Inference Methods for Correlated Left Truncated Lifetimes: Parent and Offspring Relations in an Adoption Study

The associations in mortality of adult adoptees and their biological or adoptive parents have been studied in order to separate genetic and environmental influences. The 1003 Danish adoptees born 1924–26 have previously been analysed in a Cox regression model, using dichotomised versions of the parents’ lifetimes as covariates. This model will be referred to as the conditional Cox model, as it analyses lifetimes of adoptees conditional on parental lifetimes. Shared frailty models may be more satisfactory by using the entire observed lifetime of the parents. In a simulation study, sample size, distribution of lifetimes, truncation- and censoring patterns were chosen to illustrate aspects of the adoption dataset, and were generated from the conditional Cox model or a shared frailty model with gamma distributed frailties. First, efficiency was compared in the conditional Cox model and a shared frailty model, based on the conditional approach. For data with type 1 censoring the models showed no differences, whereas in data with random or no censoring, the models had different power in favour of the one from which data were generated. Secondly, estimation in the shared frailty model by a conditional approach or a two-stage copula approach was compared. Both approaches worked well, with no sign of dependence upon the truncation pattern, but some sign of bias depending on the censoring. For frailty parameters close to zero, we found bias when the estimation procedure used did not allow negative estimates. Based on this evaluation, we prefer to use frailty models allowing for negative frailty parameter estimates. The conclusions from earlier analyses of the adoption study were confirmed, though without greater precision than using the conditional Cox model. Analyses of associations between parental lifetimes are also presented.     

On a system of nonhomogeneous components sharing a common frailty

The components of a reliability system subjected to a common random environment usually have dependent lifetimes. This paper studies the stochastic properties of such a system with lifetimes of the components following multivariate frailty models and multivariate mixed proportional reversed hazard rate (PRHR) models, respectively. Through doing stochastic comparison, we devote to throwing a new light on how the random environment affects the number of working components of a reliability system and on assessing the performance of a k-out-of-n system.     

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.     

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.     

General frailty model and stochastic orderings

In this paper, we propose a general frailty model and develop its properties including some results for stochastic comparisons. More specifically, our main results lie in seeing how the well known stochastic orderings between distributions of two frailties translate into the orderings between the corresponding survival functions. These results are used to obtain the properties of the classical multiplicative frailty model and the additive frailty model. Several of the results, in the literature, are obtained as special cases.     

Stochastic comparisons in frailty models

The frailty model in survival analysis accounts for unobserved heterogeneity between individuals by assuming that the hazard rate of an individual is the product of an individual specific quantity, called “frailty” and a baseline hazard rate. It is well known that the choice of the frailty distribution strongly affects the nonparametric estimate of the baseline hazard as well as that of the conditional probabilities. This paper reviews the basic concepts of a frailty model, presents various probability inequalities and other monotonicity results which may prove useful in choosing among alternative specifications. More specifically, our main result lies in seeing how well known stochastic orderings between distributions of two frailities translate into orderings between the corresponding survival functions. Some probabilistic aspects and implications of the models resulting from competing choices of the distributions of frailty or the baseline are compared.     

On comparisons of population and subpopulations in frailty models

Abstract

This paper studies stochastic comparisons between a population and subpopulations in both multiplicative and additive frailty models. The comparisons between a population and its baseline in stochastic ordering are conducted as a special case. We build equivalent characterizations of some common stochastic orders between a population and a subpopulation, in terms of the frailty of the subpopulation and the first two moments of frailty variable. Some examples and applications are discussed as well.     

Stochastic properties and parameter estimation for a general load-sharing system

We study here a general load-sharing parallel system in which the lifetimes of the components of the system are arbitrary continuous random variables. The system functions if at least one component in the system functions and the surviving unit shares the whole load. Some sufficient conditions are obtained for the usual stochastic order between two different load-sharing systems. We then consider the optimal allocation problem of one load standby in a series system with two independent components. Finally, the maximum likelihood estimation of the parameters for some specific systems is discussed.     

A unified approach to stochastic comparisons of multivariate mixture models

In this paper, we consider a unified approach to stochastic comparisons of random vectors corresponding to two general multivariate mixture models. These stochastic comparisons are made with respect to multivariate hazard rate, reversed hazard rate and likelihood ratio orders. As an application, results are presented for stochastic comparisons of generalized multivariate frailty models.     

A new stochastic order based upon Laplace transform with applications

This paper deals with nonnegative random variables having Laplace transforms as their reliability functions. We study a new stochastic order based upon Laplace transform. Some applications in actuarial science, frailty models and reliability are presented as well.     

Frailty models and copulas: similarities and differences

Copulas and frailty models are important tools to model bivariate survival data. Equivalence between Archimedean copula models and shared frailty models, e.g. between the Clayton-Oakes copula model and the shared gamma frailty model, has often been claimed in the literature. In this note we show that, in both the models, there is indeed a well-known equivalence between the copula functions; the modeling of the marginal survival functions, however, is quite different. The latter fact leads to different joint survival functions.     

