Marginal analysis of multivariate failure time data with a surviving fraction based on semiparametric transformation cure models

In biomedical, genetic and social studies, there may exist a fraction of individuals not experiencing the event of interest such that the survival curves eventually level off to nonzero proportions. These people are referred to as “cured” or “nonsusceptible” individuals. Models that have been developed to address this issue are known as cured models. The mixture model, which consists of a model for the binary cure status and a survival model for the event times of the noncured individuals, is one of the widely used cure models. In this paper, we propose a class of semiparametric transformation cure models for multivariate survival data with a surviving fraction by fitting a logistic regression model to the cure status and a semiparametric transformation model to the event time of the noncured individual. Both models allow incorporating covariates and do not require any assumption of the association structure. The statistical inference is based on the marginal approach by constructing a system of estimating equations. The asymptotic properties of the proposed estimators are proved, and the performance of the estimation is demonstrated via simulations. In addition, the approach is illustrated by analyzing the smoking cessation data.     

Cure models to estimate time until hospitalization due to COVID-19

A short introduction to survival analysis and censored data is included in this paper. A thorough literature review in the field of cure models has been done. An overview on the most important and recent approaches on parametric, semiparametric and nonparametric mixture cure models is also included. The main nonparametric and semiparametric approaches were applied to a real time dataset of COVID-19 patients from the first weeks of the epidemic in Galicia (NW Spain). The aim is to model the elapsed time from diagnosis to hospital admission. The main conclusions, as well as the limitations of both the cure models and the dataset, are presented, illustrating the usefulness of cure models in this kind of studies, where the influence of age and sex on the time to hospital admission is shown.

     

Bayesian bivariate generalized Lindley model for survival data with a cure fraction

The cure fraction models have been widely used to analyze survival data in which a proportion of the individuals is not susceptible to the event of interest. In this article, we introduce a bivariate model for survival data with a cure fraction based on the three-parameter generalized Lindley distribution. The joint distribution of the survival times is obtained by using copula functions. We consider three types of copula function models, the Farlie–Gumbel–Morgenstern (FGM), Clayton and Gumbel–Barnett copulas. The model is implemented under a Bayesian framework, where the parameter estimation is based on Markov Chain Monte Carlo (MCMC) techniques. To illustrate the utility of the model, we consider an application to a real data set related to an invasive cervical cancer study.     

Semiparametric analysis of clustered interval-censored survival data with a cure fraction

A generalization of the semiparametric Cox’s proportional hazards model by means of a random effect or frailty approach to accommodate clustered survival data with a cure fraction is considered. The frailty serves as a quantification of the health condition of the subjects under study and may depend on some observed covariates like age. One single individual-specific frailty that acts on the hazard function is adopted to determine the cure status of an individual and the heterogeneity on the time to event if the individual is not cured. Under this formulation, an individual who has a high propensity to be cured would tend to have a longer time to event if he is not cured. Within a cluster, both the cure statuses and the times to event of the individuals would be correlated. In contrast to some models proposed in the literature, the model accommodates the correlations among the observations in a more natural way. A multiple imputation estimation method is proposed for both right-censored and interval-censored data. Simulation studies show that the performance of the proposed estimation method is highly satisfactory. The proposed model and method are applied to the National Aeronautics and Space Administration’s hypobaric decompression sickness data to investigate the factors associated with the occurrence and the time to onset of grade IV venous gas emboli under hypobaric environments.     

Parametric cure models of relative and cause-specific survival for grouped survival times

With parametric cure models, we can express survival parameters (e.g. cured fraction, location and scale parameters) as functions of covariates. These models can measure survival from a specific disease process, either by examining deaths due to the cause under study (cause-specific survival), or by comparing all deaths to those in a matched control population (relative survival). We present a binomial maximum likelihood algorithm to be used for actuarial data, where follow-up times are grouped into specific intervals. Our algorithm provides simultaneous maximum likelihood estimates for all the parameters of a cure model and can be used for cause-specific or relative survival analysis with a variety of survival distributions. Current software does not provide the flexibility of this unified approach.     

