Modelling Converging Hazards in Survival Analysis

The Cox proportional hazards model has become the standard model for survival analysis. It is often seen as the null model in that "... explicit excuses are now needed to use different models" (Keiding, Proceedings of the XIXth International Biometric Conference, Cape Town, 1998). However, converging hazards also occur frequently in survival analysis. The Burr model, which may be derived as the marginal from a gamma frailty model, is one commonly used tool to model converging hazards. We outline this approach and introduce a mixed model which extends the Burr model and allows for both proportional and converging hazards. Although a semi-parametric model in its own right, we demonstrate how the mixed model can be derived via a gamma frailty interpretation, suggesting an E-M fitting procedure. We illustrate the modelling techniques using data on survival of hospice patients.     

Adjusted variable plots for Cox's proportional hazards regression model

Adjusted variable plots are useful in linear regression for outlier detection and for qualitative evaluation of the fit of a model. In this paper, we extend adjusted variable plots to Cox's proportional hazards model for possibly censored survival data. We propose three different plots: a risk level adjusted variable (RLAV) plot in which each observation in each risk set appears, a subject level adjusted variable (SLAV) plot in which each subject is represented by one point, and an event level adjusted variable (ELAV) plot in which the entire risk set at each failure event is represented by a single point. The latter two plots are derived from the RLAV by combining multiple points. In each point, the regression coefficient and standard error from a Cox proportional hazards regression is obtained by a simple linear regression through the origin fit to the coordinates of the pictured points. The plots are illustrated with a reanalysis of a dataset of 65 patients with multiple myeloma.     

Semi-parametric survival analysis via Dirichlet process mixtures of the First Hitting Time model

Time-to-event data often violate the proportional hazards assumption inherent in the popular Cox regression model. Such violations are especially common in the sphere of biological and medical data where latent heterogeneity due to unmeasured covariates or time varying effects are common. A variety of parametric survival models have been proposed in the literature which make more appropriate assumptions on the hazard function, at least for certain applications. One such model is derived from the First Hitting Time (FHT) paradigm which assumes that a subject’s event time is determined by a latent stochastic process reaching a threshold value. Several random effects specifications of the FHT model have also been proposed which allow for better modeling of data with unmeasured covariates. While often appropriate, these methods often display limited flexibility due to their inability to model a wide range of heterogeneities. To address this issue, we propose a Bayesian model which loosens assumptions on the mixing distribution inherent in the random effects FHT models currently in use. We demonstrate via simulation study that the proposed model greatly improves both survival and parameter estimation in the presence of latent heterogeneity. We also apply the proposed methodology to data from a toxicology/carcinogenicity study which exhibits nonproportional hazards and contrast the results with both the Cox model and two popular FHT models.

     

Survival models for fertility evaluation

"Two proportional hazards models for cohort fertility evaluation are constructed. Time-dependent covariates describe sources of heterogeneity between and within women regarding fertility characteristics. In the first model, U.S. birth rates specific to maternal age, race, parity, and birth cohort are used as underlying hazard rates. Covariate effects are estimated by maximizing the full likelihood. In the second model, covariate effects are estimated via Cox regression with stratified underlying hazard rates regarded as unknown nuisance parameters." The authors illustrate the models "with an evaluation of the fertility histories of the wives of workers at a manufacturing plant with potential for hazardous exposure. Adjustments to the U.S. birth rates for maternal age and parity zero experience are required with the first approach. Then, despite differences in the model-specific estimation procedures, the point estimates of the exposure effect and the estimated standard errors from the two models are practically equivalent."     

