Identifying Nonproportional Covariates in the Cox Model

This paper addresses problem of testing whether an individual covariate in the Cox model has a proportional (i.e., time-constant) effect on the hazard. Two existing methods are considered: one is based on the component of the score process, and the other is a Neyman type smooth test. Simulations show that, when the model contains both proportional and nonproportional covariates, these methods are not reliable tools for discrimination. A simple yet effective solution is proposed based on smooth modeling of the effects of the covariates not in focus.     

LIKELIHOOD APPROXIMATIONS AND DISCRETE MODELS FOR TIED SURVIVAL DATA

ABSTRACT

Ties among event times are often recorded in survival studies. For example, in a two week laboratory study where event times are measured in days, ties are very likely to occur. The proportional hazards model might be used in this setting using an approximated partial likelihood function. This approximation works well when the number of ties is small. On the other hand, discrete regression models are suggested when the data are heavily tied. However, in many situations it is not clear which approach should be used in practice. In this work, empirical guidelines based on Monte Carlo simulations are provided. These recommendations are based on a measure of the amount of tied data present and the mean square error. An example illustrates the proposed criterion.     

Proportional hazards model for discrete data: Some new developments

ABSTRACT

Many times, a product lifetime can be described through a non negative integer valued random variable. In this article, we propose a proportional hazards model for discrete data analogous to the version for continuous data. Some ageing properties of the model are discussed. Stochastic comparison of pair of random variables that follow the model are also made. A new test based on U-statistics is developed for testing that the proportionality parameter in the proposed model is 1. The asymptotic properties of the proposed test are studied. We present some numerical results to asses the performance of the test procedure.     

A partial score test for difference among heterogeneous populations

In event time data analysis, comparisons between distributions are made by the logrank test. When the data appear to contain crossing hazards phenomena, nonparametric weighted logrank statistics are usually suggested to accommodate different-weighted functions to increase the power. However, the gain in power by imposing different weights has its limits since differences before and after the crossing point may balance each other out. In contrast to the weighted logrank tests, we propose a score-type statistic based on the semiparametric-, heteroscedastic-hazards regression model of Hsieh [2001. On heteroscedastic hazards regression models: theory and application. J. Roy. Statist. Soc. Ser. B 63, 63–79.], by which the nonproportionality is explicitly modeled. Our score test is based on estimating functions derived from partial likelihood under the heteroscedastic model considered herein. Simulation results show the benefit of modeling the heteroscedasticity and power of the proposed test to two classes of weighted logrank tests (including Fleming–Harrington's test and Moreau's locally most powerful test), a Renyi-type test, and the Breslow's test for acceleration. We also demonstrate the application of this test by analyzing actual data in clinical trials.     

OPTIMALITY OF NONPARAMETRIC TESTS FOR ASSOCIATION BETWEEN SURVIVAL TIME AND CONTINUOUS COVARIATES

Abstract

Recently, Chen (Chen, Z. (2000 Chen, Z. 2000. A new two-parameter lifetime distribution with bathtub-shape or increasing failure rate function. Statistics &; Probability Letters, 49: 155–161. [Crossref], [Web of Science ®] , [Google Scholar]). A new two-parameter lifetime distribution with bathtub-shape or increasing failure rate function. Statistics &; Probability Letters 49:155–161.) proposed a two-parameter model that can be used to model bathtub-shaped failure rate. Although this model has several interesting properties, it does not contain a scale parameter and hence not flexible in modeling real data. A generalized model including the scale parameter has shown to be interesting and it has the traditional Weibull distribution as an asymptotic case. In this article, a detailed analysis of this model is presented. Shapes of the density and failure rate function are studied. The asymptotic confidence intervals for the parameters are also derived from the Fisher information matrix. The likelihood ratio test is applied to test the goodness of fit of Weibull extension model. Some examples are shown to illustrate the application of the model and analysis.     

