A semiparametric extended hazard regression model with time-dependent covariates

We introduce a general class of semiparametric hazard regression models, called extended hazard (EH) models, that are designed to accommodate various survival schemes with time-dependent covariates. The EH model contains both the Cox model and the accelerated failure time (AFT) model as its subclasses so that we can use this nested structure to perform model selection between the Cox model and the AFT model. A class of estimating equations using counting process and martingale techniques is developed to estimate the regression parameters of the proposed model. The performance of the estimating procedure and the impact of model misspecification are assessed through simulation studies. Two data examples, Stanford heart transplant data and Mediterranean fruit flies, egg-laying data, are used to demonstrate the usefulness of the EH model.     

A flexible parametric survival model which allows a bathtub-shaped hazard rate function

A new parametric (three-parameter) survival distribution, the lognormal–power function distribution, with flexible behaviour is introduced. Its hazard rate function can be either unimodal, monotonically decreasing or can exhibit a bathtub shape. Special cases include the lognormal distribution and the power function distribution, with finite support. Regions of parameter space where the various forms of the hazard-rate function prevail are established analytically. The distribution lends itself readily to accelerated life regression modelling. Applications to five data sets taken from the literature are given. Also it is shown how the distribution can behave like a Weibull distribution (with negative aging) for certain parameter values.     

Characterization of probability models by an odds ratio in prospective and retrospective studies

It is shown that, when exposure variables are continuous, the odds ratios are functions of exposure differences if and only if Cox's binary logistic models hold in a prospective framework, and if and only if the underlying distribution belongs to a family of exponential type distributions in a retrospective framework.     

An extension of the exponential distribution

A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an alternative to the gamma, Weibull and exponentiated exponential distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.     

Bayesian parametric accelerated failure time spatial model and its application to prostate cancer

Prostate cancer (PrCA) is the most common cancer diagnosed in American men and the second leading cause of death from malignancies. There are large geographical variation and racial disparities existing in the survival rate of PrCA. Much work on the spatial survival model is based on the proportional hazards (PH) model, but few focused on the accelerated failure time (AFT) model. In this paper, we investigate the PrCA data of Louisiana from the Surveillance, Epidemiology, and End Results program and the violation of the PH assumption suggests that the spatial survival model based on the AFT model is more appropriate for this data set. To account for the possible extra-variation, we consider spatially referenced independent or dependent spatial structures. The deviance information criterion is used to select a best-fitting model within the Bayesian frame work. The results from our study indicate that age, race, stage, and geographical distribution are significant in evaluating PrCA survival.     

Analysis of discrete lifetime data under middle-censoring and in the presence of covariates

‘Middle censoring’ is a very general censoring scheme where the actual value of an observation in the data becomes unobservable if it falls inside a random interval (L, R) and includes both left and right censoring. In this paper, we consider discrete lifetime data that follow a geometric distribution that is subject to middle censoring. Two major innovations in this paper, compared to the earlier work of Davarzani and Parsian [3 N. Davarzani and A. Parsian, Statistical inference for discrete middle-censored data, J. Statist. Plan. Inference 141 (2011), pp. 1455–1462. doi: 10.1016/j.jspi.2010.10.012[Crossref], [Web of Science ®] , [Google Scholar]], include (i) an extension and generalization to the case where covariates are present along with the data and (ii) an alternate approach and proofs which exploit the simple relationship between the geometric and the exponential distributions, so that the theory is more in line with the work of Iyer et al. [6 S.K. Iyer, S.R. Jammalamadaka, and D. Kundu, Analysis of middle censored data with exponential lifetime distributions, J. Statist. Plan. Inference 138 (2008), pp. 3550–3560. doi: 10.1016/j.jspi.2007.03.062[Crossref], [Web of Science ®] , [Google Scholar]]. It is also demonstrated that this kind of discretization of life times gives results that are close to the original data involving exponential life times. Maximum likelihood estimation of the parameters is studied for this middle-censoring scheme with covariates and their large sample distributions discussed. Simulation results indicate how well the proposed estimation methods work and an illustrative example using time-to-pregnancy data from Baird and Wilcox [1 D.D. Baird and A.J. Wilcox, Cigarette smoking associated with delayed conception, J, Am. Med. Assoc. 253 (1985), pp. 2979–2983. doi: 10.1001/jama.1985.03350440057031[Crossref], [Web of Science ®] , [Google Scholar]] is included.     

