Summary The regression discontinuity (RD) design is a quasi-experimental design which emulates a randomized study by exploiting situations where treatment is assigned according to a continuous variable as is common in many drug treatment guidelines. The RD design literature focuses principally on continuous outcomes. We exploit the link between the RD design and instrumental variables to obtain an estimate for the causal risk ratio for the treated when the outcome is binary. Occasionally this risk ratio for the treated estimator can give negative lower confidence bounds. In the Bayesian framework we impose prior constraints that prevent this from happening. This is novel and cannot be easily reproduced in a frequentist framework. We compare our estimators with those based on estimating equation and generalized methods-of-moments methods. On the basis of extensive simulations our methods compare favourably with both methods and we apply our method to a real example to estimate the effect of statins on the probability of low density lipoprotein cholesterol levels reaching recommended levels.