistic and lognormal as baseline hazard functions with the gamma and the inverse Gaussian frailty distributions. Data analyzed with the statistical software R.

RESULTS

The median recurrence time of the patients was about 23.96 months with a maximum recurrence time of 60.81 months, of which about 61.2% had first recurrences of gastric cancer. The clustering effect is significant in modeling the time to recurrence of gastric cancer. According to the result of the log-logistic inverse Gaussian frailty model, the sex of the patient, the tumor size, smoking habit, the treatment carried out, the vascular invasion, the stage of the disease, the helicobacter pylori infection and the histological type were the significant prognostic factors  at 5% level of significance.

CONCLUSION

Inverse Gaussian frailty model is the model that best describes the time to recurrence of the gastric cancer data set. Gender of the patients, tumor size, treatment taken, vascular invasion, disease stage, helicobacter pylori infection and histological type were the determining prognostic factors. This requires measures to improve patient health and prevent relapse based on significant risk factors, and particular attention should be paid to patients with such factors.

Flexible parametric copula modeling approaches for clustered survival data

Copula-based survival regression models, which consist of a copula function and marginal distribution (i.e., marginal survival function), have been widely used for analyzing clustered multivariate survival data. Archimedean copula functions are useful for modeling such dependence. For the likelihood inference, one-stage and two-stage estimation methods have been usually used. The two-stage procedure can give inefficient estimation results because of separate estimation of the marginal and copula's dependence parameters. The more efficient one-stage procedure has been mainly developed under a restrictive parametric assumption of marginal distribution due to complexity of the full likelihood with unknown marginal baseline hazard functions. In this paper, we propose a flexible parametric Archimedean copula modeling approach using a one-stage likelihood procedure. In order to reduce the complexity of the full likelihood, the unknown marginal baseline hazards are modeled based on a cubic M-spline basis function that does not require a specific parametric form. Simulation results demonstrate that the proposed one-stage estimation method gives a consistent estimator and also provides more efficient results over existing one- and two-stage methods. The new method is illustrated with three clinical data sets. The Appendix provides an R function so that the proposed method becomes directly accessible to interested readers.

Did Noise Pollution Really Improve during COVID-19? Evidence from Taiwan

Background and objectives: The impacts of COVID-19 are like two sides of one coin. During 2020, there were many research papers that proved our environmental and climate conditions were improving due to lockdown or large-scale restriction regulations. In contrast, the economic conditions deteriorated due to disruption in industry business activities and most people stayed at home and worked from home, which probably reduced the noise pollution. Methods: To assess whether there were differences in noise pollution before and during COVID-19. In this paper, we use various statistical methods following odds ratios, Wilcoxon and Fisher’s tests and Bayesian Markov chain Monte Carlo (MCMC) with various comparisons of prior selection. The outcome of interest for a parameter in Bayesian inference is complete posterior distribution. Roughly, the mean of the posterior will be clear with point approximation. That being said, the median is an available choice. Findings: To make the Bayesian MCMC work, we ran the sampling from the conditional posterior distributions. It is straightforward to draw random samples from these distributions if they have regular shapes using MCMC. The case of over-standard noise per time frame, number of noise petition cases, number of industry petition cases, number of motorcycles, number of cars and density of vehicles are significant at α=5%. In line with this, we prove that there were differences of noise pollution before and during COVID-19 in Taiwan. Meanwhile, the decreased noise pollution in Taiwan can improve quality of life.

Frailty modelling approaches for semi-competing risks data

In the semi-competing risks situation where only a terminal event censors a non-terminal event, observed event times can be correlated. Recently, frailty models with an arbitrary baseline hazard have been studied for the analysis of such semi-competing risks data. However, their maximum likelihood estimator can be substantially biased in the finite samples. In this paper, we propose effective modifications to reduce such bias using the hierarchical likelihood. We also investigate the relationship between marginal and hierarchical likelihood approaches. Simulation results are provided to validate performance of the proposed method. The proposed method is illustrated through analysis of semi-competing risks data from a breast cancer study.

