Time series for modelling counts from a relapsing‐remitting disease: Application to modelling disease activity in multiple sclerosis

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Abstract
Many chronic diseases are relapsing‐remitting diseases, in which subjects alternate between periods with increasing and decreasing disease activity; relapsing‐remitting multiple sclerosis is an example. This paper proposes two classes of models for sequences of counts observed from a relapsing‐remitting disease. In the first, the relapsing‐remitting nature of the data is modelled by a Poisson time series with a periodic trend in the mean. In this approach, the mean is expressed as a function of a sinusoidal trend and past observations of the time series. An algorithm that uses GLIM is developed, and it results in maximum‐likelihood estimation for the amplitude, frequency and autoregressive effects. In the second class of models, the relapsing‐remitting behaviour is described by a Poisson time series in which changes in the mean follow a latent Markov chain. An EM algorithm is developed for maximum‐likelihood estimation for this model. The two models are illustrated and compared with data from a study evaluating the use of serial magnetic resonance imaging as a measure of disease activity in relapsing‐remitting multiple sclerosis.