SAS macros for estimation of the cumulative incidence functions based on a Cox regression model for competing risks survival data

When considering competing risks survival data, the cause specific hazard functions are often modelled by the proportional hazards Cox regression model. First, we present how to estimate the parameters in this model when some of the covariates are allowed to have exactly the same effect on several causes of failure. In many cases, the focus is not on the parameter estimates, but rather on the probability of observing a failure from a specific cause for individuals with specified covariate values. These probabilities, the cumulative incidences, are not simple functions of the parameters and they are, so far, not provided by the standard statistical software packages. We present two SAS macros: a SAS macro named CumInc for estimation of the cumulative incidences and a SAS macro named CumIncV for estimation of the cumulative incidences and the variances of the estimated cumulative incidences. The use of the macros is demonstrated through an example.     

Parametric and semiparametric methods for mapping quantitative trait loci

Most statistical methods for mapping quantitative trait loci (QTLs) have been extensively developed for normally distributed and completely observed phenotypes. A survival model such as Cox’s proportional hazards model, or an accelerated failure time model, is the natural choice for regressing a phenotype onto a maker-type when the primary phenotype belongs to failure time. This study proposes parametric and semiparametric methods based on accelerated failure time models for interval mapping. In the parametric model, the EM algorithm (expectation maximization algorithm) is adopted to estimate regression parameters and search the entire chromosome for QTL. In the semiparametric model, the error distribution is left unspecified, and a rank-based inference is developed for the effect and location of QTL. Simulated data are used to compare the mapping performance of the semiparametric methodology with that of the parametric method, which separately selects the correct and incorrect error distribution. Analytical results reveal that the parametric estimators may be more efficient in determining the effect and location of QTL, but have obvious bias when selecting the incorrect error distribution. In contrast, the semiparametric inference is robust to the error distribution.     

NPHMC: An R-package for estimating sample size of proportional hazards mixture cure model

Due to advances in medical research, more and more diseases can be cured nowadays, which largely increases the need for an easy-to-use software in calculating sample size of clinical trials with cure fractions. Current available sample size software, such as PROC POWER in SAS, Survival Analysis module in PASS, powerSurvEpi package in R are all based on the standard proportional hazards (PH) model which is not appropriate to design a clinical trial with cure fractions. Instead of the standard PH model, the PH mixture cure model is an important tool in handling the survival data with possible cure fractions. However, there are no tools available that can help design a trial with cure fractions. Therefore, we develop an R package NPHMC to determine the sample size needed for such study design.     

Semiparametric bivariate Archimedean copulas

While parametric copulas often lack expressive capacity to capture the complex dependencies that are usually found in empirical data, non-parametric copulas can have poor generalization performance because of overfitting. A semiparametric copula method based on the family of bivariate Archimedean copulas is introduced as an intermediate approach that aims to provide both accurate and robust fits. The Archimedean copula is expressed in terms of a latent function that can be readily represented using a basis of natural cubic splines. The model parameters are determined by maximizing the sum of the log-likelihood and a term that penalizes non-smooth solutions. The performance of the semiparametric estimator is analyzed in experiments with simulated and real-world data, and compared to other methods for copula estimation: three parametric copula models, two semiparametric estimators of Archimedean copulas previously introduced in the literature, two flexible copula methods based on Gaussian kernels and mixtures of Gaussians and finally, standard parametric Archimedean copulas. The good overall performance of the proposed semiparametric Archimedean approach confirms the capacity of this method to capture complex dependencies in the data while avoiding overfitting.     

A stochastic EM algorithm for a semiparametric mixture model

Recently, there has been a considerable interest in finite mixture models with semi-/non-parametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. Hence, a generalization of the EM algorithm to semiparametric mixture models is proposed. The approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the proposed EM type estimators is studied numerically not only through several Monte-Carlo experiments but also through comparison with alternative methods existing in the literature. In addition to these numerical experiments, applications to real data are provided, showing that the estimation method behaves well, that it is fast and easy to be implemented.     

