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.     

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.     

Frailty Models for Arbitrarily Censored and Truncated Data

In this paper, we propose a frailty model for statistical inference in the case where we are faced with arbitrarily censored and truncated data. Our results extend those of Alioum and Commenges (1996), who developed a method of fitting a proportional hazards model to data of this kind. We discuss the identifiability of the regression coefficients involved in the model which are the parameters of interest, as well as the identifiability of the baseline cumulative hazard function of the model which plays the role of the infinite dimensional nuisance parameter. We illustrate our method with the use of simulated data as well as with a set of real data on transfusion-related AIDS.     

Estimating Survival Curves Under Proportional Hazards Model with Covariate Measurement Errors

For the Cox proportional hazards model with additive covariate measurement errors, we propose a corrected cumulative baseline hazard estimator that reduces the bias of the na]ve Breslow estimator. We also derive corresponding modified estimators for the hazard functions and the survival functions of individuals with particular covariate values. Using a Monte Carlo technique developed by Lin et al . (1994), we construct confidence bands for such hazard and survival functions.     

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.     

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.     

Proportional hazards estimate of the conditional survival function

We introduce a new estimator of the conditional survival function given some subset of the covariate values under a proportional hazards regression. The new estimate does not require estimating the base-line cumulative hazard function. An estimate of the variance is given and is easy to compute, involving only those quantities that are routinely calculated in a Cox model analysis. The asymptotic normality of the new estimate is shown by using a central limit theorem for Kaplan–Meier integrals. We indicate the straightforward extension of the estimation procedure under models with multiplicative relative risks, including non-proportional hazards, and to stratified and frailty models. The estimator is applied to a gastric cancer study where it is of interest to predict patients' survival based only on measurements obtained before surgery, the time at which the most important prognostic variable, stage, becomes known.     

Study of the bivariate survival data using frailty models based on Lévy processes

Frailty models allow us to take into account the non-observable inhomogeneity of individual hazard functions. Although models with time-independent frailty have been intensively studied over the last decades and a wide range of applications in survival analysis have been found, the studies based on the models with time-dependent frailty are relatively rare. In this paper, we formulate and prove two propositions related to the identifiability of the bivariate survival models with frailty given by a nonnegative bivariate Lévy process. We discuss parametric and semiparametric procedures for estimating unknown parameters and baseline hazard functions. Numerical experiments with simulated and real data illustrate these procedures. The statements of the propositions can be easily extended to the multivariate case.

     

Efficient Estimation of the Partly Linear Additive Hazards Model with Current Status Data

This paper focuses on efficient estimation, optimal rates of convergence and effective algorithms in the partly linear additive hazards regression model with current status data. We use polynomial splines to estimate both cumulative baseline hazard function with monotonicity constraint and nonparametric regression functions with no such constraint. We propose a simultaneous sieve maximum likelihood estimation for regression parameters and nuisance parameters and show that the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound. In addition, we show that rates of convergence for the estimators of nonparametric functions are optimal. We implement the proposed estimation through a backfitting algorithm on generalized linear models. We conduct simulation studies to examine the finite‐sample performance of the proposed estimation method and present an analysis of renal function recovery data for illustration.     

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.     

Shape Constrained Non‐parametric Estimators of the Baseline Distribution in Cox Proportional Hazards Model

Abstract. We investigate non‐parametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a non‐decreasing baseline hazard function are proposed. We derive the non‐parametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left‐hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strongly consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a non‐increasing baseline hazard and their asymptotic properties are obtained in a similar manner. Furthermore, we introduce a Grenander type estimator for a non‐increasing baseline density, defined as the left‐hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strongly consistent and derive its asymptotic distribution at a fixed point.     

Estimation of partially linear single-index additive hazards model with current status data

In this paper, we introduce a partially linear single-index additive hazards model with current status data. Both the unknown link function of the single-index term and the cumulative baseline hazard function are approximated by B-splines under a monotonicity constraint on the latter. The sieve method is applied to estimate the nonparametric and parametric components simultaneously. We show that, when the nonparametric link function is an exact B-spline, the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound and the rate of convergence of the estimator for the cumulative baseline hazard function is optimal. Simulation studies are presented to examine the finite sample performance of the proposed estimation method. For illustration, we apply the method to a clinical dataset with current status outcome.     

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.     

