Continuous-time multi-state survival models can be used to describe health-related processes over time. In the presence of interval-censored times for transitions between the living states, the likelihood is constructed using transition probabilities. Models can be specified using parametric or semi-parametric shapes for the hazards. Semi-parametric hazards can be fitted using $P$-splines and penalised maximum likelihood estimation. This paper presents a method to estimate flexible multi-state models which allows for parametric and semi-parametric hazard specifications. The estimation is based on a scoring algorithm. The method is illustrated with data from the English Longitudinal Study of Ageing.
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