Multistate Markov models for disease progression in the presence of informative examination times

Sweeting, Farewell and De Angelis have a new paper in Statistics in Medicine. This deals with the problem of a panel observed disease process where the examination times are generated by a non-ignorable mechanism. This is in general a very difficult problem. The authors consider a special case where, while the disease process is only observed at informative examination times, an auxiliary variable is able to be observed at a full set of planned (or ignorable) times. They consider a model where the disease process is missing at random (MAR) conditional on the values of the auxiliary variable. The likelihood then becomes of the form of a (partially) hidden Markov model. An alternative missing not at random model (MNAR) with missingness dependent on the current state is also considered. However the comparison between the MAR and MNAR models is somewhat unfair. In the simulations and the application, the disease process is only observed at a minority of planned examination times. The auxiliary variable has a fairly strong correlation with the disease process, meaning significant information about the process can be obtained through using the auxiliary variable in a hidden Markov model. Indeed, even without the suspicion of informative missingness the HMM would be worth using (if the strong assumptions about the relationship with the auxiliary variable could be reliably assumed) for improved efficiency. However, the MNAR model does not use the auxiliary variable at all. A better model to compare would be a hybrid of the two where a partially HMM is used in conjunction with the logistic model for the MNAR (although this wouldn't necessarily be consistent under AD-MAR unless the logistic model was correctly specified). This does not seem like a difficult extension.

A further shortcoming of the paper is the lack of any suggestions for model checking. Measurements from the auxiliary variable are assumed to be Normally distributed and independent conditional on the underlying disease states at the examination times. This seems fairly unrealistic and should at least be supported by the data.