Mann–Whitney-type causal effects are generally applicable to outcome variables with a natural ordering, have been recommended for clinical trials because of their clinical relevance and inter- pretability and are particularly useful in analysing an ordinal composite outcome that combines an original primary outcome with death and possibly treatment discontinuation. In this article, we consider robust and efficient estimation of such causal effects in observational studies and clinical trials. For observational studies, we propose and compare several estimators: regression estimators based on an outcome regression (OR) model or a generalised probabilistic index (GPI) model, an inverse probability weighted estimator based on a propensity score model and two doubly robust (DR), locally efficient estimators. One of the DR estimators involves a propensity score model and an OR model, is consistent and asymptotically normal under the union of the two models and attains the semiparametric information bound when both models are correct. The other DR estimator has the same properties with the OR model replaced by a GPI model. For clinical trials, we extend an existing augmented estimator based on a GPI model and propose a new one based on an OR model. The methods are evaluated and compared in simulation experiments and applied to a clinical trial in cardiology and an observational study in obstetrics.