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Structured Bradley-Terry models, and flat lizards fighting

By David Firth, D M Stuart-fox and N Bennett


The Bradley-Terry model (Bradley and Terry, 1952; Agresti, 2002) is a useful device for the scoring of ‘tournaments ’ for which the data are the results of ‘contests ’ between pairs of ‘players’. Applications are many, ranging from bibliometrics (Stigler, 1994) in which the ‘players ’ are academic journals, to genetics (for example, the allelic transmission/disequilibrium test of Sham and Curtis (1995) is based on a Bradley-Terry model in which the ‘players ’ are alleles). The Bradley-Terry model in its simplest form, with no ties permitted, is a logistic regression with a specially structured design matrix: in any contest between players i and j, logit[pr(i beats j)] = λi − λj. A simple elaboration allows an ‘order ’ effect, for example to allow one player in each contest to enjoy an advantage on account of ‘going first’, or ‘playing on home turf’: logit[pr(i beats j in contest t)] = λi − λj + δzt, where zt = 1 if i has the supposed advantage and zt = −1 if j has it. (If the ‘advantage ’ is in fact a disadvantage, δ will be negative.) The scores λi then relate to ability in the absence of any such advantage. Motivated by study of a large ‘tournament ’ among male flat lizards (Platysaurus broadleyi) in the wild, we consider structured forms of the Bradley-Terry model in which the ability parameters λ1,..., λK are determined by measured attributes of the K players involved, viz. p∑ λi = (i = 1,..., K). r=1 βrxir Special attention is needed in the case of contests involving any lizard i whose explanatory values xi1,..., xip are incomplete. This talk will describe the BradleyTerry package for R (Firth, 2004), which provides facilities for the specification and fitting of such models, for model selection by standard methods, and for model criticism via special-purpose residuals. Use of the package will be illustrated using the lizard data (K = 77), analysis of which reveals biologically-important evidence on the role played by bright colours on a male lizard’s body

Year: 2010
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