Penalised logistic regression and dynamic prediction for discrete-time recurrent event data

We consider methods for the analysis of discrete-time recurrent event data, when interest is mainly in prediction. The Aalen additive model provides an extremely simple and effective method for the determination of covariate effects for this type of data, especially in the presence of time-varying effects and time varying covariates, including dynamic summaries of prior event history. The method is weakened for predictive purposes by the presence of negative estimates. The obvious alternative of a standard logistic regression analysis at each time point can have problems of stability when event frequency is low and maximum likelihood estimation is used. The Firth penalised likelihood approach is stable but in removing bias in regression coefficients it introduces bias into predicted event probabilities. We propose an alterative modified penalised likelihood, intermediate between Firth and no penalty, as a pragmatic compromise between stability and bias. Illustration on two data sets is provided.