Bayesian Model Selection and Averaging in Additive and Proportional Hazards Models

Although Cox proportional hazards regression is the default analysis for time to event data, there is typically uncertainty about whether the effects of a predictor are more appropriately characterized by a multiplicative or additive model. To accommodate this uncertainty, we place a model selection prior on the coefficients in an additive-multiplicative hazards model. This prior assigns positive probability, not only to the model that has both additive and multiplicative effects for each predictor, but also to sub-models corresponding to no association, to only additive effects, and to only proportional effects. The additive component of the model is constrained to ensure non-negative hazards, a condition often violated by current methods. After augmenting the data with Poisson latent variables, the prior is conditionally conjugate, and posterior computation can proceed via an efficient Gibbs sampling algorithm. Simulation study results are presented, and the methodology is illustrated using data from the Framingham heart study.     

Cox regression analysis in presence of collinearity: an application to assessment of health risks associated with occupational radiation exposure

This paper considers the analysis of time to event data in the presence of collinearity between covariates. In linear and logistic regression models, the ridge regression estimator has been applied as an alternative to the maximum likelihood estimator in the presence of collinearity. The advantage of the ridge regression estimator over the usual maximum likelihood estimator is that the former often has a smaller total mean square error and is thus more precise. In this paper, we generalized this approach for addressing collinearity to the Cox proportional hazards model. Simulation studies were conducted to evaluate the performance of the ridge regression estimator. Our approach was motivated by an occupational radiation study conducted at Oak Ridge National Laboratory to evaluate health risks associated with occupational radiation exposure in which the exposure tends to be correlated with possible confounders such as years of exposure and attained age. We applied the proposed methods to this study to evaluate the association of radiation exposure with all-cause mortality.     

Monte-Carlo Sensitivity Analysis for Controlled Direct Effects Using Marginal Structural Models in the Presence of Confounded Mediators

In randomized trials, investigators are frequently interested in estimating the direct effect of a treatment on an outcome that is not relayed by intermediate variables, in addition to the usual intention-to-treat (ITT) effect. Even if the ITT effect is not confounded due to randomization, the direct effect is not identified when unmeasured variables affect the intermediate and outcome variables. Although the unmeasured variables cannot be adjusted for in the models, it is still important to evaluate the potential bias of these variables quantitatively. This article proposes a sensitivity analysis method for controlled direct effects using a marginal structural model that is an extension of the sensitivity analysis method of unmeasured confounding introduced in the context of observational studies. The proposed method is illustrated using a randomized trial of depression.     

What price semiparametric Cox regression?

Lifetime Data Analysis - Cox’s proportional hazards regression model is the standard method for modelling censored life-time data with covariates. In its standard form, this method relies on...     

A nonparametric test to compare survival distributions with covariate adjustment

Summary.  The analysis of covariance is a technique that is used to improve the power of a k -sample test by adjusting for concomitant variables. If the end point is the time of survival, and some observations are right censored, the score statistic from the Cox proportional hazards model is the method that is most commonly used to test the equality of conditional hazard functions. In many situations, however, the proportional hazards model assumptions are not satisfied. Specifically, the relative risk function is not time invariant or represented as a log-linear function of the covariates. We propose an asymptotically valid k -sample test statistic to compare conditional hazard functions which does not require the assumption of proportional hazards, a parametric specification of the relative risk function or randomization of group assignment. Simulation results indicate that the performance of this statistic is satisfactory. The methodology is demonstrated on a data set in prostate cancer.     

Risk patterns in drug safety study using relative times by accelerated failure time models when proportional hazards assumption is questionable: an illustrative case study of cancer risk of patients on glucose‐lowering therapies

