Influence Diagnostics in log-Birnbaum-Saunders Regression Models

In this paper we present various diagnostic methods for a linear regression model under a logarithmic Birnbaum-Saunders distribution for the errors, which may be applied for accelerated life testing or to compare the median lives of several populations. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are derived, analysed and discussed. We also present a connection between the local influence and generalized leverage methods. A discussion of the computation of the likelihood displacement as well as the normal curvature in the local influence method are presented. Finally, an example with real data is given for illustration.     

Some Comments on Hassairi et al.'s “Implicit Distributions and Estimation”

In social interaction studies, one commonly encounters repeated displays of behaviors along with their duration data. Statistical methods for the analysis of such data use either parametric (e.g., Weibull) or semi-nonparametric (e.g., Cox) proportional hazard models, modified to include random effects (frailty) which account for the correlation of repeated occurrences of behaviors within a unit (dyad). However, dyad-specific random effects by themselves are not able to account for the ordering of event occurrences within dyads. The occurrence of an event (behavior) can make further occurrences of the same behavior to be more or less likely during an interaction. This article develops event-dependent random effects models for analyzing repeated behaviors data using a Bayesian approach. The models are illustrated by a dataset relating to emotion regulation in families with children who have behavioral or emotional problems.     

INFLUENCE DIAGNOSTICS IN PROPER DISPERSION MODELS

This paper discusses the application of influence diagnostic methods in univariate proper dispersion models. This class includes, in particular, continuous generalized linear models as well as other subclasses of continuous regression models. We emphasize the study of the local influence on the likelihood displacement and predictions from the models. Some of the diagnostics are illustrated by an example on directional data.     

Multivariate Birnbaum–Saunders regression model

In this paper, we propose a multivariate log-linear Birnbaum–Saunders regression model. We discuss maximum-likelihood estimation of the model parameters and provide closed-form expressions for the score function and for Fisher's information matrix. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum-likelihood estimator. Some influence methods, such as the local influence and generalized leverage are discussed and the normal curvatures for studying local influence are derived under some perturbation schemes. Further, a test for the homogeneity of the shape parameter of the multivariate regression model is investigated. A real data set is presented for illustrative purposes.     

The negative binomial–beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial–beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.     

A log-Birnbaum–Saunders regression model with asymmetric errors

This paper introduces a skewed log-Birnbaum–Saunders regression model based on the skewed sinh-normal distribution proposed by Leiva et al. [A skewed sinh-normal distribution and its properties and application to air pollution, Comm. Statist. Theory Methods 39 (2010), pp. 426–443]. Some influence methods, such as the local influence and generalized leverage, are presented. Additionally, we derived the normal curvatures of local influence under some perturbation schemes. An empirical application to a real data set is presented in order to illustrate the usefulness of the proposed model.     

Extended Hazard Regression Model for Reliability and Survival Analysis

We propose an extended hazard regression model which allows the spread parameter to be dependent on covariates. This allows a broad class of models which includes the most common hazard models, such as the proportional hazards model, the accelerated failure time model and a proportional hazards/accelerated failure time hybrid model with constant spread parameter. Simulations based on sub-classes of this model suggest that maximum likelihood performs well even when only small or moderate-size data sets are available and the censoring pattern is heavy. The methodology provides a broad framework for analysis of reliability and survival data. Two numerical examples illustrate the results.     

Local likelihood smoothing of sample extremes

Trends in sample extremes are of interest in many contexts, an example being environmental statistics. Parametric models are often used to model trends in such data, but they may not be suitable for exploratory data analysis. This paper outlines a semiparametric approach to smoothing sample extremes, based on local polynomial fitting of the generalized extreme value distribution and related models. The uncertainty of fits is assessed by using resampling methods. The methods are applied to data on extreme temperatures and on record times for the women's 3000 m race.     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

A new class of generalized Lindley distributions with applications

A new four-parameter class of generalized Lindley (GL) distribution called the beta-generalized Lindley (BGL) distribution is proposed. This class of distributions contains the beta-Lindley, GL and Lindley distributions as special cases. Expansion of the density of the BGL distribution is obtained. The properties of these distributions, including hazard function, reverse hazard function, monotonicity property, shapes, moments, reliability, mean deviations, Bonferroni and Lorenz curves are derived. Measures of uncertainty such as Renyi entropy and s-entropy as well as Fisher information are presented. Method of maximum likelihood is used to estimate the parameters of the BGL and related distributions. Finally, real data examples are discussed to illustrate the applicability of this class of models.     

Sensitivity analysis of reliability functions of the exponential power series lifetime distribution

ABSTRACT

Hazard rate functions are often used in modeling of lifetime data. The Exponential Power Series (EPS) family has a monotone hazard rate function. In this article, the influence of input factors such as time and parameters on the variability of hazard rate function is assessed by local and global sensitivity analysis. Two different indices based on local and global sensitivity indices are presented. The simulation results for two datasets show that the hazard rate functions of the EPS family are sensitive to input parameters. The results also show that the hazard rate function of the EPS family is more sensitive to the exponential distribution than power series distributions.     

The generalized log-gamma mixture model with covariates: local influence and residual analysis

In a sample of censored survival times, the presence of an immune proportion of individuals who are not subject to death, failure or relapse, may be indicated by a relatively high number of individuals with large censored survival times. In this paper the generalized log-gamma model is modified for the possibility that long-term survivors may be present in the data. The model attempts to separately estimate the effects of covariates on the surviving fraction, that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. Inference for the model parameters is considered via maximum likelihood. Some influence methods, such as the local influence and total local influence of an individual are derived, analyzed and discussed. Finally, a data set from the medical area is analyzed under the log-gamma generalized mixture model. A residual analysis is performed in order to select an appropriate model. The authors would like to thank the editor and referees for their helpful comments. This work was supported by CNPq, Brazil.     

Influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes

In this paper, we propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest follows the Negative Binomial distribution and the time to event follows a Weibull distribution. Indeed, we introduce the Weibull-Negative-Binomial (WNB) distribution, which can be used in order to model survival data when the hazard rate function is increasing, decreasing and some non-monotonous shaped. Another advantage of the proposed model is that it has some distributions commonly used in lifetime analysis as particular cases. Moreover, the proposed model includes as special cases some of the well-know cure rate models discussed in the literature. We consider a frequentist analysis for parameter estimation of a WNB model with cure rate. Then, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. Finally, the methodology is illustrated on a medical data.     

A Bayesian approach to Weibull survival models—Application to a cancer clinical trial

In this paper we outline a class of fully parametric proportional hazards models, in which the baseline hazard is assumed to be a power transform of the time scale, corresponding to assuming that survival times follow a Weibull distribution. Such a class of models allows for the possibility of time varying hazard rates, but assumes a constant hazard ratio. We outline how Bayesian inference proceeds for such a class of models using asymptotic approximations which require only the ability to maximize the joint log posterior density. We apply these models to a clinical trial to assess the efficacy of neutron therapy compared to conventional treatment for patients with tumors of the pelvic region. In this trial there was prior information about the log hazard ratio both in terms of elicited clinical beliefs and the results of previous studies. Finally, we consider a number of extensions to this class of models, in particular the use of alternative baseline functions, and the extension to multi-state data.     

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.     

Direct parametric inference for the cumulative incidence function

Summary.  In survival data that are collected from phase III clinical trials on breast cancer, a patient may experience more than one event, including recurrence of the original cancer, new primary cancer and death. Radiation oncologists are often interested in comparing patterns of local or regional recurrences alone as first events to identify a subgroup of patients who need to be treated by radiation therapy after surgery. The cumulative incidence function provides estimates of the cumulative probability of locoregional recurrences in the presence of other competing events. A simple version of the Gompertz distribution is proposed to parameterize the cumulative incidence function directly. The model interpretation for the cumulative incidence function is more natural than it is with the usual cause-specific hazard parameterization. Maximum likelihood analysis is used to estimate simultaneously parametric models for cumulative incidence functions of all causes. The parametric cumulative incidence approach is applied to a data set from the National Surgical Adjuvant Breast and Bowel Project and compared with analyses that are based on parametric cause-specific hazard models and nonparametric cumulative incidence estimation.     

Heteroscedastic log-exponentiated Weibull regression model

We introduce a new class of heteroscedastic log-exponentiated Weibull (LEW) regression models. The class of regression models can be applied to censored data and be used more effectively in survival analysis. Maximum likelihood estimation of the model parameters with censored data as well as influence diagnostics for the new regression model is investigated. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the heteroscedastic LEW regression model. The normal curvatures for studying local influence are derived under various perturbation schemes. An empirical application to a real data set is provided to illustrate the usefulness of the new class of heteroscedastic regression models.     

A case-base sampling method for estimating recurrent event intensities

Case-base sampling provides an alternative to risk set sampling based methods to estimate hazard regression models, in particular when absolute hazards are also of interest in addition to hazard ratios. The case-base sampling approach results in a likelihood expression of the logistic regression form, but instead of categorized time, such an expression is obtained through sampling of a discrete set of person-time coordinates from all follow-up data. In this paper, in the context of a time-dependent exposure such as vaccination, and a potentially recurrent adverse event outcome, we show that the resulting partial likelihood for the outcome event intensity has the asymptotic properties of a likelihood. We contrast this approach to self-matched case-base sampling, which involves only within-individual comparisons. The efficiency of the case-base methods is compared to that of standard methods through simulations, suggesting that the information loss due to sampling is minimal.     

A Bayes comparison of Weibull extension and modified Weibull models for data showing bathtub hazard rate

A number of models have been proposed in the literature to model data reflecting bathtub-shaped hazard rate functions. Mixture distributions provide the obvious choice for modelling such data sets but these contain too many parameters and hamper the accuracy of the inferential procedures particularly when the data are meagre. Recently, a few distributions have been proposed which are simply generalizations of the two-parameter Weibull model and are capable of producing bathtub behaviour of the hazard rate function. The Weibull extension and the modified Weibull models are two such families. This study focuses on comparing these two distributions for data sets exhibiting bathtub shape of the hazard rate. Bayesian tools are preferred due to their wide range of applicability in various nested and non-nested model comparison problems. Real data illustrations are provided so that a particular model can be recommended based on various tools of model comparison discussed in the paper.     

A Bayesian approach for parameter estimation in multi-stage models

Multi-stage time evolving models are common statistical models for biological systems, especially insect populations. In stage-duration distribution models, parameter estimation for the models use the Laplace transform method. This method involves assumptions such as known constant shapes, known constant rates or the same overall hazard rate for all stages. These assumptions are strong and restrictive. The main aim of this paper is to weaken these assumptions by using a Bayesian approach. In particular, a Metropolis-Hastings algorithm based on deterministic transformations is used to estimate parameters. We will use two models, one which has no hazard rates, and the other has stage-wise constant hazard rates. These methods are validated in simulation studies followed by a case study of cattle parasites. The results show that the proposed methods are able to estimate the parameters comparably well, as opposed to using the Laplace transform methods.