Final published version
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Modeling Intransitivity in Pairwise Comparisons with Application to Baseball Data
AU - Spearing, Jess
AU - Tawn, Jonathan
AU - Irons, David
AU - Paulden, Tim
PY - 2023/10/2
Y1 - 2023/10/2
N2 - The seminal Bradley-Terry model exhibits transitivity, that is, the property that the probabilities of player A beating B and B beating C give the probability of A beating C, with these probabilities determined by a skill parameter for each player. Such transitive models do not account for different strategies of play between each pair of players, which gives rise to intransitivity. Various intransitive parametric models have been proposed but they lack the flexibility to cover the different strategies across n players, with the (Formula presented.) values of intransitivity modeled using (Formula presented.) parameters, while they are not parsimonious when the intransitivity is simple. We overcome their lack of adaptability by allocating each pair of players to one of a random number of K intransitivity levels, each level representing a different strategy. Our novel approach for the skill parameters involves having the n players allocated to a random number of (Formula presented.) distinct skill levels, to improve efficiency and avoid false rankings. Although we may have to estimate up to (Formula presented.) unknown parameters for (Formula presented.) we anticipate that in many practical contexts (Formula presented.). Our semiparametric model, which gives the Bradley-Terry model when (Formula presented.), is shown to have an improved fit relative to the Bradley-Terry, and the existing intransitivity models, in out-of-sample testing when applied to simulated and American League baseball data. Supplementary materials for the article areavailable online.
AB - The seminal Bradley-Terry model exhibits transitivity, that is, the property that the probabilities of player A beating B and B beating C give the probability of A beating C, with these probabilities determined by a skill parameter for each player. Such transitive models do not account for different strategies of play between each pair of players, which gives rise to intransitivity. Various intransitive parametric models have been proposed but they lack the flexibility to cover the different strategies across n players, with the (Formula presented.) values of intransitivity modeled using (Formula presented.) parameters, while they are not parsimonious when the intransitivity is simple. We overcome their lack of adaptability by allocating each pair of players to one of a random number of K intransitivity levels, each level representing a different strategy. Our novel approach for the skill parameters involves having the n players allocated to a random number of (Formula presented.) distinct skill levels, to improve efficiency and avoid false rankings. Although we may have to estimate up to (Formula presented.) unknown parameters for (Formula presented.) we anticipate that in many practical contexts (Formula presented.). Our semiparametric model, which gives the Bradley-Terry model when (Formula presented.), is shown to have an improved fit relative to the Bradley-Terry, and the existing intransitivity models, in out-of-sample testing when applied to simulated and American League baseball data. Supplementary materials for the article areavailable online.
KW - Baseball
KW - Bayesian hierarchical modeling
KW - Bradley-Terry
KW - Clustering
KW - Intransitivity
KW - Pairwise Comparisons
KW - Ranking
KW - Reversible jump Markov chain Monte Carlo
KW - Tournament structure
U2 - 10.1080/10618600.2023.2177299
DO - 10.1080/10618600.2023.2177299
M3 - Journal article
VL - 32
SP - 1383
EP - 1392
JO - Journal of Computational and Graphical Statistics
JF - Journal of Computational and Graphical Statistics
SN - 1061-8600
IS - 4
ER -