Health economics studies with missing data are increasingly using approaches such as multiple imputation that assume that the data are "missing at random." This assumption is often questionable, as-even given the observed data-the probability that data are missing may reflect the true, unobserved outcomes, such as the patients' true health status. In these cases, methodological guidelines recommend sensitivity analyses to recognise data may be "missing not at random" (MNAR), and call for the development of practical, accessible approaches for exploring the robustness of conclusions to MNAR assumptions. Little attention has been paid to the problem that data may be MNAR in health economics in general and in cost-effectiveness analyses (CEA) in particular. In this paper, we propose a Bayesian framework for CEA where outcome or cost data are missing. Our framework includes a practical, accessible approach to sensitivity analysis that allows the analyst to draw on expert opinion. We illustrate the framework in a CEA comparing an endovascular strategy with open repair for patients with ruptured abdominal aortic aneurysm, and provide software tools to implement this approach.