Establishing the effectiveness of treatments for psychopathology requires accurate models of its progression over time and the factors that impact it. Longitudinal data is however fraught with missingness, hindering accurate modeling. We re-analyse data on schizophrenia severity in a clinical trial using hidden Markov models (HMMs). We consider missing data in HMMs with a focus on situations where data is missing not at random (MNAR) and missingness depends on the latent states, allowing symptom severity to indirectly impact probability of missingness. 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, a model that allows missingness to be both state- and time-dependent is unbiased. Overall, these results show that modelling missingness as state-dependent and including relevant covariates is a useful strategy in applications of HMMs to time-series with missing data. Applying the model to data from a clinical trial, we find that drop-out is more likely for patients with less severe symptoms, which may lead to a biased assessment of treatment effectiveness.