Goodness of Fit Tests for Proportional Hazards Regression Models

We present a test of goodness of fit for the proportional hazard regression model. The test is based on a score statistic for testing against local mixture alternatives. Contrary to the findings of several other authors, we detect a significant lack of fit in Freireich's leukemia data.     

Fit and Frailties in Proportional Hazards Regression

We exploit a conjectured equivalence between proportional hazards models with frailties and a particular subclass of non proportional hazards models, specifically those with declining effects, to address the question of fit. A goodness of fit test of the proportional hazards assumption against an alternative of declining regression effect is equivalent to a test for the presence of frailties. Such tests are now widely available in standard software. Although a number of tests of the proportional hazards assumption have been developed there is no test that directly formulates the alternative in terms of a non‐specified monotonic decline in regression effect and that enables a quantification of this in terms of a simple index. The index we obtain lies between zero and one such that, for any given set of covariates, values of the index close to one indicate that the fit cannot essentially be improved by allowing the possibility of regression effects to decline. Values closer to zero and away from one indicate that the fit can be improved by relaxing the proportional hazards constraint in this particular direction. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relating Cox's Proportional Hazard Model to the Exponential Model for Survival

An approximate representation is given for the partial likelihood estimate of the regression coefficient in Cox's proportional hazard model which indicates how it measures the association between survival time and covariate. The case of a single covariate is concentrated on. The representation is closely related to the first step of a Newton-Raphson iteration, i.e. to the score test. A similar representation for the Feigl-Zelen exponential model shows that a similar type of association is being measured, if observed lifetimes are interpreted as expected lifetimes of ordered exponentials. Necessary and sufficient conditions for the existence of Cox's estimate in the simple case are also written down.     

Proportional Hazards Regression for the Analysis of Clustered Survival Data from Case–Cohort Studies

Summary Case–cohort sampling is a commonly used and efficient method for studying large cohorts. Most existing methods of analysis for case–cohort data have concerned the analysis of univariate failure time data. However, clustered failure time data are commonly encountered in public health studies. For example, patients treated at the same center are unlikely to be independent. In this article, we consider methods based on estimating equations for case–cohort designs for clustered failure time data. We assume a marginal hazards model, with a common baseline hazard and common regression coefficient across clusters. The proposed estimators of the regression parameter and cumulative baseline hazard are shown to be consistent and asymptotically normal, and consistent estimators of the asymptotic covariance matrices are derived. The regression parameter estimator is easily computed using any standard Cox regression software that allows for offset terms. The proposed estimators are investigated in simulation studies, and demonstrated empirically to have increased efficiency relative to some existing methods. The proposed methods are applied to a study of mortality among Canadian dialysis patients.     

An Asymptotically Optimal Adaptive Selection Procedure in the Proportional Hazards Model with Conditionally Independent Censoring

Assume k independent populations are given which are distributed according to R, …,R (ϑ_i ∈ Θ ⊆ R ). Taking samples of size n the population with the smallest ϑ-value is to be selected. Using the framework of Le Cam's decision theory (Le Cam , 1986; Strasser , 1985) under mild regularity assumptions, an asymptotically optimal selection procedure is derived for the sequence of localized models. In the proportional hazards model with conditionally independent censoring, an asymptotically optimal adaptive selection procedure is constructed by substituting the unknown nuisance parameter by a kernel estimator.     

Bayesian Approach to Proportional Odds Models for Survival Analysis

In this paper Bayesian approach is adopted to develop inferences about parameters in proportional odds models. Bayesian posterior intervals for coefficients in proportional odds models are derived by using approximation given in Pregibon (1981). The results are illustrated by using the lung cancer survival data reported by Prentice (1973).     

Cox回归模型在肝癌病人生存分析中应用的研究

生存时间是癌症患者和临床医师共同关心的焦点,也是临床癌症诊治工作的重要指标之一．生存分析是研究多种因素与生存时间的关系以及关系程度的大小．Cox回归模型是生存分析中常用的方法之一．本文利用Cox回归模型对786名肝癌患者进行生存分析,确定影响肝癌患者预后的主要因素是癌栓、肝癌部位、治疗方式、肝脏储备功能、端粒酶活性、细胞增殖活性、γ-GT(γ-谷氮酰转肽酶)、术后复发等．为临床研究延长肝癌病人的生存期,提高其生存率提供了有力的依据．     

A Measure of Dependence for the Stratified Cox Proportional Hazards Regression Model

Kent and O'Quigley (1988) apply the concept of information gain to measure both global and partial dependence between explanatory variables and a censored response within the framework of the proportional hazards regression model of Cox (1972). The definition of this measure is extended to cover also the stratified Cox model.     

