  An essential feature of stochastic order is its invariance against increasing
maps. In this paper, we analyze a family of invariant indices of disagreement
with respect to stochastic dominance. The indices in this family admit the
representation $\theta(F,G)=P(X>Y)$, where $(X,Y)$ is a random vector with
marginal distribution functions $F$ and $G$. This includes the case of
independent marginals, but also other interesting indices related to a
contamination model or to a joint quantile representation. For some choices of
$\theta$ the condition $\theta(F,G)=0$ is equivalent to stochastic dominance of
$G$ over $F$. We show that the index associated to the contamination model
achieves the minimal value within this family. The plug-in sample-based
versions of these indices lead to the Mann-Whitney, the one-sided
Kolmogorov-Smirnov, and the Galton statistics. For some of the most interesting
indices this fact provides sufficient theoretical support for asymptotic
inference. However, this is not the case for Galton's statistic, for which we
provide additional theory for its resampling behaviour. We stress on the
complementary roles of some of these indices, which beyond measuring
disagreement with respect to stochastic order allow to describe the maximum
possible difference in status of a value $x\in \mathbb{R}$ under $F$ or $G$. We
apply these indices to some real data sets.
