Marginal Likelihood Estimation for Proportional Odds Models with Right Censored Data

One majoraspect in medical research is to relate the survival times ofpatients with the relevant covariates or explanatory variables.The proportional hazards model has been used extensively in thepast decades with the assumption that the covariate effects actmultiplicatively on the hazard function, independent of time.If the patients become more homogeneous over time, say the treatmenteffects decrease with time or fade out eventually, then a proportionalodds model may be more appropriate. In the proportional oddsmodel, the odds ratio between patients can be expressed as afunction of their corresponding covariate vectors, in which,the hazard ratio between individuals converges to unity in thelong run. In this paper, we consider the estimation of the regressionparameter for a semiparametric proportional odds model at whichthe baseline odds function is an arbitrary, non-decreasing functionbut is left unspecified. Instead of using the exact survivaltimes, only the rank order information among patients is used.A Monte Carlo method is used to approximate the marginal likelihoodfunction of the rank invariant transformation of the survivaltimes which preserves the information about the regression parameter.The method can be applied to other transformation models withcensored data such as the proportional hazards model, the generalizedprobit model or others. The proposed method is applied to theVeteran's Administration lung cancer trial data.     

A Class of Parametric Dynamic Survival Models

A class of parametric dynamic survival models are explored in which only limited parametric assumptions are made, whilst avoiding the assumption of proportional hazards. Both the log-baseline hazard and covariate effects are modelled by piecewise constant and correlated processes. The method of estimation is to use Markov chain Monte Carlo simulations Gibbs sampling with a Metropolis–Hastings step. In addition to standard right censored data sets, extensions to accommodate interval censoring and random effects are included. The model is applied to two well known and illustrative data sets, and the dynamic variability of covariate effects investigated.     

Choice of parametric models in survival analysis: applications to monotherapy for epilepsy and cerebral palsy

Summary. In the analysis of medical survival data, semiparametric proportional hazards models are widely used. When the proportional hazards assumption is not tenable, these models will not be suitable. Other models for covariate effects can be useful. In particular, we consider accelerated life models, in which the effect of covariates is to scale the quantiles of the base-line distribution. Solomon and Hutton have suggested that there is some robustness to misspecification of survival regression models. They showed that the relative importance of covariates is preserved under misspecification with assumptions of small coefficients and orthogonal transformation of covariates. We elucidate these results by applications to data from five trials which compare two common anti-epileptic drugs (carbamazepine versus sodium valporate monotherapy for epilepsy) and to survival of a cohort of people with cerebral palsy. Results on the robustness against model misspecification depend on the assumptions of small coefficients and on the underlying distribution of the data. These results hold in cerebral palsy but do not hold in epilepsy data which have early high hazard rates. The orthogonality of coefficients is not important. However, the choice of model is important for an estimation of the magnitude of effects, particularly if the base-line shape parameter indicates high initial hazard rates.     

A dependent Dirichlet process model for survival data with competing risks

In this paper, we first propose a dependent Dirichlet process (DDP) model using a mixture of Weibull models with each mixture component resembling a Cox model for survival data. We then build a Dirichlet process mixture model for competing risks data without regression covariates. Next we extend this model to a DDP model for competing risks regression data by using a multiplicative covariate effect on subdistribution hazards in the mixture components. Though built on proportional hazards (or subdistribution hazards) models, the proposed nonparametric Bayesian regression models do not require the assumption of constant hazard (or subdistribution hazard) ratio. An external time-dependent covariate is also considered in the survival model. After describing the model, we discuss how both cause-specific and subdistribution hazard ratios can be estimated from the same nonparametric Bayesian model for competing risks regression. For use with the regression models proposed, we introduce an omnibus prior that is suitable when little external information is available about covariate effects. Finally we compare the models’ performance with existing methods through simulations. We also illustrate the proposed competing risks regression model with data from a breast cancer study. An R package “DPWeibull” implementing all of the proposed methods is available at CRAN.

     

On proportional hazards assumption under the random effects models

The proportional hazards mixed-effects model (PHMM) was proposed to handle dependent survival data. Motivated by its application in genetic epidemiology, we study the interpretation of its parameter estimates under violations of the proportional hazards assumption. The estimated fixed effect turns out to be an averaged regression effect over time, while the estimated variance component could be unaffected, inflated or attenuated depending on whether the random effect is on the baseline hazard, and whether the non-proportional regression effect decreases or increases over time. Using the conditional distribution of the covariates we define the standardized covariate residuals, which can be used to check the proportional hazards assumption. The model checking technique is illustrated on a multi-center lung cancer trial.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

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.     

