In many occasions, particularly in biomedical studies, data are unavailable for some responses and covariates. This leads to biased inference in the analysis when a substantial proportion of responses or a covariate or both are missing. Except a few situations, methods for missing data have earlier been considered either for missing response or for missing covariates, but comparatively little attention has been directed to account for both missing responses and missing covariates, which is partly attributable to complexity in modeling and computation. This seems to be important as the precise impact of substantial missing data depends on the association between two missing data processes as well. The real difficulty arises when the responses are ordinal by nature. We develop a joint model to take into account simultaneously the association between the ordinal response variable and covariates and also that between the missing data indicators. Such a complex model has been analyzed here by using the Markov chain Monte Carlo approach and also by the Monte Carlo relative likelihood approach. Their performance on estimating the model parameters in finite samples have been looked into. We illustrate the application of these two methods using data from an orthodontic study. Analysis of such data provides some interesting information on human habit. © The Author(s) 2013.