Stochastic comparisons of multivariate frailty models

A multivariate frailty model in which survival function depends on baseline distributions of components and the frailty random variable is considered. Since misspecification in choice of frailty distribution and/or baseline distribution may affect the distribution of multivariate frailty model, using theory of stochastic orders, we compare multivariate frailty models arising from different choices of frailty distribution.     

Impact of misspecified residual correlation structure on the parameter estimates in a shared spatial frailty model

In practice, survival data are often collected over geographical regions. Shared spatial frailty models have been used to model spatial variation in survival times, which are often implemented using the Bayesian Markov chain Monte Carlo method. However, this method comes at the price of slow mixing rates and heavy computational cost, which may render it impractical for data-intensive application. Alternatively, a frailty model assuming an independent and identically distributed (iid) random effect can be easily and efficiently implemented. Therefore, we used simulations to assess the bias and efficiency loss in the estimated parameters, if residual spatial correlation is present but using an iid random effect. Our simulations indicate that a shared frailty model with an iid random effect can estimate the regression coefficients reasonably well, even with residual spatial correlation present, when the percentage of censoring is not too high and the number of clusters and cluster size are not too low. Therefore, if the primary goal is to assess the covariate effects, one may choose the frailty model with an iid random effect; whereas if the goal is to predict the hazard, additional care needs to be given due to the efficiency loss in the parameter(s) for the baseline hazard.     

Residual-based specification of the random-effects distribution for cluster data

We propose a method for specifying the distribution of random effects included in a model for cluster data. The class of models we consider includes mixed models and frailty models whose random effects and explanatory variables are constant within clusters. The method is based on cluster residuals obtained by assuming that the random effects are equal between clusters. We exhibit an asymptotic relationship between the cluster residuals and variations of the random effects as the number of observations increases and the variance of the random effects decreases. The asymptotic relationship is used to specify the random-effects distribution. The method is applied to a frailty model and a model used to describe the spread of plant diseases.     

Assessing gamma frailty models for clustered failure time data

Proportional hazards frailty models use a random effect, so called frailty, to construct association for clustered failure time data. It is customary to assume that the random frailty follows a gamma distribution. In this paper, we propose a graphical method for assessing adequacy of the proportional hazards frailty models. In particular, we focus on the assessment of the gamma distribution assumption for the frailties. We calculate the average of the posterior expected frailties at several followup time points and compare it at these time points to 1, the known mean frailty. Large discrepancies indicate lack of fit. To aid in assessing the goodness of fit, we derive and estimate the standard error of the mean of the posterior expected frailties at each time point examined. We give an example to illustrate the proposed methodology and perform sensitivity analysis by simulations.     

Continuous covariate frailty models for censored and truncated clustered data

Using some logarithmic and integral transformation we transform a continuous covariate frailty model into a polynomial regression model with a random effect. The responses of this mixed model can be ‘estimated’ via conditional hazard function estimation. The random error in this model does not have zero mean and its variance is not constant along the covariate and, consequently, these two quantities have to be estimated. Since the asymptotic expression for the bias is complicated, the two-large-bandwidth trick is proposed to estimate the bias. The proposed transformation is very useful for clustered incomplete data subject to left truncation and right censoring (and for complex clustered data in general). Indeed, in this case no standard software is available to fit the frailty model, whereas for the transformed model standard software for mixed models can be used for estimating the unknown parameters in the original frailty model. A small simulation study illustrates the good behavior of the proposed method. This method is applied to a bladder cancer data set.     

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Estimation and variable selection for partially functional linear models

In this paper, a new estimation procedure based on composite quantile regression and functional principal component analysis (PCA) method is proposed for the partially functional linear regression models (PFLRMs). The proposed estimation method can simultaneously estimate both the parametric regression coefficients and functional coefficient components without specification of the error distributions. The proposed estimation method is shown to be more efficient empirically for non-normal random error, especially for Cauchy error, and almost as efficient for normal random errors. Furthermore, based on the proposed estimation procedure, we use the penalized composite quantile regression method to study variable selection for parametric part in the PFLRMs. Under certain regularity conditions, consistency, asymptotic normality, and Oracle property of the resulting estimators are derived. Simulation studies and a real data analysis are conducted to assess the finite sample performance of the proposed methods.     

Modeling multiple‐response categorical data from complex surveys

Although “choose all that apply” questions are common in modern surveys, methods for analyzing associations among responses to such questions have only recently been developed. These methods are generally valid only for simple random sampling, but these types of questions often appear in surveys conducted under more complex sampling plans. The purpose of this article is to provide statistical analysis methods that can be applied to “choose all that apply” questions in complex survey sampling situations. Loglinear models are developed to incorporate the multiple responses inherent in these types of questions. Statistics to compare models and to measure association are proposed and their asymptotic distributions are derived. Monte Carlo simulations show that tests based on adjusted Pearson statistics generally hold their correct size when comparing models. These simulations also show that confidence intervals for odds ratios estimated from loglinear models have good coverage properties, while being shorter than those constructed using empirical estimates. Furthermore, the methods are shown to be applicable to more general problems of modeling associations between elements of two or more binary vectors. The proposed analysis methods are applied to data from the National Health and Nutrition Examination Survey. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

Multilevel Mixed Linear Models for Survival Data

For the analysis of correlated survival data mixed linear models are useful alternatives to frailty models. By their use the survival times can be directly modelled, so that the interpretation of the fixed and random effects is straightforward. However, because of intractable integration involved with the use of marginal likelihood the class of models in use has been severely restricted. Such a difficulty can be avoided by using hierarchical-likelihood, which provides a statistically efficient and fast fitting algorithm for multilevel models. The proposed method is illustrated using the chronic granulomatous disease data. A simulation study is carried out to evaluate the performance.     

