EM algorithm-based likelihood estimation for a generalized Gompertz regression model in presence of survival data with long-term survivors: an application to uterine cervical cancer data

In this paper we develop a regression model for survival data in the presence of long-term survivors based on the generalized Gompertz distribution introduced by El-Gohary et al. [The generalized Gompertz distribution. Appl Math Model. 2013;37:13–24] in a defective version. This model includes as special case the Gompertz cure rate model proposed by Gieser et al. [Modelling cure rates using the Gompertz model with covariate information. Stat Med. 1998;17:831–839]. Next, an expectation maximization algorithm is then developed for determining the maximum likelihood estimates (MLEs) of the parameters of the model. In addition, we discuss the construction of confidence intervals for the parameters using the asymptotic distributions of the MLEs and the parametric bootstrap method, and assess their performance through a Monte Carlo simulation study. Finally, the proposed methodology was applied to a database on uterine cervical cancer.     

A flexible semiparametric transformation model for survival data

I suggest an extension of the semiparametric transformation model that specifies a time-varying regression structure for the transformation, and thus allows time-varying structure in the data. Special cases include a stratified version of the usual semiparametric transformation model. The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline. Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given. The method is illustrated by a worked example and a small simulation study. A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggested.     

A robust approach to longitudinal data analysis

The author introduces robust techniques for estimation, inference and variable selection in the analysis of longitudinal data. She first addresses the problem of the robust estimation of the regression and nuisance parameters, for which she derives the asymptotic distribution. She uses weighted estimating equations to build robust quasi‐likelihood functions. These functions are then used to construct a class of test statistics for variable selection. She derives the limiting distribution of these tests and shows its robustness properties in terms of stability of the asymptotic level and power under contamination. An application to a real data set allows her to illustrate the benefits of a robust analysis.     

Frailty modeling for clustered survival data: an application to birth interval in Bangladesh

The present work demonstrates an application of random effects model for analyzing birth intervals that are clustered into geographical regions. Observations from the same cluster are assumed to be correlated because usually they share certain unobserved characteristics between them. Ignoring the correlations among the observations may lead to incorrect standard errors of the estimates of parameters of interest. Beside making the comparisons between Cox's proportional hazards model and random effects model for analyzing geographically clustered time-to-event data, important demographic and socioeconomic factors that may affect the length of birth intervals of Bangladeshi women are also reported in this paper.     

A compound Rayleigh survival model and its application to randomly censored data

Received: May 5, 1999; revised version: June 15, 2000     

The Poisson-exponential model for recurrent event data: an application to bowel motility data

This paper presents a new parametric model for recurrent events, in which the time of each recurrence is associated to one or multiple latent causes and no information is provided about the responsible cause for the event. This model is characterized by a rate function and it is based on the Poisson-exponential distribution, namely the distribution of the maximum among a random number (truncated Poisson distributed) of exponential times. The time of each recurrence is then given by the maximum lifetime value among all latent causes. Inference is based on a maximum likelihood approach. A simulation study is performed in order to observe the frequentist properties of the estimation procedure for small and moderate sample sizes. We also investigated likelihood-based tests procedures. A real example from a gastroenterology study concerning small bowel motility during fasting state is used to illustrate the methodology. Finally, we apply the proposed model to a real data set and compare it with the classical Homogeneous Poisson model, which is a particular case.     

Joint modeling of multivariate censored longitudinal and event time data with application to the Genetic Markers of Inflammation Study

The Genetic Markers of Inflammation Study (GenIMS) was conceived to investigate the role of severe sepsis, which is typically defined as system-wide multi-organ failure, on survival. One major hypothesis for this systemic collapse, and reduction in survival, is a cascade of pro-inflammatory and anti-inflammatory cytokines. In this paper, we devised a novel joint modeling strategy to evaluate the joint effect of longitudinal anti-inflammatory marker IL-6 and pro-inflammatory marker IL-10 on 90-day survival. We found that, on average, patients with high initial values of both IL-6 and IL-10, that tend to increase over time, are associated with a reduction in survival expectancy and that accounting for their assumed correlation was justified.     

A parametric model fitting time to first event for overdispersed data: application to time to relapse in multiple sclerosis

In this article, we propose a parametric model for the distribution of time to first event when events are overdispersed and can be properly fitted by a Negative Binomial distribution. This is a very common situation in medical statistics, when the occurrence of events is summarized as a count for each patient and the simple Poisson model is not adequate to account for overdispersion of data. In this situation, studying the time of occurrence of the first event can be of interest. From the Negative Binomial distribution of counts, we derive a new parametric model for time to first event and apply it to fit the distribution of time to first relapse in multiple sclerosis (MS). We develop the regression model with methods for covariate estimation. We show that, as the Negative Binomial model properly fits relapse counts data, this new model matches quite perfectly the distribution of time to first relapse, as tested in two large datasets of MS patients. Finally we compare its performance, when fitting time to first relapse in MS, with other models widely used in survival analysis (the semiparametric Cox model and the parametric exponential, Weibull, log-logistic and log-normal models).     

A 3-parameter Gompertz distribution for survival data with competing risks,with an application to breast cancer data

The cumulative incidence function is of great importance in the analysis of survival data when competing risks are present. Parametric modeling of such functions, which are by nature improper, suggests the use of improper distributions. One frequently used improper distribution is that of Gompertz, which captures only monotone hazard shapes. In some applications, however, subdistribution hazard estimates have been observed with unimodal shapes. An extension to the Gompertz distribution is presented which can capture unimodal as well as monotone hazard shapes. Important properties of the proposed distribution are discussed, and the proposed distribution is used to analyze survival data from a breast cancer clinical trial.     

