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)     

Cause‐Specific Cumulative Incidence Estimation and the Fine and Gray Model Under Both Left Truncation and Right Censoring

Summary The standard estimator for the cause‐specific cumulative incidence function in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is a weighted empirical cumulative distribution function and the other a product‐limit estimator. This equivalence suggests an alternative view of the analysis of time‐to‐event data with left truncation and right censoring: individuals who are still at risk or experienced an earlier competing event receive weights from the censoring and truncation mechanisms. As a consequence, inference on the cumulative scale can be performed using weighted versions of standard procedures. This holds for estimation of the cause‐specific cumulative incidence function as well as for estimation of the regression parameters in the Fine and Gray proportional subdistribution hazards model. We show that, with the appropriate filtration, a martingale property holds that allows deriving asymptotic results for the proportional subdistribution hazards model in the same way as for the standard Cox proportional hazards model. Estimation of the cause‐specific cumulative incidence function and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time‐dependent weights. We show the implementation in the R statistical package. The proportional subdistribution hazards model is used to investigate the effect of calendar period as a deterministic external time varying covariate, which can be seen as a special case of left truncation, on AIDS related and non‐AIDS related cumulative mortality.     

Confidence bands for cumulative incidence curves under the additive risk model

In the context of competing risks, the cumulative incidence function is often used to summarize the cause-specific failure-time data. As an alternative to the proportional hazards model, the additive risk model is used to investigate covariate effects by specifying that the subject-specific hazard function is the sum of a baseline hazard function and a regression function of covariates. Based on such a formulation, we present an approach to constructing simultaneous confidence intervals for the cause-specific cumulative incidence function of patients with given risk factors. A melanoma data set is used for the purpose of illustration.     

Score tests for homogeneity of regression effect in the proportional hazards model

A simple model, containing the proportional hazards regression model as a special case, is presented. The purpose of the model is to provide a framework in which specific alternatives to the proportional hazards assumption may be tested. Rank-invariant score tests for linear, quadratic, or exponential trends can, for instance, all be undertaken within this framework. In the case of the two-sample problem the required calculations are shown to take a particularly simple form. Special consideration is given to the two-sample case in which there is an inversion of the regression effect, i.e., where the hazard functions cross at some given point. Both of the motivating examples are concerned with this problem. Computational aspects are relatively straightforward and some discussion on this is provided.     

On robustness and model flexibility in survival analysis: transformed hazard models and average effects

Yin and Ibrahim (2005a, Biometrics 61, 208-216) use a Box-Cox transformed hazard model to acknowledge uncertainty about how a linear predictor acts upon the hazard function of a failure-time response. Particularly, additive and proportional hazards models arise for particular values of the transformation parameter. As is often the case, however, this added model flexibility is obtained at the cost of lessened parameter interpretability. Particularly, the interpretation of the coefficients in the linear predictor is intertwined with the value of the transformation parameter. Moreover, some data sets contain very little information about this parameter. To shed light on the situation, we consider average effects based on averaging (over the joint distribution of the explanatory variables and the failure-time response) the partial derivatives of the hazard, or the log-hazard, with respect to the explanatory variables. First, we consider fitting models which do assume a particular form of covariate effects, for example, proportional hazards or additive hazards. In some such circumstances, average effects are seen to be inferential targets which are robust to the effect form being misspecified. Second, we consider average effects as targets of inference when using the transformed hazard model. We show that in addition to being more interpretable inferential targets, average effects can sometimes be estimated more efficiently than the corresponding regression coefficients.     

Statistical analysis of survival data from radiation countermeasure experiments

We present an introduction to, and examples of, Cox proportional hazards regression in the context of animal lethality studies of potential radioprotective agents. This established method is seldom used to analyze survival data collected in such studies, but is appropriate in many instances. Presenting a hypothetical radiation study that examines the efficacy of a potential radioprotectant both in the absence and presence of a potential modifier, we detail how to implement and interpret results from a Cox proportional hazards regression analysis used to analyze the survival data, and we provide relevant SAS? code. Cox proportional hazards regression analysis of survival data from lethal radiation experiments (1) considers the whole distribution of survival times rather than simply the commonly used proportions of animals that survived, (2) provides a unified analysis when multiple factors are present, and (3) can increase statistical power by combining information across different levels of a factor. Cox proportional hazards regression should be considered as a potential statistical method in the toolbox of radiation researchers.     

