  We consider missing data in the context of hidden Markov models with a focus
on situations where data is missing not at random (MNAR) and missingness
depends on the identity of the hidden states. In simulations, we show that
including a submodel for state-dependent missingness reduces bias when data is
MNAR and state-dependent, whilst not reducing accuracy when data is missing at
random (MAR). When missingness depends on time but not the hidden states, a
model which only allows for state-dependent missingness is biased, whilst a
model that allows for both state- and time-dependent missingness is not.
Overall, these results show that modelling missingness as state-dependent, and
including other relevant covariates, is a useful strategy in applications of
hidden Markov models to time-series with missing data. We conclude with an
application of the state- and time-dependent MNAR hidden Markov model to a real
dataset, involving severity of schizophrenic symptoms in a clinical trial.