Observational drug safety studies may be susceptible to confounding or protopathic bias. This bias may cause a spurious relationship between drug exposure and adverse side effect when none exists and may lead to unwarranted safety alerts. The spurious relationship may manifest itself through substantially different risk levels between exposure groups at the start of follow‐up when exposure is deemed too short to have any plausible biological effect of the drug. The restrictive proportional hazards assumption with its arbitrary choice of baseline hazard function renders the commonly used Cox proportional hazards model of limited use for revealing such potential bias. We demonstrate a fully parametric approach using accelerated failure time models with an illustrative safety study of glucose‐lowering therapies and show that its results are comparable against other methods that allow time‐varying exposure effects. Our approach includes a wide variety of models that are based on the flexible generalized gamma distribution and allows direct comparisons of estimated hazard functions following different exposure‐specific distributions of survival times. This approach lends itself to two alternative metrics, namely relative times and difference in times to event, allowing physicians more ways to communicate patient's prognosis without invoking the concept of risks, which some may find hard to grasp. In our illustrative case study, substantial differences in cancer risks at drug initiation followed by a gradual reduction towards null were found. This evidence is compatible with the presence of protopathic bias, in which undiagnosed symptoms of cancer lead to switches in diabetes medication. Copyright © 2015 John Wiley & Sons, Ltd.     

A New Estimator for Cox Proportional Hazard Regression Model in Presence of Collinearity

We propose a new approach to estimate the parameters of the Cox proportional hazards model in the presence of collinearity. Generally, a maximum partial likelihood estimator is used to estimate parameters for the Cox proportional hazards model. However, the maximum partial likelihood estimators can be seriously affected by the presence of collinearity since the parameter estimates result in large variances.

In this study, we develop a Liu-type estimator for Cox proportional hazards model parameters and compare it with a ridge regression estimator based on the scalar mean squared error (MSE). Finally, we evaluate its performance through a simulation study.     

Sequential tests for non-proportional hazards data

In clinical trials survival endpoints are usually compared using the log-rank test. Sequential methods for the log-rank test and the Cox proportional hazards model are largely reported in the statistical literature. When the proportional hazards assumption is violated the hazard ratio is ill-defined and the power of the log-rank test depends on the distribution of the censoring times. The average hazard ratio was proposed as an alternative effect measure, which has a meaningful interpretation in the case of non-proportional hazards, and is equal to the hazard ratio, if the hazards are indeed proportional. In the present work we prove that the average hazard ratio based sequential test statistics are asymptotically multivariate normal with the independent increments property. This allows for the calculation of group-sequential boundaries using standard methods and existing software. The finite sample characteristics of the new method are examined in a simulation study in a proportional and a non-proportional hazards setting.     

Inference for proportional hazard model with propensity score

Since the publication of the seminal paper by Cox (1972), proportional hazard model has become very popular in regression analysis for right censored data. In observational studies, treatment assignment may depend on observed covariates. If these confounding variables are not accounted for properly, the inference based on the Cox proportional hazard model may perform poorly. As shown in Rosenbaum and Rubin (1983), under the strongly ignorable treatment assignment assumption, conditioning on the propensity score yields valid causal effect estimates. Therefore we incorporate the propensity score into the Cox model for causal inference with survival data. We derive the asymptotic property of the maximum partial likelihood estimator when the model is correctly specified. Simulation results show that our method performs quite well for observational data. The approach is applied to a real dataset on the time of readmission of trauma patients. We also derive the asymptotic property of the maximum partial likelihood estimator with a robust variance estimator, when the model is incorrectly specified.     

Partial Partial Likelihood

The maximum likelihood and maximum partial likelihood approaches to the proportional hazards model are unified. The purpose is to give a general approach to the analysis of the proportional hazards model, whether the baseline distribution is absolutely continuous, discrete, or a mixture. The advantage is that heavily tied data will be analyzed with a discrete time model, while data with no ties is analyzed with ordinary Cox regression. Data sets in between are treated by a compromise between the discrete time model and Efron's approach to tied data in survival analysis, and the transitions between modes are automatic. A simulation study is conducted comparing the proposed approach to standard methods of handling ties. A recent suggestion, that revives Breslow's approach to tied data, is finally discussed.     

