Two-sample inference procedures under nonproportional hazards

We introduce a new two-sample inference procedure to assess the relative performance of two groups over time. Our model-free method does not assume proportional hazards, making it suitable for scenarios where nonproportional hazards may exist. Our procedure includes a diagnostic tau plot to identify changes in hazard timing and a formal inference procedure. The tau-based measures we develop are clinically meaningful and provide interpretable estimands to summarize the treatment effect over time. Our proposed statistic is a U-statistic and exhibits a martingale structure, allowing us to construct confidence intervals and perform hypothesis testing. Our approach is robust with respect to the censoring distribution. We also demonstrate how our method can be applied for sensitivity analysis in scenarios with missing tail information due to insufficient follow-up. Without censoring, Kendall's tau estimator we propose reduces to the Wilcoxon-Mann–Whitney statistic. We evaluate our method using simulations to compare its performance with the restricted mean survival time and log-rank statistics. We also apply our approach to data from several published oncology clinical trials where nonproportional hazards may exist.     

Bayesian Analysis of the Proportional Hazards Model with Time‐Varying Coefficients

We study a Bayesian analysis of the proportional hazards model with time‐varying coefficients. We consider two priors for time‐varying coefficients – one based on B‐spline basis functions and the other based on Gamma processes – and we use a beta process prior for the baseline hazard functions. We show that the two priors provide optimal posterior convergence rates (up to the term) and that the Bayes factor is consistent for testing the assumption of the proportional hazards when the two priors are used for an alternative hypothesis. In addition, adaptive priors are considered for theoretical investigation, in which the smoothness of the true function is assumed to be unknown, and prior distributions are assigned based on B‐splines.     

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.     

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.     

Bayesian analysis of the Box-Cox transformation model based on left-truncated and right-censored data

In this paper, we discuss the inference problem about the Box-Cox transformation model when one faces left-truncated and right-censored data, which often occur in studies, for example, involving the cross-sectional sampling scheme. It is well-known that the Box-Cox transformation model includes many commonly used models as special cases such as the proportional hazards model and the additive hazards model. For inference, a Bayesian estimation approach is proposed and in the method, the piecewise function is used to approximate the baseline hazards function. Also the conditional marginal prior, whose marginal part is free of any constraints, is employed to deal with many computational challenges caused by the constraints on the parameters, and a MCMC sampling procedure is developed. A simulation study is conducted to assess the finite sample performance of the proposed method and indicates that it works well for practical situations. We apply the approach to a set of data arising from a retirement center.     

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.

     

An Estimator of the Survival Function Basedon the Semi-Markov Model Under Dependent Censorship

Lee and Wolfe (Biometrics vol. 54 pp. 1176–1178, 1998) proposed the two-stage sampling design for testing the assumption of independent censoring, which involves further follow-up of a subset of lost-to-follow-up censored subjects. They also proposed an adjusted estimator for the survivor function for a proportional hazards model under the dependent censoring model. In this paper, a new estimator for the survivor function is proposed for the semi-Markov model under the dependent censorship on the basis of the two-stage sampling data. The consistency and the asymptotic distribution of the proposed estimator are derived. The estimation procedure is illustrated with an example of lung cancer clinical trial and simulation results are reported of the mean squared errors of estimators under a proportional hazards and two different nonproportional hazards models.     

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.     

Dynamic Random Effects Models for Times Between Repeated Events

We consider recurrent event data when the duration or gap times between successive event occurrences are of intrinsic interest. Subject heterogeneity not attributed to observed covariates is usually handled by random effects which result in an exchangeable correlation structure for the gap times of a subject. Recently, efforts have been put into relaxing this restriction to allow non-exchangeable correlation. Here we consider dynamic models where random effects can vary stochastically over the gap times. We extend the traditional Gaussian variance components models and evaluate a previously proposed proportional hazards model through a simulation study and some examples. Besides, semiparametric estimation of the proportional hazards models is considered. Both models are easily used. The Gaussian models are easily interpreted in terms of the variance structure. On the other hand, the proportional hazards models would be more appropriate in the context of survival analysis, particularly in the interpretation of the regression parameters. They can be sensitive to the choice of model for random effects but not to the choice of the baseline hazard function.     

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.     