A semiparametric estimate of treatment effects with censored data

A semiparametric estimate of an average regression effect with right-censored failure time data has recently been proposed under the Cox-type model where the regression effect beta(t) is allowed to vary with time. In this article, we derive a simple algebraic relationship between this average regression effect and a measurement of group differences in k-sample transformation models when the random error belongs to the G(rho) family of Harrington and Fleming (1982, Biometrika 69, 553-566), the latter being equivalent to the conditional regression effect in a gamma frailty model. The models considered here are suitable for the attenuating hazard ratios that often arise in practice. The results reveal an interesting connection among the above three classes of models as alternatives to the proportional hazards assumption and add to our understanding of the behavior of the partial likelihood estimate under nonproportional hazards. The algebraic relationship provides a simple estimator under the transformation model. We develop a variance estimator based on the empirical influence function that is much easier to compute than the previously suggested resampling methods. When there is truncation in the right tail of the failure times, we propose a method of bias correction to improve the coverage properties of the confidence intervals. The estimate, its estimated variance, and the bias correction term can all be calculated with minor modifications to standard software for proportional hazards regression.     

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)     

The Mizon-Richard encompassing test for the Cox and Aalen additive hazards models.

The Cox hazards model (Cox, 1972, Journal of the Royal Statistical Society, Series B34, 187-220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model (Aalen, 1980, in Lecture Notes in Statistics-2, 1-25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon-Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model (Martinussen and Scheike, 2002, Biometrika 89, 283-298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.     

Modeling multivariate survival data by a semiparametric random effects proportional odds model

In this article, the focus is on the analysis of multivariate survival time data with various types of dependence structures. Examples of multivariate survival data include clustered data and repeated measurements from the same subject, such as the interrecurrence times of cancer tumors. A random effect semiparametric proportional odds model is proposed as an alternative to the proportional hazards model. The distribution of the random effects is assumed to be multivariate normal and the random effect is assumed to act additively to the baseline log-odds function. This class of models, which includes the usual shared random effects model, the additive variance components model, and the dynamic random effects model as special cases, is highly flexible and is capable of modeling a wide range of multivariate survival data. A unified estimation procedure is proposed to estimate the regression and dependence parameters simultaneously by means of a marginal-likelihood approach. Unlike the fully parametric case, the regression parameter estimate is not sensitive to the choice of correlation structure of the random effects. The marginal likelihood is approximated by the Monte Carlo method. Simulation studies are carried out to investigate the performance of the proposed method. The proposed method is applied to two well-known data sets, including clustered data and recurrent event times data.     

A proportional hazards model for interval-censored failure time data

This paper develops a method for fitting the proportional hazards regression model when the data contain left-, right-, or interval-censored observations. Results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log-rank test for comparison of several survival curves. The method is used to analyze data from an animal tumorigenicity study and also a clinical trial.     

Model misspecification in proportional hazards regression

The proportional hazards model is frequently used to evaluatethe effect of treatment on failure time events in randomisedclinical trials. Concomitant variables are usually availableand may be considered for use in the primary analyses underthe assumption that incorporating them may reduce bias or improveefficiency. In this paper we consider two approaches to includingcovariate information: regression modelling and stratification.We focus on the setting where covariate effects are nonproportionaland we compare the bias, efficiency and coverage propertiesof these approaches. These results indicate that our intuitionbased on linear model analysis of covariance is misleading.Covariate adjustment in proportional hazards models has littleeffect on the variance but may significantly improve the accuracyof the treatment effect estimator.     

Applying a data duplication technique in linear regression analysis of waiting time to pregnancy

This analysis demonstrates the application of a data duplication technique in linear regression with censored observations of the waiting time to third pregnancy ending in two outcome types, using data from Malaysia. The linear model not only confirmed the results obtained by the Cox proportional hazards model, but also identified two additional significant factors. The method provides a useful alternative when Cox proportionality assumption of the hazards is violated.     

Tree-augmented Cox proportional hazards models

We study a hybrid model that combines Cox proportional hazards regression with tree-structured modeling. The main idea is to use step functions, provided by a tree structure, to 'augment' Cox (1972) proportional hazards models. The proposed model not only provides a natural assessment of the adequacy of the Cox proportional hazards model but also improves its model fitting without loss of interpretability. Both simulations and an empirical example are provided to illustrate the use of the proposed method.     

