Node-Crossing Vectors ("NXV") feasibly extends Mann Whitney U to handle ties and small samples for clinical rating scales

Keith Strang Reid^1,2*

• ¹Northumbria University Faculty of Health and Life Sciences, Newcastle upon Tyne, United Kingdom
• ²Cumbria Northumberland Tyne and Wear NHS Foundation Trust, Newcastle upon Tyne, United Kingdom

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 inexactly. 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 CGI and Brøset BVC. This paper introduces Node-Crossing Vectors ("NXV"), an extension of MWU which solves those problems. Whilst 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 an 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 acccessible and simulation data are shown. NXV arose during analyses aiming to protect rights of restrained persons with protected characteristics and other small samples, but is generalisable.