# Statistical inference for the mixed Bradley-Terry model

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Microsoft Word - NCRM working paper series template_front.doc Statistical inference for the mixed Bradley-Terry model ESRC National Centre for Research Methods NCRM Working Paper Series 9/06 Mohand Feddag University of Warwick (UK) Ongoing joint work: D. Firth (Warwick) & C. Varin (Padova, Italy) Statistical inference for the mixed Bradley-Terry model Mohand Feddag University of Warwick (UK) Ongoing joint work: D. Firth (Warwick) & C. Varin (Padova, Italy) Stat. Latent Variables Models in the Health Sciences, Perugia 06-08 Sept. 2006 Page 1 Plan 1. Introduction 1.1. Model 1.2. Motivation 2. Estimation by the pairwise likelihood approach 3. Illustrations 3.1. Simulation study 3.2. Real data 4. Concluding remarks Page 2 1- Introduction Bradley-Terry models Data structure • N players • Compare “player” i with “player” j in contest ijt, t = 1, . . . , Tij Simplest version: Binary yijt = 8< : 1 if i beats j 0 if j beats i Elaborations: ties; margin of victory (ordinal or continuous) Page 3 1.1- Bradley-Terry models Linear predictor: ηijt = g [Pr(yijt = 1)] • Pure B.T: ηijt = αi − αj • General: ηijt = αi − αj + z T ijtγ e.g. zijt = 1 i is at home −1 j is at home 0 otherwise Some typical aims 1. Rank the players according to their ability score αi 2. Explain ability in terms of player-specific covariates xi: αi = x T i β + ui, ui iid ∼ Fu(σ) Page 4 1.1- Bradley-Terry models Random effects • Aim 1 : ui = αi ensures appropriate shrinkage of ability estimates to take account of imprecision of estimation (e.g., Efron and Morris, JASA, 1975) • Aim 2 : ui = αi − x T i β to represent unexplained variation in ability; often not recognised in the literature (e.g., Springall, 1973) Ge

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