A Generalized Log-Normal Model for Grouped Survival Data

It is common to have experiments in which it is not possible to observe the exact lifetimes but only the interval where they occur. This sort of data presents a high number of ties and it is called grouped or interval-censored survival data. Regression methods for grouped data are available in the statistical literature. The regression structure considers modeling the probability of a subject's survival past a visit time conditional on his survival at the previous visit. Two approaches are presented: assuming that lifetimes come from (1) a continuous proportional hazards model and (2) a logistic model. However, there may be situations in which none of the models are adequate for a particular data set. This article proposes the generalized log-normal model as an alternative model for discrete survival data. This model was introduced by Chen (1995 Chen , G. ( 1995 ). Generalized Log-normal distributions with reliability application . Comput. Stat. Data Anal. 19 : 300 – 319 . [Google Scholar]) and it is extended in this article for grouped survival data. A real example related to a Chagas disease illustrates the proposed model.     

Maximum likelihood estimation in the proportional hazards model of random censorship

The maximum likelihood estimator (MLE) for the survival function S_Tunder the proportional hazards model of censorship is derived and shown to differ from the Abdushukurov-Cheng-Lin estimator when the class of allowable distributions includes all continuous and discrete distributions. The estimators are compared via an example. The MLE is calculated using a Newton-Raphson iterative procedure and implemented via a FORTRAN algorithm.     

A proportional hazards model for the analysis of doubly censored competing risks data

ABSTRACT

Competing risks data are common in medical research in which lifetime of individuals can be classified in terms of causes of failure. In survival or reliability studies, it is common that the patients (objects) are subjected to both left censoring and right censoring, which is refereed as double censoring. The analysis of doubly censored competing risks data in presence of covariates is the objective of this study. We propose a proportional hazards model for the analysis of doubly censored competing risks data, using the hazard rate functions of Gray (1988 Gray, R.J. (1988). A class of k-sample tests for comparing the cumulative incidence of a competing risk. Ann. Statist. 16:1141–1154.[Crossref], [Web of Science ®] , [Google Scholar]), while focusing upon one major cause of failure. We derive estimators for regression parameter vector and cumulative baseline cause specific hazard rate function. Asymptotic properties of the estimators are discussed. A simulation study is conducted to assess the finite sample behavior of the proposed estimators. We illustrate the method using a real life doubly censored competing risks data.     

A corrected likelihood method for the proportional hazards model with covariates subject to measurement error

There has been extensive interest in discussing inference methods for survival data when some covariates are subject to measurement error. It is known that standard inferential procedures produce biased estimation if measurement error is not taken into account. With the Cox proportional hazards model a number of methods have been proposed to correct bias induced by measurement error, where the attention centers on utilizing the partial likelihood function. It is also of interest to understand the impact on estimation of the baseline hazard function in settings with mismeasured covariates. In this paper we employ a weakly parametric form for the baseline hazard function and propose simple unbiased estimating functions for estimation of parameters. The proposed method is easy to implement and it reveals the connection between the naive method ignoring measurement error and the corrected method with measurement error accounted for. Simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error in covariates. As an illustration we apply the proposed methods to analyze a data set arising from the Busselton Health Study [Knuiman, M.W., Cullent, K.J., Bulsara, M.K., Welborn, T.A., Hobbs, M.S.T., 1994. Mortality trends, 1965 to 1989, in Busselton, the site of repeated health surveys and interventions. Austral. J. Public Health 18, 129–135].     

Bayesian partial likelihood approach for tied observations

The Bayesian analysis based on the partial likelihood for Cox's proportional hazards model is frequently used because of its simplicity. The Bayesian partial likelihood approach is often justified by showing that it approximates the full Bayesian posterior of the regression coefficients with a diffuse prior on the baseline hazard function. This, however, may not be appropriate when ties exist among uncensored observations. In that case, the full Bayesian and Bayesian partial likelihood posteriors can be much different. In this paper, we propose a new Bayesian partial likelihood approach for many tied observations and justify its use.     

