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The author's recent research papers, "Cumulative deviation of a subpopulationfrom the full population" and "A graphical method of cumulative differencesbetween two subpopulations" (both published in volume 8 of Springer'sopen-access "Journal of Big Data" during 2021), propose graphical methods andsummary statistics, without extensively calibrating formal significance tests.The summary metrics and methods can measure the calibration of probabilisticpredictions and can assess differences in responses between a subpopulation andthe full population while controlling for a covariate or score via conditioningon it. These recently published papers construct significance tests based onthe scalar summary statistics, but only sketch how to calibrate the attainedsignificance levels (also known as "P-values") for the tests. The presentarticle reviews and synthesizes work spanning many decades in order to detailhow to calibrate the P-values. The present paper presents computationallyefficient, easily implemented numerical methods for evaluating properlycalibrated P-values, together with rigorous mathematical proofs guaranteeingtheir accuracy, and illustrates and validates the methods with open-sourcesoftware and numerical examples.

Calibration of P-values for calibration and for deviation of a subpopulation from the full population