In the Lung Health Study (LHS), compliance with the use of inhaled medication was assessed at each follow-up visit both by self-report and by weighing the used medication canisters. One or both of these assessments were missing if the participant failed to attend the visit or to return all canisters. Approximately 30% of canister-weight data and 5% to 15% of self-report data were missing at different visits. We use Gibbs sampling with data augmentation and a multivariate Hastings update step to implement a Bayesian hierarchical model for LHS inhaler compliance. Incorporating individual-level random effects to account for correlations among repeated measures on the same participant, our model is a longitudinal extension of the Tobit models used in econometrics to deal with partially unobservable data. It enables (a) assessment of the relationships among visit attendance, canister return, self-reported compliance level, and canister weight compliance, and (b) determination of demographic, physiological, and behavioral predictors of compliance. In addition to addressing the estimation and prediction questions of substantive interest, we use sampling-based methods for covariate screening and model selection and investigate a range of informative priors on missing data.