Reducing bias in parameter estimates from stepwise regression in proportional hazards regression with right-censored data

When variable selection with stepwise regression and model fitting are conducted on the same data set, competition for inclusion in the model induces a selection bias in coefficient estimators away from zero. In proportional hazards regression with right-censored data, selection bias inflates the absolute value of parameter estimate of selected parameters, while the omission of other variables may shrink coefficients toward zero. This paper explores the extent of the bias in parameter estimates from stepwise proportional hazards regression and proposes a bootstrap method, similar to those proposed by Miller (Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, 2002) for linear regression, to correct for selection bias. We also use bootstrap methods to estimate the standard error of the adjusted estimators. Simulation results show that substantial biases could be present in uncorrected stepwise estimators and, for binary covariates, could exceed 250% of the true parameter value. The simulations also show that the conditional mean of the proposed bootstrap bias-corrected parameter estimator, given that a variable is selected, is moved closer to the unconditional mean of the standard partial likelihood estimator in the chosen model, and to the population value of the parameter. We also explore the effect of the adjustment on estimates of log relative risk, given the values of the covariates in a selected model. The proposed method is illustrated with data sets in primary biliary cirrhosis and in multiple myeloma from the Eastern Cooperative Oncology Group.     

Heterogeneity and varying effect in hazards regression

In the analysis of survival data, when nonproportional hazards are encountered, the Cox model is often extended to allow for a time-dependent effect by accommodating a varying coefficient. This extension, however, cannot resolve the nonproportionality caused by heterogeneity. In contrast, the heteroscedastic hazards regression (HHR) model is capable of modeling heterogeneity and thus can be applied when dealing with nonproportional hazards. In this paper, we study the application of the HHR model possibly equipped with varying coefficients. An LRR (logarithm of relative risk) plot is suggested when investigating the need to impose varying coefficients. Constancy and degeneration in the plot are used as diagnostic criteria. For the HHR model, a ‘piecewise effect’ (PE) analysis and an ‘average effect’ (AE) analysis are introduced. For the PE setting, we propose a score-type test for covariate-specific varying coefficients. The Stanford Heart Transplant data are analyzed for illustration. In the case of degeneration being destroyed by a polynomial covariate, piecewise constancy and/or monotonicity of the LRRs is considered as an alternative criterion based on the PE analysis. Finally, under the framework of the varying-coefficient HHR model, the meanings of the PE and AE analyses, along with their dynamic interpretation, are discussed.     

Bayesian analysis of generalized odds-rate hazards models for survival data

In the analysis of censored survival data Cox proportional hazards model (1972) is extremely popular among the practitioners. However, in many real-life situations the proportionality of the hazard ratios does not seem to be an appropriate assumption. To overcome such a problem, we consider a class of nonproportional hazards models known as generalized odds-rate class of regression models. The class is general enough to include several commonly used models, such as proportional hazards model, proportional odds model, and accelerated life time model. The theoretical and computational properties of these models have been re-examined. The propriety of the posterior has been established under some mild conditions. A simulation study is conducted and a detailed analysis of the data from a prostate cancer study is presented to further illustrate the proposed methodology.     

Exponential tilt models for two-group comparison with censored data

We study application of the Exponential Tilt Model (ETM) to compare survival distributions in two groups. The ETM assumes a parametric form for the density ratio of the two distributions. It accommodates a broad array of parametric models such as the log-normal and gamma models and can be sufficiently flexible to allow for crossing hazard and crossing survival functions. We develop a nonparametric likelihood approach to estimate ETM parameters in the presence of censoring and establish related asymptotic results. We compare the ETM to the Proportional Hazards Model (PHM) in simulation studies. When the proportional hazards assumption is not satisfied but the ETM assumption is, the ETM has better power for testing the hypothesis of no difference between the two groups. And, importantly, when the ETM relation is not satisfied but the PHM assumption is, the ETM can still have power reasonably close to that of the PHM. Application of the ETM is illustrated by a gastrointestinal tumor study.     

Semiparametric inference on the penetrances of rare genetic mutations based on a case-family design

A formal semiparametric statistical inference framework is proposed for the evaluation of the age-dependent penetrance of a rare genetic mutation, using family data generated under a case-family design, where phenotype and genotype information are collected from first-degree relatives of case probands carrying the targeted mutation. The proposed approach allows for unobserved risk factors that are correlated among family members. Some rigorous large sample properties are established, which show that the proposed estimators were asymptotically semiparametric efficient. A simulation study is conducted to evaluate the performance of the new approach, which shows the robustness of the proposed semiparametric approach and its advantage over the corresponding parametric approach. As an illustration, the proposed approach is applied to estimating the age-dependent cancer risk among carriers of the MSH2 or MLH1 mutation.     