Analysis of Dichotomized Factorial Data Using Cox's Proportional Hazards Model and Related Tests

This paper describes how Cox's Proportional Hazards model may be used to analyze dichotomized factorial data obtained from a right-censored epidemiological study where time to response is of interest. Exact maximum likelihood estimates of the relative mortality rates are derived for any number of prognostic factors, but for the sake of simplicity, the mathematical details are presented for the case of two factors. This method is not based on the life table procedure. Kaplan-Meier estimates are obtained for the survival function of the internal control population, Which are in turn used to determine the expected number of deaths in the study population. The asymptotic (large sample) joint sampling distribution of the relative mortality rates is derived and some relevant simultaneous and conditional statistical tests are discussed. The relative mortality rates of several prognostic factors may be jointly considered as the multivariate extension of the familiar standard mortality ratio (SMR) of epidemiological studies. A numerical example is discussed to illustrate the method.     

A Multiple Imputation Approach to Regression Analysis for Doubly Censored Data with Application to AIDS Studies

Sun, Liao, and Pagano (1999) proposed an interesting estimating equation approach to Cox regression with doubly censored data. Here we point out that a modification of their proposal leads to a multiple imputation approach, where the double censoring is reduced to single censoring by imputing for the censored initiating times. For each imputed data set one can take advantage of many existing techniques and software for singly censored data. Under the general framework of multiple imputation, the proposed method is simple to implement and can accommodate modeling issues such as model checking, which has not been adequately discussed previously in the literature for doubly censored data. Here we illustrate our method with an application to a formal goodness-of-fit test and a graphical check for the proportional hazards model for doubly censored data. We reanalyze a well-known AIDS data set.     

Analysis of Survival Data with Two Dependent Competing Risks

Here we consider a competing risks model where the two risks of interest are not independent. The dependence is due to the additive effect of an independent contaminating risk on two initially independent risks. The problem is identifiable when the three risks fllow independent exponential distributions and also when the two initial risks follow proportional hazards model. Procedures are suggested for estimation and testing hypotheses regarding the parameters of the three exponentials in the first can and the constant of proportionality in the second case, when the information available consists of the times to death and the causes of death of the individuals.     

Merits of Modelling Multivariate Survival Data Using Random Effects Proportional Odds Model

Recently, there has been a great deal of interest in the analysis of multivariate survival data. In most epidemiological studies, survival times of the same cluster are related because of some unobserved risk factors such as the environmental or genetic factors. Therefore, modelling of dependence between events of correlated individuals is required to ensure a correct inference on the effects of treatments or covariates on the survival times. In the past decades, extension of proportional hazards model has been widely considered for modelling multivariate survival data by incorporating a random effect which acts multiplicatively on the hazard function. In this article, we consider the proportional odds model, which is an alternative to the proportional hazards model at which the hazard ratio between individuals converges to unity eventually. This is a reasonable property particularly when the treatment effect fades out gradually and the homogeneity of the population increases over time. The objective of this paper is to assess the influence of the random effect on the within‐subject correlation and the population heterogeneity. We are particularly interested in the properties of the proportional odds model with univariate random effect and correlated random effect. The correlations between survival times are derived explicitly for both choices of mixing distributions and are shown to be independent of the covariates. The time path of the odds function among the survivors are also examined to study the effect of the choice of mixing distribution. Modelling multivariate survival data using a univariate mixing distribution may be inadequate as the random effect not only characterises the dependence of the survival times, but also the conditional heterogeneity among the survivors. A robust estimate for the correlation of the logarithm of the survival times within a cluster is obtained disregarding the choice of the mixing distributions. The sensitivity of the estimate of the regression parameter under a misspecification of the mixing distribution is studied through simulation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improved Logrank-Type Tests for Survival Data Using Adaptive Weights

Summary .  For testing for treatment effects with time-to-event data, the logrank test is the most popular choice and has some optimality properties under proportional hazards alternatives. It may also be combined with other tests when a range of nonproportional alternatives are entertained. We introduce some versatile tests that use adaptively weighted logrank statistics. The adaptive weights utilize the hazard ratio obtained by fitting the model of Yang and Prentice (2005,  Biometrika   92 , 1–17). Extensive numerical studies have been performed under proportional and nonproportional alternatives, with a wide range of hazard ratios patterns. These studies show that these new tests typically improve the tests they are designed to modify. In particular, the adaptively weighted logrank test maintains optimality at the proportional alternatives, while improving the power over a wide range of nonproportional alternatives. The new tests are illustrated in several real data examples.     