The Mizon–Richard Encompassing Test for the Cox and Aalen Additive Hazards Models

Summary .   The Cox hazards model ( Cox, 1972 , Journal of the Royal Statistical Society, Series B 34, 187–220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model ( Aalen, 1980 , in Lecture Notes in Statistics-2 , 1–25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon–Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model ( Martinussen and Scheike, 2002 , Biometrika 89, 283–298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.     

Residuals for proportional hazards models with interval-censored survival data

We develop diagnostic tools for use with proportional hazards models for interval-censored survival data. We propose counterparts to the Cox-Snell, Lagakos (or martingale), deviance, and Schoenfeld residuals. Many of the properties of these residuals carry over to the interval-censored case. In particular, the interval-censored versions of the Lagakos and Schoenfeld residuals may be derived as components of suitable score statistics. The Lagakos residuals may be used to check regression relationships, while the Schoenfeld residuals can help to detect nonproportional hazards in semiparametric models. The methods apply to parametric models and to the semiparametric model with discrete observation times.     

Bayesian inferences on umbrella orderings

In regression applications with categorical predictors, interest often focuses on comparing the null hypothesis of homogeneity to an ordered alternative. This article proposes a Bayesian approach for addressing this problem in the setting of normal linear and probit regression models. The regression coefficients are assigned a conditionally conjugate prior density consisting of mixtures of point masses at 0 and truncated normal densities, with a (possibly unknown) changepoint parameter included to accommodate umbrella ordering. Two strategies of prior elicitation are considered: (1) a Bayesian Bonferroni approach in which the probability of the global null hypothesis is specified and local hypotheses are considered independent; and (2) an approach which treats these probabilities as random. A single Gibbs sampling chain can be used to obtain posterior probabilities for the different hypotheses and to estimate regression coefficients and predictive quantities either by model averaging or under the preferred hypothesis. The methods are applied to data from a carcinogenesis study.     

Regression analysis of grouped survival data with application to breast cancer data

Use of the proportional hazards regression model (Cox 1972) substantially liberalized the analysis of censored survival data with covariates. Available procedures for estimation of the relative risk parameter, however, do not adequately handle grouped survival data, or large data sets with many tied failure times. The grouped data version of the proportional hazards model is proposed here for such estimation. Asymptotic likelihood results are given, both for the estimation of the regression coefficient and the survivor function. Some special results are given for testing the hypothesis of a zero regression coefficient which leads, for example, to a generalization of the log-rank test for the comparison of several survival curves. Application to breast cancer data, from the National Cancer Institute-sponsored End Results Group, indicates that previously noted race differences in breast cancer survival times are explained to a large extent by differences in disease extent and other demographic characteristics at diagnosis.     

Multi-state survival models with treatment effects and biomarkers: Simulations for study design assessment

BackgroundComparative effectiveness studies of cancer therapeutics in observational data face confounding by patterns of clinical treatment over time. The validity of survival analysis in longitudinal health records depends on study design choices including index date definition and model specification for covariate adjustment.MethodsOverall survival in cancer is a multi-state transition process with mortality and treatment switching as competing risks. Parametric Weibull regression quantifies proportionality of hazards across lines of therapy in real-world cohorts of 12 solid tumor types. Study design assessments compare alternative analytic models in simulations with realistic disproportionality. The multi-state simulation framework is adaptable to alternative treatment effect profiles and exposure patterns.ResultsEvent-specific hazards of treatment-switching and death are not proportional across lines of therapy in 12 solid tumor types. Study designs that include all eligible lines of therapy per subject showed lower bias and variance than designs that select one line per subject. Confounding by line number was effectively mitigated across a range of simulation scenarios by Cox proportional hazards models with stratified baseline hazards and inverse probability of treatment weighting.ConclusionQuantitative study design assessment can inform the planning of observational research in clinical oncology by demonstrating the potential impact of model misspecification. Use of empirical parameter estimates in simulation designs adapts analytic recommendations to the clinical population of interest.     

A group sequential test for survival trials: an alternative to rank-based procedures

A method of interim monitoring is described for survival trials in which the proportional hazards assumption may not hold. This method extends the test statistics based on the cumulative weighted difference in the Kaplan-Meier estimates (Pepe and Fleming, 1989, Biometrics 45, 497-507) to the sequential setting. Therefore, it provides a useful alternative to the group sequential linear rank tests. With an appropriate weight function, the test statistic itself provides an estimator for the cumulative weighted difference in survival probabilities, which is an interpretable measure for the treatment difference, especially when the proportional hazards model fails. The method is illustrated based on the design of a real trial. The operating characteristics are studied through a small simulation.