The purpose of this research was to develop appropriate methods for analysing repeated ordinal categorical data that arose in an intervention trial to prevent oesophageal cancer. The measured response was the degree of oesophageal dysplasia at 2.5 and 6 years after randomization. An important feature was that some response measurements were missing, and the missingness was not 'completely at random' (MCAR). We show that standard likelihood-based methods and standard methods based on marginal estimating equations yield biased results, and we propose adaptations to both these approaches that yield valid inference under the weaker 'missing at random' (MAR) assumption. On the basis of efficiency calculations, simulation studies of finite sample properties, ease of computation, and flexibility for testing and exploring a range of treatment models, we recommend the adapted likelihood-based approach for problems of this type, in which there are abundant data for estimating parameters.