Regression models have widescale application in empirical research for assessing the association between a response and covariates. Traditional regression models focus on the average response. They are therefore less well suited to the analysis of skewly distributed or outlying responses. Probabilistic index models (PIM) form a new class of models for the probability that a change in covariate pattern is associated with an increase in the response. They are models for the probabilistic index, which expresses how likely it is for the first of two randomly picked subjects with given covariate patterns, to have a higher response than the second subject. This class of semiparametric statistical models is useful when not only the mean of the response is affected by the covariates. Moreover, inference for these models is robust to outliers and has a natural link with the theory on classical rank tests. For instance, score tests under a PIM generalize the Mann-Whitney test by enabling covariate adjustment and by providing interpretable effect size parameters.