The exponentiated exponential mixture and non-mixture cure rate model in the presence of covariates

This paper presents estimates for the parameters included in long-term mixture and non-mixture lifetime models, applied to analyze survival data when some individuals may never experience the event of interest. We consider the case where the lifetime data have a two-parameters exponentiated exponential distribution. The two-parameter exponentiated exponential or the generalized exponential distribution is a particular member of the exponentiated Weibull distribution introduced by [31]. Classical and Bayesian procedures are used to get point and confidence intervals of the unknown parameters. We consider a general survival model where the scale, shape and cured fraction parameters of the exponentiated exponential distribution depends on covariates.     

Multiple imputation method for the semiparametric accelerated failure time mixture cure model

There are few discussions on the semiparametric accelerated failure time mixture cure model due to its complexity in estimation. In this paper, we propose a multiple imputation method for the semiparametric accelerated failure time mixture cure model based on the rank estimation method and the profile likelihood method. Both approaches can be easily implemented in R environment. However, the computation time for the rank estimation method is longer than that from the profile likelihood method. Simulation studies demonstrate that the performances of estimated parameters from the proposed methods are comparable to those from the expectation maximization (EM) algorithm, and the estimated variances are comparable to those from the empirical approach. For illustration, we apply the proposed method to a data set of failure times from the bone marrow transplantation.     

A SAS macro for estimation of direct adjusted survival curves based on a stratified Cox regression model

Often in biomedical research the aim of a study is to compare the outcomes of several treatment arms while adjusting for multiple clinical prognostic factors. In this paper we focus on computation of the direct adjusted survival curves for different treatment groups based on an unstratified or a stratified Cox model. The estimators are constructed by taking the average of the individual predicted survival curves. The method of direct adjustment controls for possible confounders due to an imbalance of patient characteristics between treatment groups. This adjustment is especially useful for non-randomized studies. We have written a SAS macro to estimate and compare the direct adjusted survival curves. We illustrate the SAS macro through the examples analyzing stem cell transplant data and Ewing's sarcoma data.     

Semiparametric Bayesian approaches to joinpoint regression for population-based cancer survival data

According to the American Cancer Society report (1999), cancer surpasses heart disease as the leading cause of death in the United States of America (USA) for people of age less than 85. Thus, medical research in cancer is an important public health interest. Understanding how medical improvements are affecting cancer incidence, mortality and survival is critical for effective cancer control. In this paper, we study the cancer survival trend on the population level cancer data. In particular, we develop a parametric Bayesian joinpoint regression model based on a Poisson distribution for the relative survival. To avoid identifying the cause of death, we only conduct analysis based on the relative survival. The method is further extended to the semiparametric Bayesian joinpoint regression models wherein the parametric distributional assumptions of the joinpoint regression models are relaxed by modeling the distribution of regression slopes using Dirichlet process mixtures. We also consider the effect of adding covariates of interest in the joinpoint model. Three model selection criteria, namely, the conditional predictive ordinate (CPO), the expected predictive deviance (EPD), and the deviance information criteria (DIC), are used to select the number of joinpoints. We analyze the grouped survival data for distant testicular cancer from the Surveillance, Epidemiology, and End Results (SEER) Program using these Bayesian models.     

A SAS macro for the analysis of cell survival curves

This paper describes a SAS macro for the statistical analyses of cell survival data obtained after radiation treatment using the methods of R.E. Tarone et al. (Mutation Research 111 (1983) 79-96). These analyses are usually required on a routine basis by all biomedical research laboratories involved in cell survival assays generating dose-response curves aimed at characterizing radiosensitive mutant cell strains or individuals whose body cells exhibit enhanced sensitivity to radiation and other genotoxic agents. Statistical methods of linear regression are applied to data from repeated experiments with a cell line/strain and weighted estimates of a common slope and its variance are obtained. The methods are currently implemented in two APL programs. These programs are not easily accessible to most biomedical statisticians and researchers because APL is not a common software tool for statistical analysis. Implementation of these methods in SAS, a widely used commercial software for statistical analysis, is expected to help resolve this issue. We illustrate the application of the macro using an example data set obtained in our laboratory, and hope that other investigators may find it useful in analyzing their data.     

A SAS macro for a clustered logrank test

The clustered logrank test is a nonparametric method of significance testing for correlated survival data. Examples of its application include cluster randomized trials where groups of patients rather than individuals are randomized to either a treatment or a control intervention. We describe a SAS macro that implements the 2-sample clustered logrank test for data where the entire cluster is randomized to the same treatment group. We discuss the theory and applications behind this test as well as details of the SAS code.     

A Bayesian semi-parametric bivariate failure time model

In this paper we introduce a Bayesian semiparametric model for bivariate and multivariate survival data. The marginal densities are well-known nonparametric survival models and the joint density is constructed via a mixture. Our construction also defines a copula and the properties of this new copula are studied. We also consider the model in the presence of covariates and, in particular, we find a simple generalisation of the widely used frailty model, which is based on a new bivariate gamma distribution.     

A Bayesian semi-parametric bivariate failure time model

In this paper we introduce a Bayesian semiparametric model for bivariate and multivariate survival data. The marginal densities are well-known nonparametric survival models and the joint density is constructed via a mixture. Our construction also defines a copula and the properties of this new copula are studied. We also consider the model in the presence of covariates and, in particular, we find a simple generalisation of the widely used frailty model, which is based on a new bivariate gamma distribution.     

CANSURV: A Windows program for population-based cancer survival analysis

Patient survival is one of the most important measures of cancer patient care (the diagnosis and treatment of cancer). The optimal method for monitoring the progress of patient care across the full spectrum of provider settings is through the population-based study of cancer patient survival, which is only possible using data collected by population-based cancer registries. The probability of cure, “statistical cure”, is defined for a cohort of cancer patients as the percent of patients whose annual death rate equals the death rate of general cancer-free population. Mixture cure models have been widely used to model failure time data. The models provide simultaneous estimates of the proportion of the patients cured from cancer and the distribution of the failure times for the uncured patients (latency distribution). CANSURV (CAN-cer SURVival) is a Windows software fitting both the standard survival models and the cure models to population-based cancer survival data. CANSURV can analyze both cause-specific survival data and, especially, relative survival data, which is the standard measure of net survival in population-based cancer studies. It can also fit parametric (cure) survival models to the individual data. The program is available at http://srab.cancer.gov/cansurv. The colorectal cancer survival data from the Surveillance, Epidemiology and End Results (SEER) program [Surveillance, Epidemiology and End Results Program, The Portable Survival System/Mainframe Survival System, National Cancer Institute, Bethesda, 1999.] of the National Cancer Institute, NIH is used to demonstrate the use of CANSURV program.     

Mixture cure models for multivariate survival data

Mixture cure models (MCMs) have been widely used to analyze survival data with a cure fraction. The MCMs postulate that a fraction of the patients are cured from the disease and that the failure time for the uncured patients follows a proper survival distribution, referred to as latency distribution. The MCMs have been extended to bivariate survival data by modeling the marginal distributions. In this paper, the marginal MCM is extended to multivariate survival data. The new model is applicable to the survival data with varied cluster size and interval censoring. The proposed model allows covariates to be incorporated into both the cure fraction and the latency distribution for the uncured patients. The primary interest is to estimate the marginal parameters in the mean structure, where the correlation structure is treated as nuisance parameters. The marginal parameters are estimated consistently by treating the observations within the cluster as independent. The variances of the parameters are estimated by the one-step jackknife method. The proposed method does not depend on the specification of correlation structure. Simulation studies show that the new method works well when the marginal model is correct. The performance of the MCM is also examined when the clustered survival times share common random effect. The MCM is applied to the data from a smoking cessation study.