Non-parametric estimation and model checking procedures for marginal gap time distributions for recurrent events

For recurrent events there is evidence that misspecification of the frailty distribution can cause severe bias in estimated regression coefficients (Am. J. Epidemiol 1998; 149:404-411; Statist. Med. 2006; 25:1672-1684). In this paper we adapt a procedure originally suggested in (Biometrika 1999; 86:381-393) for parallel data for checking the gamma frailty to recurrent events. To apply the model checking procedure, a consistent non-parametric estimator for the marginal gap time distributions is needed. This is in general not possible due to induced dependent censoring in the recurrent events setting, however, in (Biometrika 1999; 86:59-70) a non-parametric estimator for the joint gap time distributions based on the principle of inverse probability of censoring weights is suggested. Here, we attempt to apply this estimator in the model checking procedure and the performance of the method is investigated with simulations and applied to Danish registry data. The method is further investigated using the usual Kaplan-Meier estimator and a marginalized estimator for the marginal gap time distributions. We conclude that the procedure only works when the recurrent event is common and when the intra-individual association between gap times is weak.