Impact of model misspecification in shared frailty survival models

Survival models incorporating random effects to account for unmeasured heterogeneity are being increasingly used in biostatistical and applied research. Specifically, unmeasured covariates whose lack of inclusion in the model would lead to biased, inefficient results are commonly modeled by including a subject-specific (or cluster-specific) frailty term that follows a given distribution (eg, gamma or lognormal). Despite that, in the context of parametric frailty models, little is known about the impact of misspecifying the baseline hazard or the frailty distribution or both. Therefore, our aim is to quantify the impact of such misspecification in a wide variety of clinically plausible scenarios via Monte Carlo simulation, using open-source software readily available to applied researchers. We generate clustered survival data assuming various baseline hazard functions, including mixture distributions with turning points, and assess the impact of sample size, variance of the frailty, baseline hazard function, and frailty distribution. Models compared include standard parametric distributions and more flexible spline-based approaches; we also included semiparametric Cox models. The resulting bias can be clinically relevant. In conclusion, we highlight the importance of fitting models that are flexible enough and the importance of assessing model fit. We illustrate our conclusions with two applications using data on diabetic retinopathy and bladder cancer. Our results show the importance of assessing model fit with respect to the baseline hazard function and the distribution of the frailty: misspecifying the former leads to biased relative and absolute risk estimates, whereas misspecifying the latter affects absolute risk estimates and measures of heterogeneity.

H-likelihood approach for joint modeling of longitudinal outcomes and time-to-event data

In longitudinal studies, a subject may have different types of outcomes that could be correlated. For example, a response variable of interest would be measured repeatedly over time on the same subject and at the same time, an event time representing a single event or competing-risks event is also observed. In this paper, we propose a joint modeling framework that accounts for the inherent association between such multiple outcomes via frailties (unobserved random effects). Among outcomes, at least one outcome is an event time that has a type of a single event or competing-risks event. For inference we use the hierarchical likelihood (h-likelihood) that provides an unified efficient fitting procedure for the joint models. Numerical studies are provided to show the performance of the proposed method and two data examples are shown.

Analysis of clustered competing risks data using subdistribution hazard models with multivariate frailties

Competing risks data often exist within a center in multi-center randomized clinical trials where the treatment effects or baseline risks may vary among centers. In this paper, we propose a subdistribution hazard regression model with multivariate frailty to investigate heterogeneity in treatment effects among centers from multi-center clinical trials. For inference, we develop a hierarchical likelihood (or h-likelihood) method, which obviates the need for an intractable integration over the frailty terms. We show that the profile likelihood function derived from the h-likelihood is identical to the partial likelihood, and hence it can be extended to the weighted partial likelihood for the subdistribution hazard frailty models. The proposed method is illustrated with a dataset from a multi-center clinical trial on breast cancer as well as with a simulation study. We also demonstrate how to present heterogeneity in treatment effects among centers by using a confidence interval for the frailty for each individual center and how to perform a statistical test for such heterogeneity using a restricted h-likelihood.

Frailty modeling for clustered competing risks data with missing cause of failure

Competing risks data often occur within a center in multi-center clinical trials where the event times within a center may be correlated due to unobserved factors across individuals. In this paper, we consider the cause-specific proportional hazards model with a shared frailty to model the association between the event times within a center in the framework of competing risks. We use a hierarchical likelihood approach, which does not require any intractable integration over the frailty terms. In a clinical trial, cause of death information may not be observed for some patients. In such a case, analyses through exclusion of cases with missing cause of death may lead to biased inferences. We propose a hierarchical likelihood approach for fitting the cause-specific proportional hazards model with a shared frailty in the presence of missing cause of failure. We use multiple imputation methods to address missing cause of death information under the assumption of missing at random. Simulation studies show that the proposed procedures perform well, even if the imputation model is misspecified. The proposed methods are illustrated with data from EORTC trial 30791 conducted by European Organization for Research and Treatment of Cancer (EORTC).