Solution to the hazard function of some two-stage carcinogenesis models when normal cell growth is piecewise linear

A closed form expression is obtained for the hazard function for three stochastic two-stage carcinogenesis models, when the normal cell growth is assumed to be piecewise linear.     

A hybrid method to estimate the full parametric hazard model

Abstract

In this paper, we propose a hybrid method to estimate the baseline hazard for Cox proportional hazard model. In the proposed method, the nonparametric estimate of the survival function by Kaplan Meier, and the parametric estimate of the logistic function in the Cox proportional hazard by partial likelihood method are combined to estimate a parametric baseline hazard function. We compare the estimated baseline hazard using the proposed method and the Cox model. The results show that the estimated baseline hazard using hybrid method is improved in comparison with estimated baseline hazard using the Cox model. The performance of each method is measured based on the estimated parameters of the baseline distribution as well as goodness of fit of the model. We have used real data as well as simulation studies to compare performance of both methods. Monte Carlo simulations carried out in order to evaluate the performance of the proposed method. The results show that the proposed hybrid method provided better estimate of the baseline in comparison with the estimated values by the Cox model.     

Estimating the turning point of the log-logistic hazard function in the presence of long-term survivors with an application for uterine cervical cancer data

The hazard function plays an important role in cancer patient survival studies, as it quantifies the instantaneous risk of death of a patient at any given time. Often in cancer clinical trials, unimodal hazard functions are observed, and it is of interest to detect (estimate) the turning point (mode) of hazard function, as this may be an important measure in patient treatment strategies with cancer. Moreover, when patient cure is a possibility, estimating cure rates at different stages of cancer, in addition to their proportions, may provide a better summary of the effects of stages on survival rates. Therefore, the main objective of this paper is to consider the problem of estimating the mode of hazard function of patients at different stages of cervical cancer in the presence of long-term survivors. To this end, a mixture cure rate model is proposed using the log-logistic distribution. The model is conveniently parameterized through the mode of the hazard function, in which cancer stages can affect both the cured fraction and the mode. In addition, we discuss aspects of model inference through the maximum likelihood estimation method. A Monte Carlo simulation study assesses the coverage probability of asymptotic confidence intervals.     

Deficiency of the mle of a smooth survival function under the proportional hazard model

The problem of estimating a smooth distribution function F at a point t is treated under the proportional hazard model of random censorship. It is shown that a certain class of properly chosen kernel type estimator of F asymptotically perform better than the maximum likelihood estimator. It is shown that the relative deficiency of the maximum likelihood estimator of F under the proportional hazard model with respect to the properly chosen kernel type estimator tends to infinity as the sample size tends to infinity.     

An extension of the generalized exponential distribution

The two-parameter generalized exponential distribution has been used recently quite extensively to analyze lifetime data. In this paper the two-parameter generalized exponential distribution has been embedded in a larger class of distributions obtained by introducing another shape parameter. Because of the additional shape parameter, more flexibility has been introduced in the family. It is observed that the new family is positively skewed, and has increasing, decreasing, unimodal and bathtub shaped hazard functions. It can be observed as a proportional reversed hazard family of distributions. This new family of distributions is analytically quite tractable and it can be used quite effectively to analyze censored data also. Analyses of two data sets are performed and the results are quite satisfactory.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

Analytical hazard representations for use in reliability,mortality, and simulation studies

A simple parametric model is proposed to represent data of non-standard distributional form. An example is the "bathtub" hazard of reliability. The application of the approach to simulation and to data analysis is discussed, and will be further explored in later reports.     

