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Comparing the differences in outcomes (that is, in "dependent variables")between two subpopulations is often most informative when comparing outcomesonly for individuals from the subpopulations who are similar according to"independent variables." The independent variables are generally known as"scores," as in propensity scores for matching or as in the probabilitiespredicted by statistical or machine-learned models, for example. If theoutcomes are discrete, then some averaging is necessary to reduce the noisearising from the outcomes varying randomly over those discrete values in theobserved data. The traditional method of averaging is to bin the data accordingto the scores and plot the average outcome in each bin against the averagescore in the bin. However, such binning can be rather arbitrary and yet greatlyimpacts the interpretation of displayed deviation between the subpopulationsand assessment of its statistical significance. Fortunately, such binning isentirely unnecessary in plots of cumulative differences and in the associatedscalar summary metrics that are analogous to the workhorse statistics ofcomparing probability distributions -- those due to Kolmogorov and Smirnov andtheir refinements due to Kuiper. The present paper develops such cumulativemethods for the common case in which no score of any member of thesubpopulations being compared is exactly equal to the score of any other memberof either subpopulation.

A graphical method of cumulative differences between two subpopulations