A general piecewise multi-state survival model: application to breast cancer

Multi-state models are considered in the field of survival analysis for modelling illnesses that evolve through several stages over time. Multi-state models can be developed by applying several techniques, such as non-parametric, semi-parametric and stochastic processes, particularly Markov processes. When the development of an illness is being analysed, its progression is tracked periodically. Medical reviews take place at discrete times, and a panel data analysis can be formed. In this paper, a discrete-time piecewise non-homogeneous Markov process is constructed for modelling and analysing a multi-state illness with a general number of states. The model is built, and relevant measures, such as survival function, transition probabilities, mean total times spent in a group of states and the conditional probability of state change, are determined. A likelihood function is built to estimate the parameters and the general number of cut-points included in the model. Time-dependent covariates are introduced, the results are obtained in a matrix algebraic form and the algorithms are shown. The model is applied to analyse the behaviour of breast cancer. A study of the relapse and survival times of 300 breast cancer patients who have undergone mastectomy is developed. The results of this paper are implemented computationally with MATLAB and R.