# PROFESSOR JENNIFER ROGERS

# STATISTICIAN

## JOINT MODELLING

A key characteristic of heart failure is that an increase in hospitalisations is associated with a worsening condition and a subsequent elevated risk of cardiovascular death, meaning that subjects may die during follow-up. Consequently, any censoring due to CV death is not independent of the recurrent event process. A comparison of heart failure hospitalisation rates, between treatment groups, can be confounded by this competing risk of death and ignoring the dependent censoring can result in bias in estimated treatment effects. Therefore, any analyses of recurrent events must take into consideration informative censoring that may be present.

One simple strategy for incorporating CV death into analyses of recurrent heart failure hospitalisations is to consider this outcome as an additional event in the recurrent event process. That is, one considers a composite of recurrent heart failure hospitalisations and CV death. This updated recurrent event process can then be analysed using all standard recurrent event techniques and the subsequent estimated treatment effect that is obtained is a hazard/rate ratio for the composite of repeat heart failure hospitalisations and death. Note that any death that occurs during a heart failure hospitalisation would be treated as a single event.

The Ghosh and Lin non-parametric analysis of heart failure hospitalisations takes mortality into account whilst also adjusting for different follow-up times and multiple hospitalisations per patient. This method considers the marginal expected number of recurrent heart failure hospitalisations up to some particular time, t, acknowledging the fact that death is a terminal event after which no further recurrent hospitalisations can be experienced. This means that although a patient stays in the risk set beyond time to death, their associated recurrent hospitalisation count stays constant, fixed at whatever value it was just prior to death. The stochastic structure of the recurrent hospitalisations process is left completely unspecified and we make no assumptions regarding the dependence between the recurrent hospitalisations and death. We define the mean frequency function as the marginal expected number of recurrent heart failure hospitalisations up to some time point, t, acknowledging that no further recurrences occur after death.

An alternative approach is the use of joint modelling techniques to obtain estimates of treatment effects on heart failure hospitalisation rates whilst allowing for informative censoring. Joint modelling techniques are appropriate when analysing rates of recurrent events whilst allowing for association with a potentially informative dropout time, or when each of the outcomes is of scientific importance to the investigators and the dependence between the two processes needs to be accounted for. One approach to joint modelling is random effects models, which assume that the recurrent hospitalisations and time-to-death are conditionally independent given a latent variable. Models of this kind are intuitively appealing as they can give a tangible interpretation that an individual’s independent frailty term measures their underlying, unobserved severity of illness, which proportionately affects both their heart failure hospitalisation rate and their time-to-death (or CV death). Joint models allow distinct treatment effects to be estimated for each of the processes, whilst taking into account the association between the two.