Node-crossing vectors (“NXV”) feasibly extends Mann Whitney U to handle ties and small samples for clinical rating scales - Reid, Keith (Frontiers in Applied Mathematics and Statistics, 2026)
The Mann-Whitney U test (“MWU”) is a canonical test of order used to compare two groups. A longstanding gap is that MWU handles ties and small samples incorrectly. In the present author's clinical field, this impairs analyses of ward incident counts, or “small” rating scales used for comparing clinical populations such as Clinical Global Impression (“CGI”) and Brøset Violence Checklist (“BVC”). This paper introduces Node-Crossing Vectors (“NXV”), an extension of MWU that solves those problems. While MWU uses sample size and U-scores of order to calculate probability, NXV furthermore proceeds from first principles to handle expectations of ties combinatorially. MWU is briefly outlined in NXV's terms, then proven to be inexact for small clinical rating scales. Hitherto, ties were sometimes ignored due to computational difficulty. No settled method had been extant. Here, NXV was applied a priori to real CGI scores in a real n = 13 problem which featured seven tied elements. Ties were decisive: the population parameter λ (expected τ tied elements per individual sample of size n) critically affects power. NXV with a grounded expectation of seven tied elements gives pλ7 = 0.0280(3sf), above two-tailed α = 0.025, suggesting no difference. Alternatively expecting six tied elements gives pλ6 = 0.0241(3sf), changing the result. NXV is more exact than MWU, has viable code, and improves the analysis of two small samples. The explanation is visually accessible, and simulation data are shown. NXV arose during analyses aiming to protect the rights of restrained persons with protected characteristics and other small samples, but it is generalizable.