The Analysis of Life Table and Follow-up Data with Covariates Using Poisson Regression Model

The Poisson regression model for the analysis of life table and follow-up data with covariates is presented. An example is presented to show how this technique can be used to construct a parsimonious model which describes a set of survival data. All parameters in the model, the hazard and survival functions are estimated by maximum likelihood.     

Weighted Likelihood Method for Grouped Survival Data in Case–Cohort Studies with Application to HIV Vaccine Trials

Summary Grouped failure time data arise often in HIV studies. In a recent preventive HIV vaccine efficacy trial, immune responses generated by the vaccine were measured from a case–cohort sample of vaccine recipients, who were subsequently evaluated for the study endpoint of HIV infection at prespecified follow‐up visits. Gilbert et al. (2005, Journal of Infectious Diseases 191 , 666–677) and Forthal et al. (2007, Journal of Immunology 178, 6596–6603) analyzed the association between the immune responses and HIV incidence with a Cox proportional hazards model, treating the HIV infection diagnosis time as a right‐censored random variable. The data, however, are of the form of grouped failure time data with case–cohort covariate sampling, and we propose an inverse selection probability‐weighted likelihood method for fitting the Cox model to these data. The method allows covariates to be time dependent, and uses multiple imputation to accommodate covariate data that are missing at random. We establish asymptotic properties of the proposed estimators, and present simulation results showing their good finite sample performance. We apply the method to the HIV vaccine trial data, showing that higher antibody levels are associated with a lower hazard of HIV infection.     

L1 Penalized Estimation in the Cox Proportional Hazards Model

This article presents a novel algorithm that efficiently computes L₁ penalized (lasso) estimates of parameters in high‐dimensional models. The lasso has the property that it simultaneously performs variable selection and shrinkage, which makes it very useful for finding interpretable prediction rules in high‐dimensional data. The new algorithm is based on a combination of gradient ascent optimization with the Newton–Raphson algorithm. It is described for a general likelihood function and can be applied in generalized linear models and other models with an L₁ penalty. The algorithm is demonstrated in the Cox proportional hazards model, predicting survival of breast cancer patients using gene expression data, and its performance is compared with competing approaches. An R package, penalized , that implements the method, is available on CRAN.     

Confidence Intervals for the Difference of Median Survival Times Using the Stratified Cox Proportional Hazards Model

The stratified Cox proportional hazards model is introduced to incorporate covariates and involve nonproportional treatment effect of two groups into the analysis and then the confidence interval estimators for the difference in median survival times of two treatments in stratified Cox model are proposed. The one is based on baseline survival functions of two groups, and the other on average survival functions of two groups. I illustrate the proposed methods with an example from a study conducted by the Radiation Therapy Oncology Group in cancer of the mouth and throat. Simulations are carried out to investigate the small‐sample properties of proposed methods in terms of coverage rates.     

A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data

Joint models for a time-to-event (e.g., survival) and a longitudinal response have generated considerable recent interest. The longitudinal data are assumed to follow a mixed effects model, and a proportional hazards model depending on the longitudinal random effects and other covariates is assumed for the survival endpoint. Interest may focus on inference on the longitudinal data process, which is informatively censored, or on the hazard relationship. Several methods for fitting such models have been proposed, most requiring a parametric distributional assumption (normality) on the random effects. A natural concern is sensitivity to violation of this assumption; moreover, a restrictive distributional assumption may obscure key features in the data. We investigate these issues through our proposal of a likelihood-based approach that requires only the assumption that the random effects have a smooth density. Implementation via the EM algorithm is described, and performance and the benefits for uncovering noteworthy features are illustrated by application to data from an HIV clinical trial and by simulation.