A simple hidden markov model for bayesian modeling with time dependent data

Consider a set of real valued observations collected over time. We pro¬pose a simple hidden Markow model for these realizations in which the the predicted distribution of the next future observation given the past is easily computed. The hidden or unobservable set of parameters is assumed to have a Markov structure of a special type. The model is quite flexible and can be used to incorporate different types of prior information in straightforward and sensible ways.     

A copula model for bivariate hybrid censored survival data with application to the MACS study

A copula model for bivariate survival data with hybrid censoring is proposed to study the association between survival time of individuals infected with HIV and persistence time of infection with an additional virus. Survival with HIV is right censored and the persistence time of the additional virus is subject to interval censoring case 1. A pseudo-likelihood method is developed to study the association between the two event times under such hybrid censoring. Asymptotic consistency and normality of the pseudo-likelihood estimator are established based on empirical process theory. Simulation studies indicate good performance of the estimator with moderate sample size. The method is applied to a motivating HIV study which investigates the effect of GB virus type C (GBV-C) co-infection on survival time of HIV infected individuals.     

A joint model of binary and longitudinal data with non-ignorable missingness,with application to marital stress and late-life major depression in women

Understanding how long-term marital stress affects major depressive disorder (MDD) in older women has clinical implications for the treatment of women at risk. In this paper, we consider the problem of predicting MDD in older women (mean age 60) from a marital stress scale administered four times during the preceding 20-year period, with a greater dropout by women experiencing marital stress or MDD. To analyze these data, we propose a Bayesian joint model consisting of: (1) a linear mixed effects model for the longitudinal measurements, (2) a generalized linear model for the binary primary endpoint, and (3) a shared parameter model for the missing data mechanism. Our analysis indicates that MDD in older women is significantly associated with higher levels of prior marital stress and increasing marital stress over time, although there is a generally decreasing trend in marital stress. This is the first study to propose a joint model for incompletely observed longitudinal measurements, a binary primary endpoint, and non-ignorable missing data; a comparison shows that the joint model yields better predictive accuracy than a two-stage model. These findings suggest that women who experience marital stress in mid-life need treatment to help prevent late-life MDD, which has serious consequences for older persons.     

A bivariate regression model for matched paired survival data: local influence and residual analysis

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we consider a location scale model for bivariate survival times based on the proposal of a copula to model the dependence of bivariate survival data. For the proposed model, we consider inferential procedures based on maximum likelihood. Gains in efficiency from bivariate models are also examined in the censored data setting. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the bivariate regression model for matched paired survival data. Sensitivity analysis methods such as local and total influence are presented and derived under three perturbation schemes. The martingale marginal and the deviance marginal residual measures are used to check the adequacy of the model. Furthermore, we propose a new measure which we call modified deviance component residual. The methodology in the paper is illustrated on a lifetime data set for kidney patients.     

A semi-analytical solution to the maximum-likelihood fit of Poisson data to a linear model using the Cash statistic

The Cash statistic, also known as the C statistic, is commonly used for the analysis of low-count Poisson data, including data with null counts for certain values of the independent variable. The use of this statistic is especially attractive for low-count data that cannot be combined, or re-binned, without loss of resolution. This paper presents a new maximum-likelihood solution for the best-fit parameters of a linear model using the Poisson-based Cash statistic. The solution presented in this paper provides a new and simple method to measure the best-fit parameters of a linear model for any Poisson-based data, including data with null counts. In particular, the method enforces the requirement that the best-fit linear model be non-negative throughout the support of the independent variable. The method is summarized in a simple algorithm to fit Poisson counting data of any size and counting rate with a linear model, by-passing entirely the use of the traditional χ2 statistic.     

A nested frailty model for survival data,with an application to the study of child survival in northeast Brazil

"This article presents a multivariate hazard model for survival data that are clustered at two hierarchical levels.... We apply the model to an analysis of the covariates of child survival using survey data from northeast Brazil collected via a hierarchically clustered sampling scheme. We find that family and community frailty effects are fairly small in magnitude but are of importance because they alter the results in a systematic pattern."     

Jackknifing the kaplan-meier survival estimator for censored data: Simulation results and asymptotic analysis

It is shown by simulation t h a t the (arc-sine transformation of the) Kaplan-Meier survival estimator for censored data can be usefully jackknifed to give conservative confidence limits for survival probabilities when samples are small (25 and 50). Mathematical demonstration of the asymptotic, large-sample, validity of the jackknife is included.     

A model of uniform association plus two-diagonalsparameter and its application to occupational mobility table data

For the analysis of square contingency tables with ordered categories, Agresti (1988) introduced a model having the structure of uniform association plus a main-diagonal parameter. This paper extends that model. The extended model has the structure of uniform association plus two-diagonals-parameter, and it is a special case of the quasi-uniform association model introduced by Goodman (1979). The Danish occupational mobility table data are analyzed using the models introduced here.     

A bivariate generalized linear model with an application to meteorological data analysis

A bivariate generalized linear model is developed as a mixture distribution with one component of the mixture being discrete with probability mass only at the origin. The use of the proposed model is illustrated by analyzing local area meteorological measurements with constant correlation structure that incorporates predictor variables. The Monte Carlo study is performed to evaluate the inferential efficiency of model parameters for two types of true models. These results suggest that the estimates of regression parameters are consistent and the efficiency of the inference increases for the proposed model for ρ≥0.50 especially in larger samples. As an illustration of a bivariate generalized linear model, we analyze a precipitation monitoring data of adjacent local stations for Tokyo and Yokohama.