Abstract
For paired comparisons, we propose a measure to represent the degree of departure from the extended Bradley-Terry model. The measure is expressed by using the Kullback-Leibler information and it ranges between $0$ and $1$.The measure is applied to the win-loss standings of professional baseball league in Japan.
Highlights
Consider the athletic competitions with the outcome for the play of any two teams of R teams, namely a set of data from R(R − 1)/2 paired comparison
We propose a measure to represent the degree of departure from the extended Bradley-Terry model
Note that we can consider a generalized measure for representing the degree of departure from the extended Bradley-Terry (EBT) (EQS) model by using the power-divergence (Cressie and Read, 1984) including the Kullback-Leibler information as follows: for λ > −1, Ψ(λ) λ(λ + 1) 2(2λ − 1)
Summary
Consider the athletic competitions with the outcome for the play of any two teams of R teams, namely a set of data from R(R − 1)/2 paired comparison. A special case of this model obtained by putting γ = 1 is the BT model This model indicates that for the plays of any two teams of teams i, j and k, the probability that team i defeats team j, team j defeats team k, and team k defeats team i, is γ times higher than the probability that k defeats j, j defeats i, and i defeats k. For square tables with nominal categories, in which cells on the main diagonal are empty, Tahata et al (2004) proposed the measures to represent the degree of departure from the BT model. We are interested in considering a measure which represents the degree of departure from the EBT model for square tables with ordered categories.
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