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 atria l fibrillation (PAF) relies on the assumption that the events are distribut ed according to a Poisson distribution. We contend that the occurrence of P AF events are clearly not Poisson and tend to occur in clusters. A candidat e parametric model of the inter-event interval, the Weibull distribution, i s presented. When the events are distributed according to a Poisson distrib ution, the time to the first event (TFE) has the same distribution as the i nterevent intervals (IEI) due to the 'memoryless' property of the Poisson d istribution, 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 distribu tion 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 bia sed. The bias in the survivor function is a function both of time and the p arameters of the distribution. Therefore when two groups have different par ameters for their distributions (as in the case of different treatment effe cts), 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 ka ppa < 1.0. It is likely that this problem will arise for other clustered ev ent processes. This suggests that careful empirical investigation of the di stribution of IEI for recurrent events is necessary before choosing to anal yse the data using the TFE. Copyright (C) 1999 John Wiley & Sons, Ltd.