scholarly journals A tutorial on frailty models

2020 ◽  
Vol 29 (11) ◽  
pp. 3424-3454 ◽  
Author(s):  
Theodor A Balan ◽  
Hein Putter
Keyword(s):  
Random Effects ◽  
Recurrent Events ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Unobserved Heterogeneity ◽  
Frailty Models ◽  
Survival Outcomes ◽  
Time To Event ◽  
Homogeneous Population ◽  
Hazards Model

The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.

scholarly journals Parameter Estimation of Shared Frailty Models Based on Particle Swarm Optimization

2016 ◽  
Vol 6 (1) ◽  
pp. 48 ◽  
Author(s):  
Oykum Esra Askin ◽  
Deniz Inan ◽  
Ali Hakan Buyuklu
Keyword(s):  
Particle Swarm Optimization ◽  
Survival Data ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Unobserved Heterogeneity ◽  
Particle Swarm ◽  
Likelihood Estimation ◽  
Real Data ◽  
Frailty Models ◽  
Swarm Optimization

Standard survival techniques such as proportional hazards model are suffering from the unobserved heterogeneity. Frailty models provide an alternative way in order to account for heterogeneity caused by unobservable risk factors. Although vast studies have been done on estimation procedures, Evolutionary Algorithms (EAs) haven't received much attention in frailty studies. In this paper, we investigate the estimation performance of maximum likelihood estimation (MLE) via Particle Swarm Optimization (PSO) in modelling multivariate survival data with shared gamma frailty. Simulation studies and real data application are performed in order to assess the performance of MLE via PSO, quasi-Newton  and conjugate gradient method.

A censored quantile regression approach for the analysis of time to event data

2016 ◽  
Vol 27 (3) ◽  
pp. 955-965 ◽  
Author(s):  
Xiaonan Xue ◽  
Xianhong Xie ◽  
Howard D Strickler
Keyword(s):  
Quantile Regression ◽  
Regression Model ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Time To Event ◽  
Censored Quantile Regression ◽  
Quantile Regression Model ◽  
Hazards Model

The commonly used statistical model for studying time to event data, the Cox proportional hazards model, is limited by the assumption of a constant hazard ratio over time (i.e., proportionality), and the fact that it models the hazard rate rather than the survival time directly. The censored quantile regression model, defined on the quantiles of time to event, provides an alternative that is more flexible and interpretable. However, the censored quantile regression model has not been widely adopted in clinical research, due to the complexity involved in interpreting its results properly and consequently the difficulty to appreciate its advantages over the Cox proportional hazards model, as well as the absence of adequate validation procedure. In this paper, we addressed these limitations by (1) using both simulated examples and data from National Wilms’ Tumor clinical trials to illustrate proper interpretation of the censored quantile regression model and the differences and the advantages of the model compared to the Cox proportional hazards model; and (2) developing a validation procedure for the predictive censored quantile regression model. The performance of this procedure was examined using simulation studies. Overall, we recommend the use of censored quantile regression model, which permits a more sensitive analysis of time to event data together with the Cox proportional hazards model.

scholarly journals A new modeling and inference approach for the Systolic Blood Pressure Intervention Trial outcomes

2018 ◽  
Vol 15 (3) ◽  
pp. 305-312 ◽  
Author(s):  
Song Yang ◽  
Walter T Ambrosius ◽  
Lawrence J Fine ◽  
Adam P Bress ◽  
William C Cushman ◽  
...  
Keyword(s):  
Treatment Effect ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Rank Test ◽  
Time To Event ◽  
New Approach ◽  
New Methods ◽  
Log Rank Test ◽  
Hazards Model ◽  
Average Hazard Ratio

Background/aims In clinical trials with time-to-event outcomes, usually the significance tests and confidence intervals are based on a proportional hazards model. Thus, the temporal pattern of the treatment effect is not directly considered. This could be problematic if the proportional hazards assumption is violated, as such violation could impact both interim and final estimates of the treatment effect. Methods We describe the application of inference procedures developed recently in the literature for time-to-event outcomes when the treatment effect may or may not be time-dependent. The inference procedures are based on a new model which contains the proportional hazards model as a sub-model. The temporal pattern of the treatment effect can then be expressed and displayed. The average hazard ratio is used as the summary measure of the treatment effect. The test of the null hypothesis uses adaptive weights that often lead to improvement in power over the log-rank test. Results Without needing to assume proportional hazards, the new approach yields results consistent with previously published findings in the Systolic Blood Pressure Intervention Trial. It provides a visual display of the time course of the treatment effect. At four of the five scheduled interim looks, the new approach yields smaller p values than the log-rank test. The average hazard ratio and its confidence interval indicates a treatment effect nearly a year earlier than a restricted mean survival time–based approach. Conclusion When the hazards are proportional between the comparison groups, the new methods yield results very close to the traditional approaches. When the proportional hazards assumption is violated, the new methods continue to be applicable and can potentially be more sensitive to departure from the null hypothesis.

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