Estimating the parameters of multi-state models with time-dependent covariates through likelihood decomposition

Multi-state models become complex when the number of states is large, when back and forth transitions between states are allowed, and when time-dependent covariates are inevitable. However, these conditions are sometimes necessary in the context of medical issues. For instance, they were needed for modelling the future treatments of patients with end-stage renal disease according to age and to various treatments.

The available modelling tools do not allow an easy handling of all issues; we designed thus a specific multi-state model that takes into account the complexity of the research question. Parameter estimation relied on decomposition of the likelihood and separate maximisations of the resulting likelihoods. This was possible because there were no interactions between patient treatment courses and because all exact times of transition from any state to another were known. Poisson likelihoods were calculated using the time spent at risk in each state and the observed transitions between each state and all others. The likelihoods were calculated on short time intervals during which age was considered as constant.

The method was not limited by the number of parameters to estimate; it could be applied to a multi-state model with 10 renal replacement therapies. Supposing the parameters of the model constant over each of seven time intervals, this method was able to estimate one hundred age-dependent transitions.

The method is easy to adapt to any disease with numerous states or grades as long as the disease does not imply interactions between patient courses.