RESUMEN
In this paper, we extend the linear M-quantile random intercept model (MQRE) to discrete data and use the proposed model to evaluate the effect of selected covariates on two count responses: the number of generic medical examinations and the number of specialised examinations for health districts in three regions of central Italy. The new approach represents an outlier-robust alternative to the generalised linear mixed model with Gaussian random effects and it allows estimating the effect of the covariates at various quantiles of the conditional distribution of the target variable. Results from a simulation experiment, as well as from real data, confirm that the method proposed here presents good robustness properties and can be in certain cases more efficient than other approaches.