Regression analysis of competing risks data via semi-parametric additive hazard model

When the subjects in a study possess different demographic and disease characteristics and are exposed to more than one types of failure, a practical problem is to assess the covariate effects on each type of failure as well as on all-cause failure. The most widely used method is to employ the Cox models on each cause-specific hazard and the all-cause hazard. It has been pointed out that this method causes the problem of internal inconsistency. To solve such a problem, the additive hazard models have been advocated. In this paper, we model each cause-specific hazard with the additive hazard model that includes both constant and time-varying covariate effects. We illustrate that the covariate effect on all-cause failure can be estimated by the sum of the effects on all competing risks. Using data from a longitudinal study on breast cancer patients, we show that the proposed method gives simple interpretation of the final results, when the primary covariate effect is constant in the additive manner on each cause-specific hazard. Based on the given additive models on the cause-specific hazards, we derive the inferences for the adjusted survival and cumulative incidence functions.     

Partial orders with respect to continuous covariates and tests for the proportional hazards model

Several omnibus tests of the proportional hazards assumption have been proposed in the literature. In the two-sample case, tests have also been developed against ordered alternatives like monotone hazard ratio and monotone ratio of cumulative hazards. Here we propose a natural extension of these partial orders to the case of continuous and potentially time varying covariates, and develop tests for the proportional hazards assumption against such ordered alternatives. The work is motivated by applications in biomedicine and economics where covariate effects often decay over lifetime. The proposed tests do not make restrictive assumptions on the underlying regression model, and are applicable in the presence of time varying covariates, multiple covariates and frailty. Small sample performance and an application to real data highlight the use of the framework and methodology to identify and model the nature of departures from proportionality.     

Nonparametric Tests for Stratum Effects in the Cox Model

Thispaper considers the stratified proportional hazards model witha focus on the assessment of stratum effects. The assessmentof such effects is often of interest, for example, in clinicaltrials. In this case, two relevant tests are the test of stratuminteraction with covariates and the test of stratum interactionwith baseline hazard functions. For the test of stratum interactionwith covariates, one can use the partial likelihood method (Kalbfleischand Prentice, 1980; Lin, 1994). For the test of stratum interactionwith baseline hazard functions, however, there seems to be noformal test available. We consider this problem and propose aclass of nonparametric tests. The asymptotic distributions ofthe tests are derived using the martingale theory. The proposedtests can also be used for survival comparisons which need tobe adjusted for covariate effects. The method is illustratedwith data from a lung cancer clinical trial.     

Comparison of goodness of fit tests for the cox proportional hazards model

The proportional hazards regression model of Cox(1972) is widely used in analyzing survival data. We examine several goodness of fit tests for checking the proportionality of hazards in the Cox model with two-sample censored data, and compare the performance of these tests by a simulation study. The strengths and weaknesses of the tests are pointed out. The effects of the extent of random censoring on the size and power are also examined. Results of a simulation study demonstrate that Gill and Schumacher's test is most powerful against a broad range of monotone departures from the proportional hazards assumption, but it may not perform as well fail for alternatives of nonmonotone hazard ratio. For the latter kind of alternatives, Andersen's test may detect patterns of irregular changes in hazards.     

A parametric dynamic survival model applied to breast cancer survival times

Summary. Much current analysis of cancer registry data uses the semiparametric proportional hazards Cox model. In this paper, the time-dependent effect of various prognostic indicators on breast cancer survival times from the West Midlands Cancer Intelligence Unit are investigated. Using Bayesian methodology and Markov chain Monte Carlo estimation methods, we develop a parametric dynamic survival model which avoids the proportional hazards assumption. The model has close links to that developed by both Gamerman and Sinha and co-workers: the log-base-line hazard and covariate effects are piecewise constant functions, related between intervals by a simple stochastic evolution process. Here this evolution is assigned a parametric distribution, with a variance that is further included as a hyperparameter. To avoid problems of convergence within the Gibbs sampler, we consider using a reparameterization. It is found that, for some of the prognostic indicators considered, the estimated effects change with increasing follow-up time. In general those prognostic indicators which are thought to be representative of the most hazardous groups (late-staged tumour and oldest age group) have a declining effect.     

Stochastic properties of the mixed accelerated hazard models

The accelerated hazard model in survival analysis assumes that the covariate effect acts the time scale of the baseline hazard rate. In this paper, we study the stochastic properties of the mixed accelerated hazard model since the covariate is considered basically unobservable. We build dependence structure between the population variable and the covariate, and also present some preservation properties. Using some well-known stochastic orders, we compare two mixed accelerated hazards models arising out of different choices of distributions for unobservable covariates or different baseline hazard rate functions.     

