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.
Highlights
For many decades, the Bradley-Terry model [1] on paired competition data has remained the most popular approach for ranking and estimating probabilities of possible outcomes
We look at two typical applications of the Bradley-Terry model to determine whether the linearity assumption has been met, and if the BradleyTerry model is appropriate for the data
The Bradley-Terry assumption is potentially too restrictive to apply to many data sets, and it is imperative that this assumption must be tested
Summary
The Bradley-Terry model [1] on paired competition data has remained the most popular approach for ranking and estimating probabilities of possible outcomes. The computational foundation for estimation in Bradley-Terry model is the likelihood approach built upon the logistic probability function and independence assumptions among all dyadic games This generalized linear model framework renders that individuals’ total wins and losses are the natural sufficient statistics for the vector of winning potentials. Fujii et al [5] combine empirical transitivity with the estimation of the tiered hierarchical power structure of a competitive society to produce dominance probability estimates between any two players in that society This provides two important advantages over the Deviance test statistic: 1) by adapting empirical transitivity, one of the major systemic characteristics of the game of interest is accommodated; 2) the uncertainty of game is incorporated. Few methods exist currently for the Bradley-Terry model
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