Non-parametric Bayesian analysis of clustered survival data

Clustered survival data are often modelled with frailty models which incorporate frailties to model the cluster specific heterogeneity and the dependence between observations in the same cluster. For the analysis of the frailty models, we propose Bayesian modelling with beta process prior on the cumulative hazard function and describe the details of the posterior computation. We demonstrate the method with two data sets using three different frailty distributions: gamma, log-normal and log-logistic distributions. We also empirically demonstrate the difficulty in checking the assumed frailty distribution with the posterior sample of the frailties.     

A Semi-parametric Bayesian Analysis of Survival Data Based on Lévy-driven Processes

In the presence of covariate information, the proportional hazards model is one of the most popular models. In this paper, in a Bayesian nonparametric framework, we use a Markov (Lévy-driven) process to model the baseline hazard rate. Previous Bayesian nonparametric models have been based on neutral to the right processes, which have a number of drawbacks, such as discreteness of the cumulative hazard function. We allow the covariates to be time dependent functions and develop a full posterior analysis via substitution sampling. A detailed illustration is presented.     

The Identifiability of Dependent Competing Risks Models Induced by Bivariate Frailty Models

In this paper, we propose to use a special class of bivariate frailty models to study dependent censored data. The proposed models are closely linked to Archimedean copula models. We give sufficient conditions for the identifiability of this type of competing risks models. The proposed conditions are derived based on a property shared by Archimedean copula models and satisfied by several well‐known bivariate frailty models. Compared with the models studied by Heckman and Honoré and Abbring and van den Berg, our models are more restrictive but can be identified with a discrete (even finite) covariate. Under our identifiability conditions, expectation–maximization (EM) algorithm provides us with consistent estimates of the unknown parameters. Simulation studies have shown that our estimation procedure works quite well. We fit a dependent censored leukaemia data set using the Clayton copula model and end our paper with some discussions. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Weighted estimation for multivariate shared frailty models for complex surveys

Multivariate frailty models have been used for clustered survival data to characterize the relationship between the hazard of correlated failures/events and exposure variables and covariates. However, these models can introduce serious biases of the estimation for failures from complex surveys that may depend on the sampling design (informative or noninformative). In order to consistently estimate parameters, this paper considers weighting the multivariate frailty model by the inverse of the probability of selection at each stage of sampling. This follows the principle of the pseudolikelihood approach. The estimation is carried out by maximizing the penalized partial and marginal pseudolikelihood functions. The performance of the proposed estimator is assessed through a Monte Carlo simulation study and the 4 waves of data from the 1998–1999 Early Childhood Longitudinal Study. Results show that the weighted estimator is consistent and approximately unbiased.

     

Cox regression with missing covariate data using a modified partial likelihood method

Missing covariate values is a common problem in survival analysis. In this paper we propose a novel method for the Cox regression model that is close to maximum likelihood but avoids the use of the EM-algorithm. It exploits that the observed hazard function is multiplicative in the baseline hazard function with the idea being to profile out this function before carrying out the estimation of the parameter of interest. In this step one uses a Breslow type estimator to estimate the cumulative baseline hazard function. We focus on the situation where the observed covariates are categorical which allows us to calculate estimators without having to assume anything about the distribution of the covariates. We show that the proposed estimator is consistent and asymptotically normal, and derive a consistent estimator of the variance–covariance matrix that does not involve any choice of a perturbation parameter. Moderate sample size performance of the estimators is investigated via simulation and by application to a real data example.     

A Baseline-free Procedure for Transformation Models Under Interval Censorship

An important property of Cox regression model is that the estimation of regression parameters using the partial likelihood procedure does not depend on its baseline survival function. We call such a procedure baseline-free. Using marginal likelihood, we show that an baseline-free procedure can be derived for a class of general transformation models under interval censoring framework. The baseline-free procedure results a simplified and stable computation algorithm for some complicated and important semiparametric models, such as frailty models and heteroscedastic hazard/rank regression models, where the estimation procedures so far available involve estimation of the infinite dimensional baseline function. A detailed computational algorithm using Markov Chain Monte Carlo stochastic approximation is presented. The proposed procedure is demonstrated through extensive simulation studies, showing the validity of asymptotic consistency and normality. We also illustrate the procedure with a real data set from a study of breast cancer. A heuristic argument showing that the score function is a mean zero martingale is provided.     

Shared frailty models based on reversed hazard rate for modified inverse Weibull distribution as baseline distribution

The 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.