## RECURRENT EVENTS

Many chronic diseases are characterised by non-fatal recurrent events. Examples of such include asthma attacks in asthma, epileptic seizures in epilepsy and hospitalisations for worsening condition in heart failure. Typically in clinical trials for heart failure, composite outcomes (e.g. heart failure hospitalisation or cardiovascular death) are adopted as the primary endpoint as they combine fatal and non-fatal events, which increases the event rate and avoids multiplicity issues. However, composite endpoints that only consider the first event are suboptimal for a chronic disease such as heart failure which is characterised by recurrent heart failure hospitalisations, since repeat events within individuals are ignored in analyses. Recurrent hospitalisations are also an indication of worsening condition and so analysing all of these repeat events within individuals is more representative of disease progression and more accurately estimates the effect of treatment on the true burden of disease.

Methods for analysing recurrent events are well developed and can be split into two broad categories: time-to-event approaches and methods based on event rates. The Wei-Lin-Weissfeld examines the cumulative time from randomisation to ordered events considering each ordered event in turn as the outcome in a sequence of applications of the Cox proportional-hazards model. The distinctive feature of the WLW model is that each individual's time at risk for each event (i.e. first, second, third event etc.) is considered to start at randomisation, so full randomised treatment groups are compared. So in the case of heart failure hospitalisations, time to first hospitalisation would be analysed and then total time to second hospitalisation would be analysed separately, including everyone randomised, even if they hadn't yet had a first hospitalisation. Analysis continues in this manner giving distinct estimated hazard ratios for each ordered hospitalisation and these hazard ratios can be combined to give an 'average effect'. The Prentice-Williams-Peterson model is similar to the WLW, but rather than considering total time to each ordered event, gap times (i.e. the times between consecutive events) are considered with conditional risk sets. The Andersen-Gill is an extension of the Cox proportional-hazards model which analyses gap times independently. In the Cox proportional-hazards model, each individual's time to event contributes to the partial likelihood, but in the Andersen-Gill model each gap time contributes independently. Each of these methods assume independence of events, but hospitalisations within individuals are likely to be related to each other and so robust standard errors can be used to accommodate heterogeneity.

Methods based on event rates include the Poisson and negative binomial distributions. The Poisson model is a commonly misused approach to determine if rates of an event differ between two treatment groups. The Poisson distribution assumes that the underlying event rate is the same in all subjects (and follows a Poisson process). This does not hold in the case of heart failure hospitalisations, where the observed distribution of the number of events is markedly more skew than the Poisson distribution. In heart failure, some patients are inherently more/less frail than others, subsequently presenting with increased/fewer hospitalisations respectively. If this overdispersion is ignored, standard errors are likely to be too small resulting in confidence intervals that are too narrow and an increase in the type I error rate. An alternative approach is to use the negative binomial distribution which naturally induces an association between repeat events within individuals through a random effect term which is assumed to follow a gamma distribution. The negative-binomial assumes individual-specifc Poisson event rates conditional on a random effect for each patient's true underlying rate.