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.... Show moreThe 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. Show less
Geloven, N. van; Balan, T.A.; Putter, H.; Cessie, S. le 2020
We study the effect of delaying treatment in the presence of (unobserved) heterogeneity. In a homogeneous population and assuming a proportional treatment effect, a treatment delay period will... Show moreWe study the effect of delaying treatment in the presence of (unobserved) heterogeneity. In a homogeneous population and assuming a proportional treatment effect, a treatment delay period will result in notably lower cumulative recovery percentages. We show in theoretical scenarios using frailty models that if the population is heterogeneous, the effect of a delay period is much smaller. This can be explained by the selection process that is induced by the frailty. Patient groups that start treatment later have already undergone more selection. The marginal hazard ratio for the treatment will act differently in such a more homogeneous patient group. We further discuss modeling approaches for estimating the effect of treatment delay in the presence of heterogeneity, and compare their performance in a simulation study. The conventional Cox model that fails to account for heterogeneity overestimates the effect of treatment delay. Including interaction terms between treatment and starting time of treatment or between treatment and follow up time gave no improvement. Estimating a frailty term can improve the estimation, but is sensitive to misspecification of the frailty distribution. Therefore, multiple frailty distributions should be used and the results should be compared using the Akaike Information Criterion. Non-parametric estimation of the cumulative recovery percentages can be considered if the dataset contains sufficient long term follow up for each of the delay strategies. The methods are demonstrated on a motivating application evaluating the effect of delaying the start of treatment with assisted reproductive techniques on time-to-pregnancy in couples with unexplained subfertility. Show less
Klaveren, D. van; Balan, T.A.; Steyerberg, E.W.; Kent, D.M. 2019
This dissertation presents methodological advances in the field of frailty models. Time to event data is very common in biomedical applications. Multivariate data, such as recurrent event... Show moreThis dissertation presents methodological advances in the field of frailty models. Time to event data is very common in biomedical applications. Multivariate data, such as recurrent event histories or clustered failures, typically require special modeling techniques. Frailty models extend the proportional hazards models by employing random effects. In this book, the first chapter represents an introduction to the field of frailty models. The second chapter studies the small sample properties of such models in more depth. In particular, it addresses the questions of identifiability of frailty models. The third chapter describes a score test that may be used to test whether a recurrent events process is associated with a terminal event. The fourth chapter studies the problem of event-dependent selection in the context of recurrent events data. This problem is relevant when the data are collected from registries. The fifth chapter introduces a novel R package and presents an overview of available software for estimating frailty models. Show less
Balan, T.A.; Jonker, M.A.; Johannesma, P.C.; Putter, H. 2016
The statistical analysis of recurrent events relies on the assumption of independent censoring. When random effects are used, this means, in addition, that the censoring cannot depend on the random... Show moreThe statistical analysis of recurrent events relies on the assumption of independent censoring. When random effects are used, this means, in addition, that the censoring cannot depend on the random effect. Whenever the recurrent event process is terminated by death, this assumption might not be satisfied. Because joint models arising from such situations are more difficult to fit and interpret, clinicians rarely check whether joint modeling is preferred. In this paper, we propose and compare simple, yet efficient methods for testing whether the terminal event and the recurrent events are associated or not. The performance of the proposed methods is evaluated in a simulation study, and the sensitivity to misspecification of the model is assessed. Finally, the methods are illustrated on a data set comprising repeated observations of skin tumors on T-cell lymphoma patients. Copyright (c) 2016 John Wiley & Sons, Ltd. Show less
Balan, T.A.; Boonk, S.E.; Vermeer, M.H.; Putter, H. 2016
The statistical analysis of recurrent events relies on the assumption of independent censoring. When random effects are used, this means, in addition, that the censoring cannot depend on the random... Show moreThe statistical analysis of recurrent events relies on the assumption of independent censoring. When random effects are used, this means, in addition, that the censoring cannot depend on the random effect. Whenever the recurrent event process is terminated by death, this assumption might not be satisfied. Because joint models arising from such situations are more difficult to fit and interpret, clinicians rarely check whether joint modeling is preferred. In this paper, we propose and compare simple, yet efficient methods for testing whether the terminal event and the recurrent events are associated or not. The performance of the proposed methods is evaluated in a simulation study, and the sensitivity to misspecification of the model is assessed. Finally, the methods are illustrated on a data set comprising repeated observations of skin tumors on T-cell lymphoma patients. Show less
Balan, T.A.; Jonker, M.A.; Johannesma, P.C.; Putter, H. 2016
In retrospective studies involving recurrent events, it is common to select individuals based on their event history up to the time of selection. In this case, the ascertained subjects might not be... Show moreIn retrospective studies involving recurrent events, it is common to select individuals based on their event history up to the time of selection. In this case, the ascertained subjects might not be representative for the target population, and the analysis should take the selection mechanism into account. The purpose of this paper is two-fold. First, to study what happens when the data analysis is not adjusted for the selection and second, to propose a corrected analysis. Under the Andersen–Gill and shared frailty regression models, we show that the estimators of covariate effects, incidence, and frailty variance can be biased if the ascertainment is ignored, and we show that with a simple adjustment of the likelihood, unbiased and consistent estimators are obtained. The proposed method is assessed by a simulation study and is illustrated on a data set comprising recurrent pneumothoraces. Show less
