Fundamentals of survival data

Survival data stand out as a special statistical field. This paper tries to describe what survival data is and what makes it so special. Survival data concern times to some events. A key point is the successive observation of time, which on the one hand leads to some times not being observed so that all that is known is that they exceed some given times (censoring), and on the other hand implies that predictions regarding the future course should be conditional on the present status (truncation). In the simplest case, this condition is that the individual is alive. The successive conditioning makes the hazard function, which describes the probability of an event happening during a short interval given that the individual is alive today (or more generally able to experience the event), the most relevant concept. Standard distributions available (normal, log-normal, gamma, inverse Gaussian, and so forth) can account for censoring and truncation, but this is cumbersome. Besides, they fit badly because they are either symmetric or right skewed, but survival time distributions can easily be left-skewed positive variables. A few distributions satisfying these requirements are available, but often nonparametric methods are preferable as they account better conceptually for truncation and censoring and give a better fit. Finally, we compare the proportional hazards regression models with accelerated failure time models.     

Analysis of survival data with cross-effects of survival functions

We present a model and semiparametric estimation procedures for analysis of survival data with cross-effects (CE) of survival functions. Finite sample properties of the estimators are analyzed by simulation. A goodness-of-fit test for the proportional hazards model against the CE model is proposed. The well known data concerning effects of chemotherapy and radiotherapy on the survival times of gastric cancer patients is analyzed as an example.     

Modelling paired survival data with covariates

The objective of this paper is to consider the parametric analysis of paired censored survival data when additional covariate information is available, as in the Diabetic Retinopathy Study, which assessed the effectiveness of laser photocoagulation in delaying loss of visual acuity. Our first approach is to extend the fully parametric model of Clayton (1978, Biometrika 65, 141-151) to incorporate covariate information. Our second approach is to obtain parameter estimates from an independence working model together with robust variance estimates. The approaches are compared in terms of efficiency and computational considerations. A fundamental consideration in choosing a strategy for the analysis of paired survival data is whether the correlation within a pair is a nuisance parameter or a parameter of intrinsic scientific interest. The approaches are illustrated with the Diabetic Retinopathy Study.     

Modeling heterogeneity in nest survival data

Current statistical methods for estimating nest survival rates assume that nests are identical in their propensity to succeed. However, there are several biological reasons to question this assumption. For example, experience of the nest builder, number of nest helpers, genetic fitness of individuals, and site effects may contribute to an inherent disparity between nests with respect to their daily mortality rates. Ignoring such heterogeneity can lead to incorrect survival estimates. Our results show that constant survival models can seriously underestimate overall survival in the presence of heterogeneity. This paper presents a flexible random-effects approach to model heterogeneous nest survival data. We illustrate our methods through data on redwing blackbirds.     

Distribution-free regression analysis of grouped survival data

Methods based on regression models for logarithmic hazard functions, Cox models, are given for analysis of grouped and censored survival data. By making an approximation it is possible to obtain explicitly a maximum likelihood function involving only the regression parameters. This likelihood function is a convenient analog to Cox's partial likelihood for ungrouped data. The method is applied to data from a toxicological experiment.     

A spatial scan statistic for survival data

Spatial scan statistics with Bernoulli and Poisson models are commonly used for geographical disease surveillance and cluster detection. These models, suitable for count data, were not designed for data with continuous outcomes. We propose a spatial scan statistic based on an exponential model to handle either uncensored or censored continuous survival data. The power and sensitivity of the developed model are investigated through intensive simulations. The method performs well for different survival distribution functions including the exponential, gamma, and log-normal distributions. We also present a method to adjust the analysis for covariates. The cluster detection method is illustrated using survival data for men diagnosed with prostate cancer in Connecticut from 1984 to 1995.     

Sparse kernel methods for high-dimensional survival data

Sparse kernel methods like support vector machines (SVM) have been applied with great success to classification and (standard) regression settings. Existing support vector classification and regression techniques however are not suitable for partly censored survival data, which are typically analysed using Cox's proportional hazards model. As the partial likelihood of the proportional hazards model only depends on the covariates through inner products, it can be 'kernelized'. The kernelized proportional hazards model however yields a solution that is dense, i.e. the solution depends on all observations. One of the key features of an SVM is that it yields a sparse solution, depending only on a small fraction of the training data. We propose two methods. One is based on a geometric idea, where-akin to support vector classification-the margin between the failed observation and the observations currently at risk is maximised. The other approach is based on obtaining a sparse model by adding observations one after another akin to the Import Vector Machine (IVM). Data examples studied suggest that both methods can outperform competing approaches. AVAILABILITY: Software is available under the GNU Public License as an R package and can be obtained from the first author's website http://www.maths.bris.ac.uk/~maxle/software.html.     

Multivariate survival analysis for case-control family data

Multivariate survival data arise from case-control family studies in which the ages at disease onset for family members may be correlated. In this paper, we consider a multivariate survival model with the marginal hazard function following the proportional hazards model. We use a frailty-based approach in the spirit of Glidden and Self (1999) to account for the correlation of ages at onset among family members. Specifically, we first estimate the baseline hazard function nonparametrically by the innovation theorem, and then obtain maximum pseudolikelihood estimators for the regression and correlation parameters plugging in the baseline hazard function estimator. We establish a connection with a previously proposed generalized estimating equation-based approach. Simulation studies and an analysis of case-control family data of breast cancer illustrate the methodology's practical utility.     

Additive risk survival model with microarray data

Background  

Microarray techniques survey gene expressions on a global scale. Extensive biomedical studies have been designed to discover subsets of genes that are associated with survival risks for diseases such as lymphoma and construct predictive models using those selected genes. In this article, we investigate simultaneous estimation and gene selection with right censored survival data and high dimensional gene expression measurements.     

Modeling longitudinal data with nonparametric multiplicative random effects jointly with survival data

Summary .   In clinical studies, longitudinal biomarkers are often used to monitor disease progression and failure time. Joint modeling of longitudinal and survival data has certain advantages and has emerged as an effective way to mutually enhance information. Typically, a parametric longitudinal model is assumed to facilitate the likelihood approach. However, the choice of a proper parametric model turns out to be more elusive than models for standard longitudinal studies in which no survival endpoint occurs. In this article, we propose a nonparametric multiplicative random effects model for the longitudinal process, which has many applications and leads to a flexible yet parsimonious nonparametric random effects model. A proportional hazards model is then used to link the biomarkers and event time. We use B-splines to represent the nonparametric longitudinal process, and select the number of knots and degrees based on a version of the Akaike information criterion (AIC). Unknown model parameters are estimated through maximizing the observed joint likelihood, which is iteratively maximized by the Monte Carlo Expectation Maximization (MCEM) algorithm. Due to the simplicity of the model structure, the proposed approach has good numerical stability and compares well with the competing parametric longitudinal approaches. The new approach is illustrated with primary biliary cirrhosis (PBC) data, aiming to capture nonlinear patterns of serum bilirubin time courses and their relationship with survival time of PBC patients.