Hierarchical likelihood inference on clustered competing risks data

The frailty model, an extension of the proportional hazards model, is often used to model clustered survival data. However, some extension of the ordinary frailty model is required when there exist competing risks within a cluster. Under competing risks, the underlying processes affecting the events of interest and competing events could be different but correlated. In this paper, the hierarchical likelihood method is proposed to infer the cause-specific hazard frailty model for clustered competing risks data. The hierarchical likelihood incorporates fixed effects as well as random effects into an extended likelihood function, so that the method does not require intensive numerical methods to find the marginal distribution. Simulation studies are performed to assess the behavior of the estimators for the regression coefficients and the correlation structure among the bivariate frailty distribution for competing events. The proposed method is illustrated with a breast cancer dataset.

Mixed effects models for recurrent events data with partially observed time-varying covariates: Ecological momentary assessment of smoking

Cigarette smoking is a prototypical example of a recurrent event. The pattern of recurrent smoking events may depend on time-varying covariates including mood and environmental variables. Fixed effects and frailty models for recurrent events data assume that smokers have a common association with time-varying covariates. We develop a mixed effects version of a recurrent events model that may be used to describe variation among smokers in how they respond to those covariates, potentially leading to the development of individual-based smoking cessation therapies. Our method extends the modified EM algorithm of Steele (1996) for generalized mixed models to recurrent events data with partially observed time-varying covariates. It is offered as an alternative to the method of Rizopoulos, Verbeke, and Lesaffre (2009) who extended Steele's (1996) algorithm to a joint-model for the recurrent events data and time-varying covariates. Our approach does not require a model for the time-varying covariates, but instead assumes that the time-varying covariates are sampled according to a Poisson point process with known intensity. Our methods are well suited to data collected using Ecological Momentary Assessment (EMA), a method of data collection widely used in the behavioral sciences to collect data on emotional state and recurrent events in the every-day environments of study subjects using electronic devices such as Personal Digital Assistants (PDA) or smart phones.

Modeling time-to-cure from severe acute malnutrition: application of various parametric frailty models

Background

In developing countries about 3.5% of children aged 0–5 years are victims of severe acute malnutrition (SAM). Once the morbidity has developed the cure process takes variable period depending on various factors. Knowledge of time-to-cure from SAM will enable health care providers to plan resources and monitor the progress of cases with SAM. The current analysis presents modeling time-to-cure from SAM starting from the day of diagnosis in Wolisso St. Luke Catholic hospital, southwest Ethiopia.

Methods

With the aim of coming up with appropriate survival (time-to-event) model that describes the SAM dataset, various parametric clustered time-to-event (frailty) models were compared. Frailty model, which is an extension of the proportional hazards Cox survival model, was used to analyze time-to-cure from SAM. Kebeles (villages) of the children were considered as the clustering variable in all the models. We used exponential, weibull and log-logistic as baseline hazard functions and the gamma as well as inverse Gaussian for the frailty distributions and then based on AIC criteria, all models were compared for their performance.

Results

The median time-to-cure from SAM cases was 14 days with the maximum of 63 days of which about 83% were cured. The log-logistic model with inverse Gaussian frailty has the minimum AIC value among the models compared. The clustering effect was significant in modeling time-to-cure from SAM. The results showed that age of a child and co-infection were the determinant prognostic factors for SAM, but sex of the child and the type of malnutrition were not significant.

Conclusions

The log-logistic with inverse Gaussian frailty model described the SAM dataset better than other distributions used in this study. There is heterogeneity between the kebeles in the time-to-cure from SAM, indicating that one needs to account for this clustering variable using appropriate clustered time-to-event frailty models.

