Evaluation of the bias in using the time to the first event when the inter-event intervals have a Weibull distribution

Currently the analysis of clinical trials for treatment of paroxysmal atrial fibrillation (PAF) relies on the assumption that the events are distributed according to a Poisson distribution. We contend that the occurrence of PAF events are clearly not Poisson and tend to occur in clusters. A candidate parametric model of the inter-event interval, the Weibull distribution, is presented. When the events are distributed according to a Poisson distribution, the time to the first event (TFE) has the same distribution as the inter-event intervals (IEI) due to the 'memoryless' property of the Poisson distribution, hence the TFE can be used instead of the IEI. When the events do not form a Poisson distribution, the TFE does not have the same distribution as the IEI. We show that for the Weibull distribution, when the TFE is used to model the IEI, both the mean and the survivor distribution are biased. The bias in the survivor function is a function both of time and the parameters of the distribution. Therefore when two groups have different parameters for their distributions (as in the case of different treatment effects), the discrepancy between the survivor distribution of the IEI and the survivor distribution of the TFE is affected differentially. We demonstrate the low coverage probabilities of the mean and the survivor function which result when the underlying distribution is Weibull with shape parameter kappa < 1.0. It is likely that this problem will arise for other clustered event processes. This suggests that careful empirical investigation of the distribution of IEI for recurrent events is necessary before choosing to analyse the data using the TFE.