Maximum Penalized Likelihood Estimation in a Gamma-Frailty Model

The shared frailty models allow for unobserved heterogeneity or for statistical dependence between observed survival data. The most commonly used estimation procedure in frailty models is the EM algorithm, but this approach yields a discrete estimator of the distribution and consequently does not allow direct estimation of the hazard function. We show how maximum penalized likelihood estimation can be applied to nonparametric estimation of a continuous hazard function in a shared gamma-frailty model with right-censored and left-truncated data. We examine the problem of obtaining variance estimators for regression coefficients, the frailty parameter and baseline hazard functions. Some simulations for the proposed estimation procedure are presented. A prospective cohort (Paquid) with grouped survival data serves to illustrate the method which was used to analyze the relationship between environmental factors and the risk of dementia.     

Semiparametric mixed-effects models for clustered doubly censored data

The Cox proportional frailty model with a random effect has been proposed for the analysis of right-censored data which consist of a large number of small clusters of correlated failure time observations. For right-censored data, Cai et al. [3] proposed a class of semiparametric mixed-effects models which provides useful alternatives to the Cox model. We demonstrate that the approach of Cai et al. [3] can be used to analyze clustered doubly censored data when both left- and right-censoring variables are always observed. The asymptotic properties of the proposed estimator are derived. A simulation study is conducted to investigate the performance of the proposed estimator.     

Some stochastic properties of conditionally dependent frailty models

The frailty approach is commonly used in reliability theory and survival analysis to model the dependence between lifetimes of individuals or components subject to common risk factors; according to this model the frailty (an unobservable random vector that describes environmental conditions) acts simultaneously on the hazard functions of the lifetimes. Some interesting conditions for stochastic comparisons between random vectors defined in accordance with these models have been described in the literature; in particular, comparisons between frailty models have been studied by assuming independence for the baseline survival functions and the corresponding environmental parameters. In this paper, a generalization of these models is developed, which assumes conditional dependence between the components of the random vector, and some conditions for stochastic comparisons are provided. Some examples of frailty models satisfying these conditions are also described.     

The analysis of tumorigenicity data using a frailty effect

In animal tumorigenicity data, the time of occurrence of the tumor is not observed because the existence of the tumor is looked for only at either the time of death or the time of sacrifice of the animal. Such an incomplete data structure makes it difficult to investigate the impact of treatment on the occurrence of tumors. A three-state model (no tumor–tumor–death) is used to model events that occurred sequentially and to connect them. In this paper, we also employed a frailty effect to model the dependency of death on tumor occurrence. For the inference of parameters, an EM algorithm is considered. The method is applied to a real bladder tumor data set and a simulation study is performed to show the behavior of the proposed estimators.     

A resampling approach to estimate variance components of multilevel models

In a multilevel model for complex survey data, the weight‐inflated estimators of variance components can be biased. We propose a resampling method to correct this bias. The performance of the bias corrected estimators is studied through simulations using populations generated from a simple random effects model. The simulations show that, without lowering the precision, the proposed procedure can reduce the bias of the estimators, especially for designs that are both informative and have small cluster sizes. Application of these resampling procedures to data from an artificial workplace survey provides further evidence for the empirical value of this method. The Canadian Journal of Statistics 40: 150–171; 2012 © 2012 Statistical Society of Canada     

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.     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

Additive transformation models for clustered failure time data

We propose a class of additive transformation risk models for clustered failure time data. Our models are motivated by the usual additive risk model for independent failure times incorporating a frailty with mean one and constant variability which is a natural generalization of the additive risk model from univariate failure time to multivariate failure time. An estimating equation approach based on the marginal hazards function is proposed. Under the assumption that cluster sizes are completely random, we show the resulting estimators of the regression coefficients are consistent and asymptotically normal. We also provide goodness-of-fit test statistics for choosing the transformation. Simulation studies and real data analysis are conducted to examine the finite-sample performance of our estimators.     

Information attainable in some randomly incomplete multivariate response models

In a general parametric setup, a multivariate regression model is considered when responses may be missing at random while the explanatory variables and covariates are completely observed. Asymptotic optimality properties of maximum likelihood estimators for such models are linked to the Fisher information matrix for the parameters. It is shown that the information matrix is well defined for the missing-at-random model and that it plays the same role as in the complete-data linear models. Applications of the methodologic developments in hypothesis-testing problems, without any imputation of missing data, are illustrated. Some simulation results comparing the proposed method with Rubin's multiple imputation method are presented.     

Semiparametric median residual life model and inference

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada