Models For Survival Data Research Articles

Convergent stochastic algorithm for estimation in general multivariate correlated frailty models using integrated partial likelihood

The Cox model with unspecified baseline hazard is often used to model survival data. In the case of correlated event times, this model can be extended by introducing random effects, also called frailty terms, leading to the frailty model. Few methods have been put forward to estimate parameters of such frailty models, and they often consider only a particular distribution for the frailty terms and specific correlation structures. In this paper, a new efficient method is introduced to perform parameter estimation by maximizing the integrated partial likelihood. The proposed stochastic estimation procedure can deal with frailty models with a broad choice of distributions for the frailty terms and with any kind of correlation structure between the frailty components, also allowing random interaction terms between the covariates and the frailty components. The almost sure convergence of the stochastic estimation algorithm towards a critical point of the integrated partial likelihood is proved. Numerical convergence properties are evaluated through simulation studies and comparison with existing methods is performed. In particular, the robustness of the proposed method with respect to different parametric baseline hazards and misspecified frailty distributions is demonstrated through simulation. Finally, the method is applied to a mastitis and a bladder cancer dataset.

Likelihood-based inference for semi-parametric transformation cure models with interval censored data

A simple yet effective way of modeling survival data with cure fraction is by considering Box-Cox transformation cure model (BCTM) that unifies mixture and promotion time cure models. In this article, we numerically study the statistical properties of the BCTM when applied to interval censored data. Time-to-events associated with susceptible subjects are modeled through proportional hazards structure that allows for non-homogeneity across subjects, where the baseline hazard function is estimated by distribution-free piecewise linear function with varied degrees of non-parametricity. Due to missing cured statuses for right censored subjects, maximum likelihood estimates of model parameters are obtained by developing an expectation-maximization (EM) algorithm. Under the EM framework, the conditional expectation of the complete data log-likelihood function is maximized by considering all parameters (including the Box-Cox transformation parameter α) simultaneously, in contrast to conventional profile-likelihood technique of estimating α. The robustness and accuracy of the model and estimation method are established through a detailed simulation study under various parameter settings, and an analysis of real-life data obtained from a smoking cessation study.

Bias Reduction of Maximum Likelihood Estimation in the Inverse Xgamma Distribution

A distribution for modeling survival data that accounts for flexibility in modeling data with upside-down bathtub-shaped hazard rate functions is the inverse Xgamma distribution (IXG). The maximum likelihood method (MLE) is the most often used technique for parameter estimation of the IXG distribution. Conversely, the MLE is infamously biased for small sample sizes. This motivates us to produce almost unbiased estimators for IXG parameter. More precisely, we minimize MLE biases to the second degree of bias using two techniques for bias correction: bootstrap and analytical approaches. Two actual data applications and Monte Carlo simulations are used to compare the performances of these methods.

Modelling Survival Data in Medical Research: Fourth Edition

Modelling Survival Data in Medical Research: Fourth Edition

Bias reduction of maximum likelihood estimation in exponentiated Teissier distribution

The exponentiated Teissier distribution (ETD) offers an alternative for modeling survival data, taking into account flexibility in modeling data with increasing and decreasing hazard rate functions. The most popular method for parameter estimation of the ETD distribution is the maximum likelihood estimation (MLE). The MLE, on the other hand, is notoriously biased for its small sample sizes. We are therefore driven to generate virtually unbiased estimators for ETD parameters. More specifically, we focus on two methods of bias correction, bootstrapping and analytical approaches, to reduce MLE biases to the second order of bias. The performances of these approaches are compared through Monte Carlo simulations and two real-data applications.

Conditional survival in breast cancer up to 10 years in the Nordic countries.

Survival in breast cancer (BC) has developed favorably but late recurrences are still a problem. We model survival data from the NORDCAN database and analyze 1-, 5-, and 10-year relative survival and 5/1- and 10/5-year conditional survival in BC from Denmark (DK), Finland (FI), Norway (NO), and Sweden (SE) between 1971 and 2020. Conditional survival measures survival in those who had survived year 1 to reach year 5 (5/1), or in those who had survived year 5 to reach year 10 (10/5). Almost all survival metrics were best for SE but survival in all countries improved in the course of time approaching the SE levels which were 98.3% for 1-year, 92.3% for 5-year, and 87.8% for 10-year survival. Conditional 10/5-year survival, covering 5 years, was better than 5/1-year survival, covering 4 years. A contributing factor is most likely the high rate of recurrence in period 2-5 years. The difference was observed for all countries but for DK 10/5-year survival approached 1-year survival and for NO and SE 10/5-year survival was only barely better than 5/1-year survival. The explanation to this was the excellent 10/5-year survival in DK compared to SE and particularly to NO. Literature search suggested that the reason for the relatively low 10/5-year survival in NO might be stagnant survival development in old patients. We assume that late mortality is critically limiting survival in BC and either interference with the late metastatic process or effective treatment will be key to future improvements in BC survival.

Sample size reestimation and Bayesian predictive probability for single-arm clinical trials with a time-to-event endpoint using Weibull distribution with unknown shape parameter

ABSTRACT This manuscript consists of two topics. Firstly, we explore the utility of internal pilot study (IPS) approach for reestimating sample size at an interim stage when a reliable estimate of the nuisance shape parameter of the Weibull distribution for modeling survival data is unavailable during the planning phase of a study. Although IPS approach can help rescue the study power, it is noted that the adjusted sample size can be as much as twice the initially planned sample size, which may put substantial practical constraints to continue the study. Secondly, we discuss Bayesian predictive probability for conducting interim analyses to obtain preliminary evidence of efficacy or futility of an experimental treatment warranting early termination of a clinical trial. In the context of single-arm clinical trials with time-to-event endpoints following Weibull distribution, we present the calculation of the Bayesian predictive probability when the shape parameter of the Weibull distribution is unknown. Based on the data accumulated at the interim, we propose two approaches which rely on the posterior mode or the entire posterior distribution of the shape parameter. To account for uncertainty in the shape parameter, it is recommended to incorporate its entire posterior distribution in our calculation.

Fast and flexible inference for joint models of multivariate longitudinal and survival data using integrated nested Laplace approximations.

Modeling longitudinal and survival data jointly offers many advantages such as addressing measurement error and missing data in the longitudinal processes, understanding and quantifying the association between the longitudinal markers and the survival events, and predicting the risk of events based on the longitudinal markers. A joint model involves multiple submodels (one for each longitudinal/survival outcome) usually linked together through correlated or shared random effects. Their estimation is computationally expensive (particularly due to a multidimensional integration of the likelihood over the random effects distribution) so that inference methods become rapidly intractable, and restricts applications of joint models to a small number of longitudinal markers and/or random effects. We introduce a Bayesian approximation based on the integrated nested Laplace approximation algorithm implemented in the R package R-INLA to alleviate the computational burden and allow the estimation of multivariate joint models with fewer restrictions. Our simulation studies show that R-INLA substantially reduces the computation time and the variability of the parameter estimates compared with alternative estimation strategies. We further apply the methodology to analyze five longitudinal markers (3 continuous, 1 count, 1 binary, and 16 random effects) and competing risks of death and transplantation in a clinical trial on primary biliary cholangitis. R-INLA provides a fast and reliable inference technique for applying joint models to the complex multivariate data encountered in health research.

On interquantile smoothness of censored quantile regression with induced smoothing.

Quantile regression has emerged as a useful and effective tool in modeling survival data, especially for cases where noises demonstrate heterogeneity. Despite recent advancements, non-smooth components involved in censored quantile regression estimators may often yield numerically unstable results, which, in turn, lead to potentially self-contradicting conclusions. We propose an estimating equation-based approach to obtain consistent estimators of the regression coefficients of interest via the induced smoothing technique to circumvent the difficulty. Our proposed estimator can be shown to be asymptotically equivalent to its original unsmoothed version, whose consistency and asymptotic normality can be readily established. Extensions to handle functional covariate data and recurrent event data are also discussed. To alleviate the heavy computational burden of bootstrap-based variance estimation, we also propose an efficient resampling procedure that reduces the computational time considerably. Our numerical studies demonstrate that our proposed estimator provides substantially smoother model parameter estimates across different quantile levels and can achieve better statistical efficiency compared to a plain estimator under various finite-sample settings. The proposed method is also illustrated via four survival datasets, including the HMO (health maintenance organizations) HIV (human immunodeficiency virus) data, the primary biliary cirrhosis (PBC) data, and soforth.

Comparison of Some Methods for Estimating the Survival Function and Failure Rate for the Exponentiated Expanded Power Function Distribution


 We have presented the distribution of the exponentiated expanded power function (EEPF) with four parameters, where this distribution was created by the exponentiated expanded method created by the scientist Gupta to expand the exponential distribution by adding a new shape parameter to the cumulative function of the distribution, resulting in a new distribution, and this method is characterized by obtaining a distribution that belongs for the exponential family. We also obtained a function of survival rate and failure rate for this distribution, where some mathematical properties were derived, then we used the method of maximum likelihood (ML) and method least squares developed (LSD) to estimate the parameters and because of the nonlinear relationship between the parameters, numerical algorithms were used to find the estimates of the two methods. They are Newton-Raphson (NR) and Nelder mead (NM) algorithms to improve the estimators, and a Monte Carlo simulation experiment was conducted to evaluate the performance of the two algorithms' estimates, and the average integrated error criterion (IMSE) was used to compare the survival function estimates and the failure rate. The results showed the efficiency of the maximum likelihood method estimates and least squares developed using the two algorithms (NR, NM) where their results were close, and this shows the new distribution efficiency (EEPF) for modeling survival data.

Genomic and transcriptomic somatic alterations of hepatocellular carcinoma in non-cirrhotic livers

Genomic and transcriptomic somatic alterations of hepatocellular carcinoma in non-cirrhotic livers

Bayesian and frequentist approach for the generalized log-logistic accelerated failure time model with applications to larynx-cancer patients

The log-normal, log-logistic and Weibull distributions are commonly utilized to model survival data. Unimodal (or non-monotone) failure rate functions are modeled using the log-normal and the log-logistic families, whereas monotone failure rate functions are modeled using the Weibull family. The growing availability of survival data with a variety of features encourages statisticians to propose more flexible parametric models that can accommodate both monotone (increasing or decreasing), and non-monotone (unimodal or bathtub) failure rate functions. One such model is the generalized log-logistic distribution which not only accommodates unimodal failure rates but also allows for a monotone and non-monotone failure rate functions. This distribution has shown to have a lot of potential in univariate analysis of survival data. However, many studies are primarily concerned with determining the link between survival time and one or more explanatory variables. This leads to the study of hazard-based regression models in survival and reliability analysis, which can be formulated in a variety of ways. One such method concerns formulating hazard-based regression models for the accelerated failure time (AFT) family of continuous probability distributions. The log-logistic, Weibull and log-normal distributions are the most widely utilized this framework. In this paper, we show that the generalized log-logistic distribution is closed under the accelerated failure time framework. We then formulate an accelerated failure time model based on the generalized log-logistic distribution. Furthermore, we show parameter estimation for the model using Bayesian and frequentist approaches. An extensive simulation study is conducted to illustrate the inferential properties of the proposed model, more specifically, the tendency to recover the baseline hazard shapes, parameter estimation, as well as the effect of censoring proportions on inference. The simulation results demonstrate that the generalized log-logistic accelerated failure time model can be capable of modeling survival data with various hazard rate shapes. Finally, a real-life survival data relating to larynx-cancer patients was used.

Additive Trinomial Fréchet distribution with practical application

This article presents an innovative model called Additive Trinomial Fré chet (ATF) distribution using six parameters. The indicated model is worthy of modeling survival data with a non-monotonic hazard rate. The statistical characteristics of ATF model such as probability generating function, Renyi, Shannon, Tsallis and Mathai–Houbold entropy, quantile function, order statistics, maximum likelihood estimation, factorial and characteristic function, moment generating function, Stress-Strength analysis are thoroughly discussed. The effectiveness of suggested model is demonstrated by the use of a data set from real life. The suggested model has demonstrated better performance and fits the data used superior than other significant counterparts.

On a Class of Transmuted Distributions to Model Survival Data with a Cure Fraction

Modelling time to event data, when there is always a proportion of the individuals, commonly referred to as immunes who do not experience the event of interest, is of importance in many biomedical studies. Improper distributions are used to model these situations and they are generally referred to as cure rate models. In the literature, two main families of cure rate models have been proposed, namely the mixture cure models and promotion time cure models. Here we propose a new model by extending the mixture model via a generating function by considering a shifted Bernoulli distribution. This gives rise to a new class of popular distributions called the transmuted class of distributions to model survival data with a cure fraction. The properties of the proposed model are investigated and parameters estimated. The Bayesian approach to the estimation of parameters is also adopted. The complexity of the likelihood function is handled through the Metropolis-Hasting algorithm. The proposed method is illustrated with few examples using different baseline distributions. A real life data set is also analysed. AMS subject classifications: 62N02, 62F15

A Comparison of Different Bayesian Models for Leukemia Data

Different probability models are used to model survival data. However, it is important to know which model describe best the data because if the assumptions for parametric methods hold, the resulting estimates have smaller standard errors and are easier to interpret and helps in predictions. This article presents the Bayesian censored data modeling assuming Gumbel, double exponential, exponentially modified Gaussian, Weibull, and lognormal distributions as sampling models. In particular, a historical Leukemia data set is used to show the comparison among different models. Markov Chain Monte Carlo (MCMC) methods are used to compute the posterior summaries. Different model selection criteria, like, Akaike Information Criterion (AIC), Deviance Information Criterion (DIC), Leave-one-out Cross-Validation (LOOCV), and Watanabe-Akaike Information Criterion (WAIC) are used for model selection. It is observed from the comparative study that the lognormal model has the minimum values of different model selection criteria and considered to be the best for this Leukemia data.

A Transmuted Survival Model With Application

In this paper, a new survival model is being defined. This model is adaptable for produce a new lifetime distribution for analyzing positive data. We define a new lifetime models called the Transmuted-Survival-Exponential distribution (TSE). Essential functions associated with the proposed distribution are obtained. Some structural properties of the new models are investigated. The new formula may serve as a flexible transmuted to other distributions for modeling survival data arising in several fields of science such as hydrology, biostatistics, metrology and engineering

Utilization of a Mixture Cure Rate Model based on the Generalized Modified Weibull Distribution for the Analysis of Leukemia Patients.

Cure rate models are survival models, commonly applied to model survival data with a cured fraction. In the existence of a cure rate, if the distribution of survival times for susceptible patients is specified, researchers usually prefer cure models to parametric models. Different distributions can be assumed for the survival times, for instance, generalized modified Weibull (GMW), exponentiated Weibull (EW), and log-beta Weibull. The purpose of this study is to select the best distribution for uncured patients' survival times by comparing the mixture cure models based on the GMW distribution and its particular cases. A data set of 91 patients with high-risk acute lymphoblastic leukemia (ALL) followed for five years from 1982 to 1987 was chosen for fitting the mixture cure model. We used the maximum likelihood estimation technique via R software 3.6.2 to obtain the estimates for parameters of the proposed model in the existence of cure rate, censored data, and covariates. For the best model choice, the Akaike information criterion (AIC) was implemented. After comparing different parametric models fitted to the data, including or excluding cure fraction, without covariates, the smallest AIC values were obtained by the EW and the GMW distributions, (953.31/969.35) and (955.84/975.99), respectively. Besides, assuming a mixture cure model based on GMW with covariates, an estimated ratio between cure fractions for allogeneic and autologous bone marrow transplant groups (and its 95% confidence intervals) were 1.42972 (95% CI: 1.18614 - 1.72955). The results of this study reveal that the EW and the GMW distributions are the best choices for the survival times of Leukemia patients.<br />.

A New Class of Survival Regression Models with Cure Fraction

In this paper, we propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest follows the negative binomial distribution and the time to event follows a generalized gamma distribution. We define the negative binomial-generalized gamma distribution, which can be used to model survival data. The new model includes as special cases some of the well-known cure rate models discussed in the literature. We consider a frequentist analysis and nonparametric bootstrap for parameter estimation of a negative binomial-generalized gamma regression model with cure rate. Then, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. Finally, we analyze a real data set from the medical area.

Revisiting methods for modeling longitudinal and survival data: Framingham Heart Study

BackgroundStatistical methods for modeling longitudinal and time-to-event data has received much attention in medical research and is becoming increasingly useful. In clinical studies, such as cancer and AIDS, longitudinal biomarkers are used to monitor disease progression and to predict survival. These longitudinal measures are often missing at failure times and may be prone to measurement errors. More importantly, time-dependent survival models that include the raw longitudinal measurements may lead to biased results. In previous studies these two types of data are frequently analyzed separately where a mixed effects model is used for the longitudinal data and a survival model is applied to the event outcome.MethodsIn this paper we compare joint maximum likelihood methods, a two-step approach and a time dependent covariate method that link longitudinal data to survival data with emphasis on using longitudinal measures to predict survival. We apply a Bayesian semi-parametric joint method and maximum likelihood joint method that maximizes the joint likelihood of the time-to-event and longitudinal measures. We also implement the Two-Step approach, which estimates random effects separately, and a classic Time Dependent Covariate Model. We use simulation studies to assess bias, accuracy, and coverage probabilities for the estimates of the link parameter that connects the longitudinal measures to survival times.ResultsSimulation results demonstrate that the Two-Step approach performed best at estimating the link parameter when variability in the longitudinal measure is low but is somewhat biased downwards when the variability is high. Bayesian semi-parametric and maximum likelihood joint methods yield higher link parameter estimates with low and high variability in the longitudinal measure. The Time Dependent Covariate method resulted in consistent underestimation of the link parameter. We illustrate these methods using data from the Framingham Heart Study in which lipid measurements and Myocardial Infarction data were collected over a period of 26 years.ConclusionsTraditional methods for modeling longitudinal and survival data, such as the time dependent covariate method, that use the observed longitudinal data, tend to provide downwardly biased estimates. The two-step approach and joint models provide better estimates, although a comparison of these methods may depend on the underlying residual variance.

Cause-specific hazard regression estimation for modified Weibull distribution under a class of non-informative priors

ABSTRACT In time to event analysis, the situation of competing risks arises when the individual (or subject) may experience p mutually exclusive causes of death (failure), where cause-specific hazard function is of great importance in this framework. For instance, in malignancy-related death, colorectal cancer is one of the leading causes of the death in the world and death due to other causes considered as competing causes. We include prognostic variables in the model through parametric Cox proportional hazards model. Mostly, in literature exponential, Weibull, etc. distributions have been used for parametric modelling of cause-specific hazard function but they are incapable to accommodate non-monotone failure rate. Therefore, in this article, we consider a modified Weibull distribution which is capable to model survival data with non-monotonic behaviour of hazard rate. For estimating the cumulative cause-specific hazard function, we utilized maximum likelihood and Bayesian methods. A class of non-informative types of prior (uniform, Jeffrey’s and half-t) is introduced for Bayes estimation under squared error (symmetric) as well as LINEX (asymmetric) loss functions. A simulation study is performed for a comprehensive comparison of Bayes and maximum likelihood estimators of cumulative cause-specific hazard function. Real data on colorectal cancer is used to demonstrate the proposed model.