Monotone Delta: An Order-Theoretic Tournament Graph Approach for Internal Consistency Assessment

Muhammad Umair Danish^1*Umair Rehman²Katarina Grolinger¹

• ¹Western University, London, Canada
• ²Wilfrid Laurier University, Waterloo, Canada

This paper introduces Monotone Delta ($\delta$), an order-theoretic measure for assessing the internal consistency of survey-based instruments. Classical coefficients such as Cronbach's Alpha and McDonald's Omega can yield misleading estimates under practical violations, including redundancy, multidimensional constructs, and correlated errors. Monotone Delta avoids parametric and factor-model assumptions by quantifying internal consistency through contradiction minimization with a weighted tournament formulation, aligning responses to an optimal unidimensional latent order. In controlled synthetic studies across four scenarios (tau-equivalence, redundancy, multidimensionality, and non-normal/correlated errors), Monotone Delta stays closest to the theoretical reliability, with absolute error \(\le 0.02\) in the challenging scenarios where Alpha and Omega deviate by as much as \(0.22\) and \(0.14\), respectively. On a 350-participant human study for AI-generated image assessment, Monotone Delta agrees with Alpha/Omega under near-ideal conditions (overall \(\delta=0.91\) vs.\ \(\alpha=0.92\), \(\omega=0.94\)) while remaining stable under redundancy and non-normal perturbations (overall \(\delta=0.84\) and \(\delta=0.81\), respectively, where Alpha drops to \(0.95\) and \(0.35\)). These results position Monotone Delta as a practical alternative for reliability assessment in socio-technical systems, human factors, healthcare, and interactive system design.