The glm log-rank test:general linear modeling of log-rank scores as a method of analysis for survival data

The use of general linear modeling (GLM) procedures based on log-rank scores is proposed for the analysis of survival data and compared to standard survival analysis procedures. For the comparison of two groups, this approach performed similarly to the traditional log-rank test. In the case of more complicated designs - without ties in the survival times - the approach was only marginally less powerful than tests from proportional hazards models, and clearly less powerful than a likelihood ratio test for a fully parametric model; however, with ties in the survival time, the approach proved more powerful than tests from Cox's semi-parametric proportional hazards procedure. The method appears to provide a reasonably powerful alternative for the analysis of survival data, is easily used in complicated study designs, avoids (semi-)parametric assumptions, and is quite computationally easy and inexpensive to employ.     

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.

     

Nonparametric survival regression using the beta-Stacy process

A novel class of hierarchical nonparametric Bayesian survival regression models for time-to-event data with uninformative right censoring is introduced. The survival curve is modeled as a random function whose prior distribution is defined using the beta-Stacy (BS) process. The prior mean of each survival probability and its prior variance are linked to a standard parametric survival regression model. This nonparametric survival regression can thus be anchored to any reference parametric form, such as a proportional hazards or an accelerated failure time model, allowing substantial departures of the predictive survival probabilities when the reference model is not supported by the data. Also, under this formulation the predictive survival probabilities will be close to the empirical survival distribution near the mode of the reference model and they will be shrunken towards its probability density in the tails of the empirical distribution.     

Jointly modeling longitudinal proportional data and survival times with an application to the quality of life data in a breast cancer trial

Motivated by the joint analysis of longitudinal quality of life data and recurrence free survival times from a cancer clinical trial, we present in this paper two approaches to jointly model the longitudinal proportional measurements, which are confined in a finite interval, and survival data. Both approaches assume a proportional hazards model for the survival times. For the longitudinal component, the first approach applies the classical linear mixed model to logit transformed responses, while the second approach directly models the responses using a simplex distribution. A semiparametric method based on a penalized joint likelihood generated by the Laplace approximation is derived to fit the joint model defined by the second approach. The proposed procedures are evaluated in a simulation study and applied to the analysis of breast cancer data motivated this research.     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

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.     

Penalized spline joint models for longitudinal and time-to-event data

The joint models for longitudinal data and time-to-event data have recently received numerous attention in clinical and epidemiologic studies. Our interest is in modeling the relationship between event time outcomes and internal time-dependent covariates. In practice, the longitudinal responses often show non linear and fluctuated curves. Therefore, the main aim of this paper is to use penalized splines with a truncated polynomial basis to parameterize the non linear longitudinal process. Then, the linear mixed-effects model is applied to subject-specific curves and to control the smoothing. The association between the dropout process and longitudinal outcomes is modeled through a proportional hazard model. Two types of baseline risk functions are considered, namely a Gompertz distribution and a piecewise constant model. The resulting models are referred to as penalized spline joint models; an extension of the standard joint models. The expectation conditional maximization (ECM) algorithm is applied to estimate the parameters in the proposed models. To validate the proposed algorithm, extensive simulation studies were implemented followed by a case study. In summary, the penalized spline joint models provide a new approach for joint models that have improved the existing standard joint models.     

A comparison of parametric and semi-parametric survival models with artificial neural networks

Survival models are used to examine data in the event of an occurrence. These are discussed in various types including parametric, non-parametric and semi-parametric models. Parametric models require a clear distribution of survival time, and semi-parametric models assume proportional hazards. Among these models, the non-parametric model of artificial neural network has the fewest assumptions and can be often replaced by other models. Given the importance of distribution Weibull survival models in this study of simulation shape parameter of the Weibull distribution have been assumed as 1, 2 and 3, and also the average rate at levels of 0%–75% have been censored. The values predicted by the neural network forecasting model with parametric survival and Cox regression models were compared. This comparison considering levels of complexity due to the hazard model using the ROC curve and the corresponding tests have been carried out.     

Predictive capability of stratified proportional hazards models

Following on from the work of O'Quigley & Flandre (1994) and, more recently, O'Quigley & Xu (2000), we develop a measure, R2, of the predictive ability of a stratified proportional hazards regression model. The extension of this earlier work to the stratified case is relatively straightforward, both conceptually and in its practical implementation. The extension is nonetheless important in that the stratified model is making weaker assumptions than the full multivariate model. Formulae are given that can be readily incorporated into standard software routines, since the component parts of the calculations are routinely provided by most packages. We give examples on the predictability of survival in breast cancer data, modelled via proportional hazards and stratified proportional hazards models, the latter being necessary in view of the effects of a non-proportional hazards nature.     

A Survey of Microcomputer Survival Analysis Software: The Need for an Integrated Framework

This paper surveys commercially available MS-DOS and Microsoft Windows based microcomputer software for survival analysis, especially for Cox proportional hazards regression and parametric survival models. Emphasis is given to functionality, documentation, generality, and flexibility of software. A discussion of the need for software integration is given, which leads to the conclusion that survival analysis software not closely tied to a well-designed package will not meet an analyst's general needs. Some standalone programs are good tools for teaching the theory of some survival analysis procedures, but they may not teach the student good data analysis techniques such as critically examining regression assumptions. We contrast typical software with a general, integrated, modeling framework that is available with S-PLUS.     