A class of semiparametric transormation frailty models with application to estimating treatment effects and heterogenity in aids clinical traials

Muitivariate failure time data are common in medical research; com¬monly used statistical models for such correlated failure-time data include frailty and marginal models. Both types of models most often assume pro¬portional hazards (Cox, 1972); but the Cox model may not fit the data well This article presents a class of linear transformation frailty models that in¬cludes, as a special case, the proportional hazards model with frailty. We then propose approximate procedures to derive the best linear unbiased es¬timates and predictors of the regression parameters and frailties. We apply the proposed methods to analyze results of a clinical trial of different dose levels of didansine (ddl) among HIV-infected patients who were intolerant of zidovudine (ZDV). These methods yield estimates of treatment effects and of frailties corresponding to patient groups defined by clinical history prior to entry into the trial.     

Some recent developments for regression analysis of multivariate failure time data

Cox's seminal 1972 paper on regression methods for possibly censored failure time data popularized the use of time to an event as a primary response in prospective studies. But one key assumption of this and other regression methods is that observations are independent of one another. In many problems, failure times are clustered into small groups where outcomes within a group are correlated. Examples include failure times for two eyes from one person or for members of the same family.This paper presents a survey of models for multivariate failure time data. Two distinct classes of models are considered: frailty and marginal models. In a frailty model, the correlation is assumed to derive from latent variables (frailties) common to observations from the same cluster. Regression models are formulated for the conditional failure time distribution given the frailties. Alternatively, marginal models describe the marginal failure time distribution of each response while separately modelling the association among responses from the same cluster.We focus on recent extensions of the proportional hazards model for multivariate failure time data. Model formulation, parameter interpretation and estimation procedures are considered.     

A Simulation Study Comparing Modeling Approaches in an Illness-Death Multi-State Model

In many medical studies, there are covariates that change their values over time and their analysis is most often modeled using the Cox regression model. However, many of these time-dependent covariates can be expressed as an intermediate event, which can be modeled using a multi-state model. Using the relationship of time-dependent (discrete) covariates and multi-state models, we compare (via simulation studies) the Cox model with time-dependent covariates with the most frequently used multi-state regression models. This article also details the procedures for generating survival data arising from all approaches, including the Cox model with time-dependent covariates.     

Maximum likelihood estimation for the proportional hazards model with partly interval-censored data

Summary. The maximum likelihood estimator (MLE) for the proportional hazards model with partly interval-censored data is studied. Under appropriate regularity conditions, the MLEs of the regression parameter and the cumulative hazard function are shown to be consistent and asymptotically normal. Two methods to estimate the variance–covariance matrix of the MLE of the regression parameter are considered, based on a generalized missing information principle and on a generalized profile information procedure. Simulation studies show that both methods work well in terms of the bias and variance for samples of moderate size. An example illustrates the methods.     

On cumulative entropies

In analogy with the cumulative residual entropy recently proposed by Wang et al. [2003a. A new and robust information theoretic measure and its application to image alignment. In: Information Processing in Medical Imaging. Lecture Notes in Computer Science, vol. 2732, Springer, Heidelberg, pp. 388–400; 2003b. Cumulative residual entropy, a new measure of information and its application to image alignment. In: Proceedings on the Ninth IEEE International Conference on Computer Vision (ICCV’03), vol. 1, IEEE Computer Society Press, Silver Spring, MD, pp. 548–553], we introduce and study the cumulative entropy, which is a new measure of information alternative to the classical differential entropy. We show that the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X. Hence, our measure is particularly suitable to describe the information in problems related to ageing properties of reliability theory based on the past and on the inactivity times. Our results include various bounds to the cumulative entropy, its connection to the proportional reversed hazards model, and the study of its dynamic version that is shown to be increasing if the mean inactivity time is increasing. The empirical cumulative entropy is finally proposed to estimate the new information measure.