Delta Plots and Coherent Distribution Ordering

Social scientists often compare subclasses of populations or manipulatons. For example, in comparing task-completion times across two levels of a manipulations, if one group has faster overall mean response, it is natural to ask if the fastest 10% of the first group has a faster mean than the fastest 10% of the second group, and so on. Delta plots, a type of quantilequantile residual plot used by psychologists, shed light on these comparisons and motivate new notions of stochastic ordering. If all percentile classes have faster mean in one group than in the other, we say that there is coherent mean ordering and that one group stochastically dominates the other in mean. A related notion of coherent variance ordering can be defined similarly. Violations of coherent orderings of means or variances are diagnostic signatures of complex effects and suggest further avenues of study. In this note, we derive necessary and sufficient conditions for stochastic dominance in mean and variance. We show that stochastic mean dominance is exactly equivalent to the usual stochastic dominance and stochastic variance dominance is equivalent to ordering of the first derivative of the quantile functions.