A Hidden Markov Model for Informative Dropout in Longitudinal Response Data with Crisis States

Spagnoli, Henderson, Boys and Houwing-Duistermaat have a new paper in Statistics and Probability Letters. The paper is concerned with the modelling of discrete-time continuous response longitudinal data in the presence of informative dropout. The process of dropout is modelled by a three-state (potentially non-homogeneous) discrete time Markov model. The three states correspond to stable, crisis and dropout. The crisis state is characterized by a higher probability of
dropout and a shift in the mean response. The shift is random but is fixed for each individual so that repeated visits to the crisis state have a cumulative effect on the mean. The model is motivated by studies in which dropout is more likely to occur after the treatment has been ineffective for a period of time. A linear-mixed model, with random slope and intercept, is taken for the responses, with the response at time m being shifted by a random quantity, d, times by the number of time periods spent in the crisis state.

The authors put the model within the standard Rubin framework of MCAR/MAR/MNAR, making a distinction between observable and latent filtration. They are careful to formulate the model so that the latent mechanism for dropout only depends on the past and not the future.

The model is applied to schizophrenia data and the Leiden 85+ data. In the former case, interest is in the mean response curves in the hypothetical situation of no dropout. However for the Leiden 85+ data dropout is death and thus such curves would have little meaning. For both cases the crisis-state model represents a substantial improvement in likelihood compared to a two-state model.