Estimation of odds of concordance based on the Aalen additive model

The Cox regression model is often used when analyzing survival data as it provides a convenient way of summarizing covariate effects in terms of relative risks. The proportional hazards assumption may not hold, however. A typical violation of the assumption is time-changing covariate effects. Under such scenarios one may use more flexible models but the results from such models may be complicated to communicate and it is desirable to have simple measures of a treatment effect, say. In this paper we focus on the odds-of-concordance measure that was recently studied by Schemper et al. (Stat Med 28:2473?C2489, 2009). They suggested to estimate this measure using weighted Cox regression (WCR). Although WCR may work in many scenarios no formal proof can be established. We suggest an alternative estimator of the odds-of-concordance measure based on the Aalen additive hazards model. In contrast to the WCR, one may derive the large sample properties for this estimator making formal inference possible. The estimator also allows for additional covariate effects.     

Flexible Bayesian Modelling for Survival Data

The analysis of failure time data often involves two strong assumptions. The proportional hazards assumption postulates that hazard rates corresponding to different levels of explanatory variables are proportional. The additive effects assumption specifies that the effect associated with a particular explanatory variable does not depend on the levels of other explanatory variables. A hierarchical Bayes model is presented, under which both assumptions are relaxed. In particular, time-dependent covariate effects are explicitly modelled, and the additivity of effects is relaxed through the use of a modified neural network structure. The hierarchical nature of the model is useful in that it parsimoniously penalizes violations of the two assumptions, with the strength of the penalty being determined by the data.     

Delayed treatment effects,treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology

In the analysis of survival times, the logrank test and the Cox model have been established as key tools, which do not require specific distributional assumptions. Under the assumption of proportional hazards, they are efficient and their results can be interpreted unambiguously. However, delayed treatment effects, disease progression, treatment switchers or the presence of subgroups with differential treatment effects may challenge the assumption of proportional hazards. In practice, weighted logrank tests emphasizing either early, intermediate or late event times via an appropriate weighting function may be used to accommodate for an expected pattern of non-proportionality. We model these sources of non-proportional hazards via a mixture of survival functions with piecewise constant hazard. The model is then applied to study the power of unweighted and weighted log-rank tests, as well as maximum tests allowing different time dependent weights. Simulation results suggest a robust performance of maximum tests across different scenarios, with little loss in power compared to the most powerful among the considered weighting schemes and huge power gain compared to unfavorable weights. The actual sources of non-proportional hazards are not obvious from resulting populationwise survival functions, highlighting the importance of detailed simulations in the planning phase of a trial when assuming non-proportional hazards.We provide the required tools in a software package, allowing to model data generating processes under complex non-proportional hazard scenarios, to simulate data from these models and to perform the weighted logrank tests.     

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.     

Statistical power to detect violation of the proportional hazards assumption when using the Cox regression model

The use of the Cox proportional hazards regression model is wide-spread. A key assumption of the model is that of proportional hazards. Analysts frequently test the validity of this assumption using statistical significance testing. However, the statistical power of such assessments is frequently unknown. We used Monte Carlo simulations to estimate the statistical power of two different methods for detecting violations of this assumption. When the covariate was binary, we found that a model-based method had greater power than a method based on cumulative sums of martingale residuals. Furthermore, the parametric nature of the distribution of event times had an impact on power when the covariate was binary. Statistical power to detect a strong violation of the proportional hazards assumption was low to moderate even when the number of observed events was high. In many data sets, power to detect a violation of this assumption is likely to be low to modest.     

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.

     

Partitioned log-rank tests for the overall homogeneity of hazard rate functions

In survival analysis, it is routine to test equality of two survival curves, which is often conducted by using the log-rank test. Although it is optimal under the proportional hazards assumption, the log-rank test is known to have little power when the survival or hazard functions cross. To test the overall homogeneity of hazard rate functions, we propose a group of partitioned log-rank tests. By partitioning the time axis and taking the supremum of the sum of two partitioned log-rank statistics over different partitioning points, the proposed test gains enormous power for cases with crossing hazards. On the other hand, when the hazards are indeed proportional, our test still maintains high power close to that of the optimal log-rank test. Extensive simulation studies are conducted to compare the proposed test with existing methods, and three real data examples are used to illustrate the commonality of crossing hazards and the advantages of the partitioned log-rank tests.