Variable selection in subdistribution hazard frailty models with competing risks data

The proportional subdistribution hazards model (i.e. Fine-Gray model) has been widely used for analyzing univariate competing risks data. Recently, this model has been extended to clustered competing risks data via frailty. To the best of our knowledge, however, there has been no literature on variable selection method for such competing risks frailty models. In this paper, we propose a simple but unified procedure via a penalized h-likelihood (HL) for variable selection of fixed effects in a general class of subdistribution hazard frailty models, in which random effects may be shared or correlated. We consider three penalty functions, least absolute shrinkage and selection operator (LASSO), smoothly clipped absolute deviation (SCAD) and HL, in our variable selection procedure. We show that the proposed method can be easily implemented using a slight modification to existing h-likelihood estimation approaches. Numerical studies demonstrate that the proposed procedure using the HL penalty performs well, providing a higher probability of choosing the true model than LASSO and SCAD methods without losing prediction accuracy. The usefulness of the new method is illustrated using two actual datasets from multi-center clinical trials.

Coping with time and space in modelling malaria incidence: a comparison of survival and count regression models

To study the effect of a mega hydropower dam in southwest Ethiopia on malaria incidence, we have set up a longitudinal study. To gain insight in temporal and spatial aspects, that is, in time (period  =  year-season combination) and location (village), we need models that account for these effects. The frailty model with periodwise constant baseline hazard (a constant value for each period) and a frailty term that models the clustering in villages provides an appropriate tool for the analysis of such incidence data. Count data can be obtained by aggregating for each period events at the village level. The mixed Poisson regression model can be used to model the count data. We show the similarities between the two models. The risk factor in both models is the distance to the dam, and we study the effect of the risk factor on malaria incidence. In the frailty model, each subject has its own risk factor, whereas in the Poisson regression model, we also need to average the risk factors of all subjects contributing to a particular count. The power loss caused by using village averaged distance instead of individual distance is studied and quantified. The loss in the malaria data example is rather small. In such a setting, it might be advantageous to use less labor-intensive sampling schemes than the weekly individual follow-up scheme used in this study; the proposed alternative sampling schemes might also avoid community fatigue, a typical problem in such research projects.

Interval estimation of random effects in proportional hazards models with frailties

Semi-parametric frailty models are widely used to analyze clustered survival data. In this article, we propose the use of the hierarchical likelihood interval for individual frailties. We study the relationship between hierarchical likelihood, empirical Bayesian, and fully Bayesian intervals for frailties. We show that our proposed interval can be interpreted as a frequentist confidence interval and Bayesian credible interval under a uniform prior. We also propose an adjustment of the proposed interval to avoid null intervals. Simulation studies show that the proposed interval preserves the nominal confidence level. The procedure is illustrated using data from a multicenter lung cancer clinical trial.

Mixed-Poisson Point Process with Partially-Observed Covariates: Ecological Momentary Assessment of Smoking

Ecological Momentary Assessment is an emerging method of data collection in behavioral research that may be used to capture the times of repeated behavioral events on electronic devices, and information on subjects' psychological states through the electronic administration of questionnaires at times selected from a probability-based design as well as the event times. A method for fitting a mixed Poisson point process model is proposed for the impact of partially-observed, time-varying covariates on the timing of repeated behavioral events. A random frailty is included in the point-process intensity to describe variation among subjects in baseline rates of event occurrence. Covariate coefficients are estimated using estimating equations constructed by replacing the integrated intensity in the Poisson score equations with a design-unbiased estimator. An estimator is also proposed for the variance of the random frailties. Our estimators are robust in the sense that no model assumptions are made regarding the distribution of the time-varying covariates or the distribution of the random effects. However, subject effects are estimated under gamma frailties using an approximate hierarchical likelihood. The proposed approach is illustrated using smoking data.

Conditional Akaike information under generalized linear and proportional hazards mixed models

We study model selection for clustered data, when the focus is on cluster specific inference. Such data are often modelled using random effects, and conditional Akaike information was proposed in Vaida & Blanchard (2005) and used to derive an information criterion under linear mixed models. Here we extend the approach to generalized linear and proportional hazards mixed models. Outside the normal linear mixed models, exact calculations are not available and we resort to asymptotic approximations. In the presence of nuisance parameters, a profile conditional Akaike information is proposed. Bootstrap methods are considered for their potential advantage in finite samples. Simulations show that the performance of the bootstrap and the analytic criteria are comparable, with bootstrap demonstrating some advantages for larger cluster sizes. The proposed criteria are applied to two cancer datasets to select models when the cluster-specific inference is of interest.

