We propose a hierarchical model for the analysis of data from several randomized trials where some outcomes are missing. The degree of departure from a missing-at-random assumption in each arm of each trial is expressed by an informative missing odds ratio (IMOR). We require a realistic prior for the IMORs, including an assessment of the prior correlation between IMORs in different arms and in different trials. The model is fitted by Monte Carlo Markov Chain techniques. By applying the method in three different data sets, we show that it is possible to appropriately capture the extra uncertainty due to missing data, and we discuss in what circumstances it is possible to learn about the IMOR.