Systemic testing on bradley-terry model against nonlinear ranking hierarchy

Abstract

We take a system point of view toward constructing any power or ranking hierarchy onto a society of human or animal players. The most common hierarchy is the linear ranking, which is habitually used in nearly all real-world problems. A stronger version of linear ranking via increasing and unvarying winning potentials, known as Bradley-Terry model, is particularly popular. Only recently non-linear ranking hierarchy is discussed and developed through recognition of dominance information contents beyond direct dyadic win-and-loss. We take this development further by rigorously arguing for the necessity of accommodating system's global pattern information contents, and then introducing a systemic testing on Bradley- Terry model. Our test statistic with an ensemble based empirical distribution favorably compares with the Deviance test equipped with a Chi-squared asymptotic approximation. Several simulated and real data sets are analyzed throughout our development.