Data-driven smooth tests of the proportional hazards assumption

A new test of the proportional hazards assumption in the Cox model is proposed. The idea is based on Neyman’s smooth tests. The Cox model with proportional hazards (i.e. time-constant covariate effects) is embedded in a model with a smoothly time-varying covariate effect that is expressed as a combination of some basis functions (e.g., Legendre polynomials, cosines). Then the smooth test is the score test for significance of these artificial covariates. Furthermore, we apply a modification of Schwarz’s selection rule to choosing the dimension of the smooth model (the number of the basis functions). The score test is then used in the selected model. In a simulation study, we compare the proposed tests with standard tests based on the score process.     

Assessing gamma frailty models for clustered failure time data

Proportional hazards frailty models use a random effect, so called frailty, to construct association for clustered failure time data. It is customary to assume that the random frailty follows a gamma distribution. In this paper, we propose a graphical method for assessing adequacy of the proportional hazards frailty models. In particular, we focus on the assessment of the gamma distribution assumption for the frailties. We calculate the average of the posterior expected frailties at several followup time points and compare it at these time points to 1, the known mean frailty. Large discrepancies indicate lack of fit. To aid in assessing the goodness of fit, we derive and estimate the standard error of the mean of the posterior expected frailties at each time point examined. We give an example to illustrate the proposed methodology and perform sensitivity analysis by simulations.     

Estimating Regression Parameters in an Extended Proportional Odds Model

The proportional odds model may serve as a useful alternative to the Cox proportional hazards model to study association between covariates and their survival functions in medical studies. In this article, we study an extended proportional odds model that incorporates the so-called "external" time-varying covariates. In the extended model, regression parameters have a direct interpretation of comparing survival functions, without specifying the baseline survival odds function. Semiparametric and maximum likelihood estimation procedures are proposed to estimate the extended model. Our methods are demonstrated by Monte-Carlo simulations, and applied to a landmark randomized clinical trial of a short course Nevirapine (NVP) for mother-to-child transmission (MTCT) of human immunodeficiency virus type-1 (HIV-1). Additional application includes analysis of the well-known Veterans Administration (VA) Lung Cancer Trial.     

A Flexible Approach to Time-varying Coefficients in the Cox Regression Setting

Research on methods for studying time-to-event data (survival analysis) has been extensive in recent years. The basic model in use today represents the hazard function for an individual through a proportional hazards model (Cox, 1972). Typically, it is assumed that a covariate's effect on the hazard function is constant throughout the course of the study. In this paper we propose a method to allow for possible deviations from the standard Cox model, by allowing the effect of a covariate to vary over time. This method is based on a dynamic linear model. We present our method in terms of a Bayesian hierarchical model. We fit the model to the data using Markov chain Monte Carlo methods. Finally, we illustrate the approach with several examples. This revised version was published online in July 2006 with corrections to the Cover Date.     

Evaluation strategies for case series: is Cox regression an alternative to the self controlled case series method for terminal events?

In this paper, we deal with the analysis of case series. The self-controlled case series method (SCCS) was developed to analyse the temporal association between time-varying exposure and an outcome event. We apply the SCCS method to the vaccination data of the German Examination Survey for Children and Adolescents (KiGGS). We illustrate that the standard SCCS method cannot be applied to terminal events such as death. In this situation, an extension of SCCS adjusted for terminal events gives unbiased point estimators. The key question of this paper is whether the general Cox regression model for time-dependent covariates may be an alternative to the adjusted SCCS method for terminal events. In contrast to the SCCS method, Cox regression is included in most software packages (SPSS, SAS, STATA, R, …) and it is easy to use. We can show that Cox regression is applicable to test the null hypothesis. In our KiGGS example without censored data, the Cox regression and the adjusted SCCS method yield point estimates almost identical to the standard SCCS method. We have conducted several simulation studies to complete the comparison of the two methods. The Cox regression shows a tendency to underestimate the true effect with prolonged risk periods and strong effects (Relative Incidence >2). If risk of the event is strongly affected by the age, the adjusted SCCS method slightly overestimates the predefined exposure effect. Cox regression has the same efficiency as the adjusted SCCS method in the simulation.     

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.