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

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.