Centre‐effect on Survival after Bone Marrow Transplantation: Application of Time‐dependent Frailty Models

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Abstract
In many biomedical investigations, multivariate or clustered failure time data are encountered. They arise for instance, when the sample consists of centres and each centre contains several patients. For this kind of data, frailty models can be used to take into account the possible correlation within centres. Most common are centre‐specific frailty models in which the centre‐effect on failure time is assumed to be constant with follow‐up. But with many applications this assumption is too restrictive. More realistic are models with time‐varying frailties. Therefore, we studied ways to extend the constant centre‐specific frailty model to allow time dependence of the frailties. To begin, we followed and adapted Paik et al. (1994) who generalized the shared frailty model by introducing additional random frailty terms for different time‐intervals. Although, the model allows the frailty to vary between intervals, it is very cumbersome to calculate and difficult to fit. We developed two much simpler centre‐specific frailty models. First, we proposed a model with a power parameter in which the “effect” of the centre‐specific frailty is allowed to vary between intervals. And secondly, we extended the model permitting the centre‐specific frailty to vary with time. Although the convenience of the gamma and positive stable frailty models is lost in both extensions of the frailty model, the computations of these models are much easier than of the adapted Paik's model. We applied these time‐dependent frailty models to a data set from the European Blood Marrow Transplant (EBMT) registry, where a decaying centre‐effect was found.

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