  Comparing the differences in outcomes (that is, in "dependent variables")
between two subpopulations is often most informative when comparing outcomes
only 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 probabilities
predicted by statistical or machine-learned models, for example. If the
outcomes are discrete, then some averaging is necessary to reduce the noise
arising from the outcomes varying randomly over those discrete values in the
observed data. The traditional method of averaging is to bin the data according
to the scores and plot the average outcome in each bin against the average
score in the bin. However, such binning can be rather arbitrary and yet greatly
impacts the interpretation of displayed deviation between the subpopulations
and assessment of its statistical significance. Fortunately, such binning is
entirely unnecessary in plots of cumulative differences and in the associated
scalar summary metrics that are analogous to the workhorse statistics of
comparing probability distributions -- those due to Kolmogorov and Smirnov and
their refinements due to Kuiper. The present paper develops such cumulative
methods for the common case in which no score of any member of the
subpopulations being compared is exactly equal to the score of any other member
of either subpopulation.