Bayes estimators of the average failure rates

In the present paper we have proposed a Bayesian approach for making inferences from accelerated life tests which do not require distributional assumptions     

Estimation and simulation of conditional hazard function in the quasi-associated framework when the observations are linked via a functional single-index structure

We consider the estimation of the conditional hazard function of a scalar response variable Y given a Hilbertian random variable X when the observations are linked via a single-index structure in the quasi-associated framework. We establish the pointwise almost complete convergence and the uniform almost complete convergence (with the rate) of the estimate of this model. A simulation is given to illustrate the good behavior in the practice of our methodology.     

Regression modelling of interval censored data based on the adaptive ridge procedure

A new method for the analysis of time to ankylosis complication on a dataset of replanted teeth is proposed. In this context of left-censored, interval-censored and right-censored data, a Cox model with piecewise constant baseline hazard is introduced. Estimation is carried out with the expectation maximisation (EM) algorithm by treating the true event times as unobserved variables. This estimation procedure is shown to produce a block diagonal Hessian matrix of the baseline parameters. Taking advantage of this interesting feature in the EM algorithm, a L0 penalised likelihood method is implemented in order to automatically determine the number and locations of the cuts of the baseline hazard. This procedure allows to detect specific areas of time where patients are at greater risks for ankylosis. The method can be directly extended to the inclusion of exact observations and to a cure fraction. Theoretical results are obtained which allow to derive statistical inference of the model parameters from asymptotic likelihood theory. Through simulation studies, the penalisation technique is shown to provide a good fit of the baseline hazard and precise estimations of the resulting regression parameters.     

Two-Sample Statistics for Testing the Equality of Survival Functions Against Improper Semi-parametric Accelerated Failure Time Alternatives: An Application to the Analysis of a Breast Cancer Clinical Trial

This paper presents two-sample statistics suited for testing equality of survival functions against improper semi-parametric accelerated failure time alternatives. These tests are designed for comparing either the short- or the long-term effect of a prognostic factor, or both. These statistics are obtained as partial likelihood score statistics from a time-dependent Cox model. As a consequence, the proposed tests can be very easily implemented using widely available software. A breast cancer clinical trial is presented as an example to demonstrate the utility of the proposed tests.     

Search boundaries of the truncated discrete sequential test

The theme of this paper is improved planning of binomial sequential probability ratio tests in the context of comparison of two objects as to their time between failures or to failure, assumed to be exponentially distributed. The authors’ earlier works established that the probabilities of I- and II- type errors (α and β) are discrete in character and do not lend themselves to analytical expression. Accordingly, the choice of the optimal parameters for the decision boundaries necessitates a search for extrema in discrete sets. The present work outlines a procedure that involves application of the continued-fractions theory, and permits finding the set of boundary positions in which the test characteristics undergo changes. It was established, that in the domains described in the earlier papers, the relationships of α and β versus these positions are close to planar and – within narrow limits – stepwise. The step sizes are highly variable, so that the standard minimum search procedures are either cumbersome or actually useless. On the basis of these relationships~– and others – a search algorithm is proposed for the optimal test boundaries. An example is presented – planning and implementation of this test in the integrated-circuit industry.     

Estimation of the generalized exponential renewal function

When the shape parameter is a non-integer of the generalized exponential (GE) distribution, the analytical renewal function (RF) usually is not tractable. To overcome this, the approximation method has been used in this paper. In the proposed model, the n-fold convolution of the GE cumulative distribution function (CDF) is approximated by n-fold convolutions of gamma and normal CDFs. We obtain the GE RF by a series approximation model. The method is very simple in the computation. Numerical examples have shown that the approximate models are accurate and robust. When the parameters are unknown, we present the asymptotic confidence interval of the RF. The validity of the asymptotic confidence interval is checked via numerical experiments.