Frailty modelling for survival data from multi-centre clinical trials

Despite the use of standardized protocols in, multi-centre, randomized clinical trials, outcome may vary between centres. Such heterogeneity may alter the interpretation and reporting of the treatment effect. Below, we propose a general frailty modelling approach for investigating, inter alia, putative treatment-by-centre interactions in time-to-event data in multi-centre clinical trials. A correlated random effects model is used to model the baseline risk and the treatment effect across centres. It may be based on shared, individual or correlated random effects. For inference we develop the hierarchical-likelihood (or h-likelihood) approach which facilitates computation of prediction intervals for the random effects with proper precision. We illustrate our methods using disease-free time-to-event data on bladder cancer patients participating in an European Organization for Research and Treatment of Cancer trial, and a simulation study. We also demonstrate model selection using h-likelihood criteria.

Robust frailty modelling using non-proportional hazards models

Correlated survival times can be modelled by introducing a random effect, or frailty component, into the hazard function. For multivariate survival data, we extend a non-proportional hazards (PH) model, the generalized time-dependent logistic survival model, to include random effects. The hierarchical likelihood procedure, which obviates the need for marginalization over the random effect distribution, is derived for this extended model and its properties are discussed. The extended model leads to a robust estimation result for the regression parameters against the misspecification of the form of the basic hazard function or frailty distribution compared to PH-based alternatives. The proposed method is illustrated by two practical examples and a simulation study which demonstrate the advantages of the new model.

The use of Gaussian quadrature for estimation in frailty proportional hazards models

In this paper, we propose a novel Gaussian quadrature estimation method in various frailty proportional hazards models. We approximate the unspecified baseline hazard by a piecewise constant one, resulting in a parametric model that can be fitted conveniently by Gaussian quadrature tools in standard software such as SAS Proc NLMIXED. We first apply our method to simple frailty models for correlated survival data (e.g. recurrent or clustered failure times), then to joint frailty models for correlated failure times with informative dropout or a dependent terminal event such as death. Simulation studies show that our method compares favorably with the well-received penalized partial likelihood method and the Monte Carlo EM (MCEM) method, for both normal and Gamma frailty models. We apply our method to three real data examples: (1) the time to blindness of both eyes in a diabetic retinopathy study, (2) the joint analysis of recurrent opportunistic diseases in the presence of death for HIV-infected patients, and (3) the joint modeling of local, distant tumor recurrences and patients survival in a soft tissue sarcoma study. The proposed method greatly simplifies the implementation of the (joint) frailty models and makes them much more accessible to general statistical practitioners.

Model selection for multi-component frailty models

Various frailty models have been developed and are now widely used for analysing multivariate survival data. It is therefore important to develop an information criterion for model selection. However, in frailty models there are several alternative ways of forming a criterion and the particular criterion chosen may not be uniformly best. In this paper, we study an Akaike information criterion (AIC) on selecting a frailty structure from a set of (possibly) non-nested frailty models. We propose two new AIC criteria, based on a conditional likelihood and an extended restricted likelihood (ERL) given by Lee and Nelder (J. R. Statist. Soc. B 1996; 58:619-678). We compare their performance using well-known practical examples and demonstrate that the two criteria may yield rather different results. A simulation study shows that the AIC based on the ERL is recommended, when attention is focussed on selecting the frailty structure rather than the fixed effects.

Dispersion frailty models and HGLMs

In medical research recurrent event times can be analysed using a frailty model in which the frailties for different individuals are independent and identically distributed. However, such a homogeneous assumption about frailties could sometimes be suspect. For modelling heterogeneity in frailties we describe dispersion frailty models arising from a new class of models, namely hierarchical generalized linear models. Using the kidney infection data we illustrate how to detect and model heterogeneity among frailties. Stratification of frailty models is also investigated.