In binary and ordinal response models outlying covariates as well as incoherent responses may affect the reliability of the maximum likelihood estimators and that of the derived inferential procedures. However the various link functions, which provide the relationship between the linear predictor and the probabilities of the response categories, differ in terms of sensitivity to anomalous data. The current paper derives conditions useful to evaluate the properties of the link functions with respect to robustness, either when the covariates are outlier free or when extreme design points may occur. The main results show that - by an appropriate choice of the link function - robust estimators, with a bounded influence function, can be easily derived from the usual likelihood function, while preserving the predictive ability of the fitted model.
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