Comparisons between parallel systems with exponentiated generalized gamma components

Consider two parallel systems with their independent components’ lifetimes following heterogeneous exponentiated generalized gamma distributions, where the heterogeneity is in both shape and scale parameters. We then obtain the usual stochastic (reversed hazard rate) order between the lifetimes of two systems by using the weak submajorization order between the vectors of shape parameters and the p-larger (weak supermajorization) order between the vectors of scale parameters, under some restrictions on the involved parameters. Further, by reducing the heterogeneity of parameters in each system, the usual stochastic (reversed hazard rate) order mentioned above is strengthened to the hazard rate (likelihood ratio) order. Finally, two characterization results concerning the comparisons of two parallel systems, one with independent heterogeneous generalized exponential components and another with independent homogeneous generalized exponential components, are derived. These characterization results enable us to find some lower and upper bounds for the hazard rate and reversed hazard rate functions of a parallel system consisting of independent heterogeneous generalized exponential components. The results established here generalize some of the known results in the literature, concerning the comparisons of parallel systems under generalized exponential and exponentiated Weibull models.     

On proportional mean past lifetimes model

ABSTRACT

In this paper, we first investigate some reliability properties in the proportional mean past lifetimes model. Specifically, some implications of stochastic orders and aging notions between random variables which have proportional mean past lifetimes are discussed. Then, as an extension, mixture model arising from the proportional mean past lifetimes model is introduced and preservation properties of some stochastic orders and aging notions concerning this mixture model are studied. We also study some negative dependence properties in the proposed mixture model.     

Estimation in the Positive Stable Shared Frailty Cox Proportional Hazards Model

Shared frailty models are of interest when one has clustered survival data and when focus is on comparing the lifetimes within clusters and further on estimating the correlation between lifetimes from the same cluster. It is well known that the positive stable model should be preferred to the gamma model in situations where the correlated survival data show a decreasing association with time. In this paper, we devise a likelihood based estimation procedure for the positive stable shared frailty Cox model, which is expected to obtain high efficiency. The proposed estimator is provided with large sample properties and also a consistent estimator of standard errors is given. Simulation studies show that the estimation procedure is appropriate for practical use, and that it is much more efficient than a recently suggested procedure. The suggested methodology is applied to a dataset concerning time to blindness for patients with diabetic retinopathy.     

Hierarchical-Likelihood Approach for Mixed Linear Models with Censored Data

Mixed linear models describe the dependence via random effects in multivariate normal survival data. Recently they have received considerable attention in the biomedical literature. They model the conditional survival times, whereas the alternative frailty model uses the conditional hazard rate. We develop an inferential method for the mixed linear model via Lee and Nelder's (1996) hierarchical-likelihood (h-likelihood). Simulation and a practical example are presented to illustrate the new method.     

Maximum likelihood from spatial random effects models via the stochastic approximation expectation maximization algorithm

We introduce a class of spatial random effects models that have Markov random fields (MRF) as latent processes. Calculating the maximum likelihood estimates of unknown parameters in SREs is extremely difficult, because the normalizing factors of MRFs and additional integrations from unobserved random effects are computationally prohibitive. We propose a stochastic approximation expectation-maximization (SAEM) algorithm to maximize the likelihood functions of spatial random effects models. The SAEM algorithm integrates recent improvements in stochastic approximation algorithms; it also includes components of the Newton-Raphson algorithm and the expectation-maximization (EM) gradient algorithm. The convergence of the SAEM algorithm is guaranteed under some mild conditions. We apply the SAEM algorithm to three examples that are representative of real-world applications: a state space model, a noisy Ising model, and segmenting magnetic resonance images (MRI) of the human brain. The SAEM algorithm gives satisfactory results in finding the maximum likelihood estimate of spatial random effects models in each of these instances.     

On hazard rate ordering of parallel systems with two independent components

This note builds a sufficient condition for the hazard rate ordering between lifetimes of parallel systems with two independent components having proportional hazard rates. Some comparisons on lifetimes of such systems with general components are also obtained.     

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.     

Chi-Squared Goodness-of-Fit Tests for Parametric Accelerated Failure Time Models

We give chi-squared goodness-of fit tests for parametric regression models such as accelerated failure time, proportional hazards, generalized proportional hazards, frailty models, transformation models, and models with cross-effects of survival functions. Random right censored data are used. Choice of random grouping intervals as data functions is considered.     

Estimation in Semiparametric Marginal Shared Gamma Frailty Models

The semiparametric marginal shared frailty models in survival analysis have the non–parametric hazard functions multiplied by a random frailty in each cluster, and the survival times conditional on frailties are assumed to be independent. In addition, the marginal hazard functions have the same form as in the usual Cox proportional hazard models. In this paper, an approach based on maximum likelihood and expectation–maximization is applied to semiparametric marginal shared gamma frailty models, where the frailties are assumed to be gamma distributed with mean 1 and variance θ. The estimates of the fixed–effect parameters and their standard errors obtained using this approach are compared in terms of both bias and efficiency with those obtained using the extended marginal approach. Similarly, the standard errors of our frailty variance estimates are found to compare favourably with those obtained using other methods. The asymptotic distribution of the frailty variance estimates is shown to be a 50–50 mixture of a point mass at zero and a truncated normal random variable on the positive axis for θ₀ = 0. Simulations demonstrate that, for θ₀ < 0, it is approximately an x −(100 − x )%, 0 ≤ x ≤ 50, mixture between a point mass at zero and a truncated normal random variable on the positive axis for small samples and small values of θ₀; otherwise, it is approximately normal.