A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

Nonparametric Bayes Stochastically Ordered Latent Class Models

Latent class models (LCMs) are used increasingly for addressing a broad variety of problems, including sparse modeling of multivariate and longitudinal data, model-based clustering, and flexible inferences on predictor effects. Typical frequentist LCMs require estimation of a single finite number of classes, which does not increase with the sample size, and have a well-known sensitivity to parametric assumptions on the distributions within a class. Bayesian nonparametric methods have been developed to allow an infinite number of classes in the general population, with the number represented in a sample increasing with sample size. In this article, we propose a new nonparametric Bayes model that allows predictors to flexibly impact the allocation to latent classes, while limiting sensitivity to parametric assumptions by allowing class-specific distributions to be unknown subject to a stochastic ordering constraint. An efficient MCMC algorithm is developed for posterior computation. The methods are validated using simulation studies and applied to the problem of ranking medical procedures in terms of the distribution of patient morbidity.     

New approaches for censored longitudinal data in joint modelling of longitudinal and survival data,with application to HIV vaccine studies

In HIV vaccine studies, longitudinal immune response biomarker data are often left-censored due to lower limits of quantification of the employed immunological assays. The censoring information is important for predicting HIV infection, the failure event of interest. We propose two approaches to addressing left censoring in longitudinal data: one that makes no distributional assumptions for the censored data—treating left censored values as a “point mass” subgroup—and the other makes a distributional assumption for a subset of the censored data but not for the remaining subset. We develop these two approaches to handling censoring for joint modelling of longitudinal and survival data via a Cox proportional hazards model fit by h-likelihood. We evaluate the new methods via simulation and analyze an HIV vaccine trial data set, finding that longitudinal characteristics of the immune response biomarkers are highly associated with the risk of HIV infection.

     

Checking the Short‐Term and Long‐Term Hazard Ratio Model for Survival Data

Abstract. The short‐term and long‐term hazard ratio model includes the proportional hazards model and the proportional odds model as submodels, and allows a wider range of hazard ratio patterns compared with some of the more traditional models. We propose two omnibus tests for checking this model, based, respectively, on the martingale residuals and the contrast between the non‐parametric and model‐based estimators of the survival function. These tests are shown to be consistent against any departure from the model. The empirical behaviours of the tests are studied in simulations, and the tests are illustrated with some real data examples.     

Semiparametric models of longitudinal and time-to-event data with applications to HIV viral dynamics and CD4 counts

We propose a semiparametric approach based on proportional hazards and copula method to jointly model longitudinal outcomes and the time-to-event. The dependence between the longitudinal outcomes on the covariates is modeled by a copula-based times series, which allows non-Gaussian random effects and overcomes the limitation of the parametric assumptions in existing linear and nonlinear random effects models. A modified partial likelihood method using estimated covariates at failure times is employed to draw statistical inference. The proposed model and method are applied to analyze a set of progression to AIDS data in a study of the association between the human immunodeficiency virus viral dynamics and the time trend in the CD4/CD8 ratio with measurement errors. Simulations are also reported to evaluate the proposed model and method.     

A semi-parametric approach to dual modeling when no replication exists

In many applications, it is of interest to simultaneously model the mean and variance of a response when no replication exists. Modeling the mean and variance simultaneously is commonly referred to as dual modeling. Parametric approaches to dual modeling are popular when the underlying mean and variance functions can be expressed explicitly. Quite often, however, nonparametric approaches are more appropriate due to the presence of unusual curvature in the underlying functions. In sparse data situations, nonparametric methods often fit the data too closely while parametric estimates exhibit problems with bias. We propose a semi-parametric dual modeling approach [Dual Model Robust Regression (DMRR)] for non-replicated data. DMRR combines parametric and nonparametric fits resulting in improved mean and variance estimation. The methodology is illustrated with a data set from the literature as well as via a simulation study.     

Chi-Squared Goodness-of-Fit Tests for Parametric Accelerated Failure Time Models

We give chi-squared goodness-of fit tests for parametric regression models such as accelerated failure time, proportional hazards, generalized proportional hazards, frailty models, transformation models, and models with cross-effects of survival functions. Random right censored data are used. Choice of random grouping intervals as data functions is considered.     

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.

     

Estimation of Survival Function in Cox Regression Model Under Random Censoring From Both Sides

In this article we present three types of parametric–non parametric estimators for conditional survival function in Cox proportional hazards regression model when the lifetime of interest is subjected to random censorship from both sides. We prove consistency and asymptotic normality of estimators.     

Report of the ASA Technical Panel on the Census Undercount

A new model is proposed for the joint distribution of paired survival times generated from clinical trials and certain reliability settings. The new model can be considered an extension to the bivariate exponential models studied in the literature. Here, a more flexible bivariate Weibull model will be derived, and two exact parametric tests for testing the equality of marginal survival distributions are developed.