A Frailty Model in Crossover Studies with Time-to-Event Response Variable

In this article, we develop a model to study treatment, period, carryover, and other applicable effects in a crossover design with a time-to-event response variable. Because time-to-event outcomes on different treatment regimens within the crossover design are correlated for an individual, we adopt a proportional hazards frailty model. If the frailty is assumed to have a gamma distribution, and the hazard rates are piecewise constant, then the likelihood function can be determined via closed-form expressions. We illustrate the methodology via an application to a data set from an asthma clinical trial and run simulations that investigate sensitivity of the model to data generated from different distributions.     

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.     

Additive hazards model with truncated and doubly censored data

In longitudinal studies, the additive hazard model is often used to analyze covariate effects on the duration time, defined as the elapsed time between the first and the second event. In this article, we consider the situation when the first event suffers partly interval censoring and the second event suffers left truncation and right-censoring. We proposed a two-step estimation procedure for estimating the regression coefficients of the additive hazards model. A simulation study is conducted to investigate the performance of the proposed estimator. The proposed method is applied to the Centers for Disease Control acquired immune deficiency syndrome blood transfusion data.     

Properties of Added Variable Plots in Cox's Regression Model

The added variable plot is useful for examining the effect of a covariate in regression models. The plot provides information regarding the inclusion of a covariate, and is useful in identifying influential observations on the parameter estimates. Hall et al. (1996) proposed a plot for Cox's proportional hazards model derived by regarding the Cox model as a generalized linear model. This paper proves and discusses properties of this plot. These properties make the plot a valuable tool in model evaluation. Quantities considered include parameter estimates, residuals, leverage, case influence measures and correspondence to previously proposed residuals and diagnostics.     

An extended semiparametric model for short-term and long-term hazard ratios

In survival analysis, one way to deal with non-proportional hazards is to model short-term and long-term hazard ratios. The existing model of this nature has no control over how fast the hazard ratio is changing over time. We add a parameter to the existing model to allow the hazard ratio to change over time at different speed. A nonparametric maximum likelihood approach is used to estimate the model parameters. The existing model is a special case of the extended model when the speed parameter is 0, which leads naturally to a way of testing the adequacy of the existing model. Simulation results show that there can be substantial bias in the estimation of the short-term and long-term hazard ratio if the speed parameter is fixed incorrectly at 0 rather than estimated. The extended model is fitted to three real data sets to shed new insights, including the observation that converging hazards does not necessarily imply the odds are proportional.     

Bayesian variable selection for the Cox regression model with missing covariates

In this paper, we develop Bayesian methodology and computational algorithms for variable subset selection in Cox proportional hazards models with missing covariate data. A new joint semi-conjugate prior for the piecewise exponential model is proposed in the presence of missing covariates and its properties are examined. The covariates are assumed to be missing at random (MAR). Under this new prior, a version of the Deviance Information Criterion (DIC) is proposed for Bayesian variable subset selection in the presence of missing covariates. Monte Carlo methods are developed for computing the DICs for all possible subset models in the model space. A Bone Marrow Transplant (BMT) dataset is used to illustrate the proposed methodology.     

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.     

Proportional hazards regression with interval-censored and left-truncated data

This paper considers the estimation of the regression coefficients in the Cox proportional hazards model with left-truncated and interval-censored data. Using the approaches of Pan [A multiple imputation approach to Cox regression with interval-censored data, Biometrics 56 (2000), pp. 199–203] and Heller [Proportional hazards regression with interval censored data using an inverse probability weight, Lifetime Data Anal. 17 (2011), pp. 373–385], we propose two estimates of the regression coefficients. The first estimate is based on a multiple imputation methodology. The second estimate uses an inverse probability weight to select event time pairs where the ordering is unambiguous. A simulation study is conducted to investigate the performance of the proposed estimators. The proposed methods are illustrated using the Centers for Disease Control and Prevention (CDC) acquired immunodeficiency syndrome (AIDS) Blood Transfusion Data.     

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 note on the asymptotic relative efficiency of estimators from the additive hazards model

In this note, the asymptotic variance formulas are explicitly derived and compared between the parametric and semiparametric estimators of a regression parameter and survival probability under the additive hazards model. To obtain explicit formulas, it is assumed that the covariate term including a regression coefficient follows a gamma distribution and the baseline hazard function is constant. The results show that the semiparametric estimator of the regression coefficient parameter is fully efficient relative to the parametric counterpart when the survival time and a covariate are independent, as in the proportional hazards model. Relative to a more realistic case of the parametric additive hazards model with a Weibull baseline, the loss of efficiency of the semiparametric estimator of survival probability is moderate.