A few Murray connections: Extended Bradley-Terry models. Bradley-Terry model. Pair-comparison studies. Some data. Some data

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1 Introduction A few Murray connections: Extended Bradley-Terry models David Firth (with Heather Turner) Department of Statistics University of Warwick psychometrics sport Lancaster missing data structured sources of variation interactive modelling software Murray s 70th celebration, London, Pair-comparison studies Bradley-Terry model The basic model: Sport: player i beats player j Psychometrics: object i is preferred to object j pr(i beats j) = with α i the relative ability of object i. α i α i + α j, Sport (etc.): interest in players and their attributes Psychometrics (etc.): interest in judges (subjects) and their attributes Work with log abilities: logit[pr(i beats j)] = log(α i ) log(α j ) = λ i λ j. Simple interpretation; Luce s axiom. Some data From (A. Agresti, Categorical Data Analysis): Loser Winner Seles Graf Sabatini Navratilova Sanchez Seles Graf Sabatini Navratilova Sanchez Bradley-Terry model scores the players in terms of their (assumed constant) relative abilities α i. Some data From 1983: Loser Winner Aitkin Firth Aitkin 1 Firth 0 MLE for the ability parameters here: ˆα Firth = 100 (by convention!) ˆα Aitkin = (!)

2 Extensions? Structured Bradley-Terry model We focus on three possible directions from the basic model: 1. (Log-)abilities λ i determined/predicted by object covariate vector x i. 2. λ i λ ik : the ability of object i varies between different comparisons k. 3. i versus j, no preference? ( tied comparisons) λ i = f i (β) + U i = r β r x ir + U i attributes of objects/players predict ability (for example) U i is random error, with variance σ 2, say needed in order to allow for imperfect prediction complex random effects model, with linear predictor (x ir x jr )β r + (U i U j ) r Ability varying between comparisons Ability varying between comparisons (continued) e.g., time-varying covariates, e.g., subject-specific abilities, λ i λ ik λ ik = r β r x ikr + U i λ ik = λ is, where s = s(k) identifies the subject who makes comparison k. e.g., abilities predicted by subject covariates, λ is = t γ it z st + E is e.g., still with abilities λ is varying between subjects, a particular form likely to be useful is multiplicative interaction, ( ) λ is = λ i exp γ t z st + E is a generalized nonlinear mixed model (with non-nested random effects, as always with structured Bradley-Terry models). R packages BradleyTerry2 and gnm t An example: Reptile social science Male Augrabies flat lizards:

3 A lizard tournament 189 male lizards were captured and various measurements made. Then released, and contests (fights) observed. Every contest has a winner and a loser (judged by the observer apparently not difficult to judge!) In all, 100 contests were observed, involving 77 of the lizards. Explanatory variables PC1throat, PC2throat, PC3throat: first 3 PCs of throat spectrum PC1FL, PC2FL, PC3FL: first 3 PCs of forelimb spectrum PC1BA, PC2BA, PC3BA: first 3 PCs of badge spectrum Bsize: badge size AdjT: blood testosterone concentration SVL: snout-vent length HL.res, HW.res, HH.res: residuals of head length, width, height on SVL condition: residuals of body mass on SVL A two-stage analysis? 1. Obtain estimates ˆα 1,..., ˆα 77 from a standard maximum-likelihood analysis of the Bradley-Terry model for the contest results. 2. Fit linear models E(log ˆα i ) = p β rx ir, using the ML estimates as response variable (and reciprocals of squared standard errors as weights). This is fine in principle. Indeed, it may be shown theoretically to be fully efficient, asymptotically as the number of fights per lizard increases. But: information on unconnected lizards is lost completely lizard 62 is massively influential many of the ML estimates ˆα i are infinite-valued.

4 A more direct approach With λ i = log α i = β r x ir + U i, estimate the coefficients of the implied logistic regression logit[p (i beats j)] = β r (x ir x jr ) + (U i U j ) directly. (But note difficulties with ML estimation here!) Advantages: Uses all the available information. The influence of a contest is now related in the usual way to its position in predictor space. logit[p (i beats j)] = Some problems: β r (x ir x jr ) + (U i U j ) model specification/search in standard software is painful. missing data. In practice some values of x ir are missing because the measurements were corrupted or could not be taken for reasons entirely unconnected with the values to be measured, or with contest outcomes. likelihood intractible, so approximate methods are needed. we need lizard-specific residuals (not contest-specific residuals) in order to criticise the linear predictor. Missing data logit[p (i beats j)] = β r (x ir x jr ) + (U i U j ) A standard approach: if a value of x ir x jr is missing (at random), omit the corresponding contest when fitting the model. But this is very wasteful potentially, and actually in the case of the lizard data. Consider an extreme example: Solution: the model really should be player 1 player 5 log α i = { p β rx ir + U i if case i has no missing values λ i if case i has missing values player 2 player 3 player 4 player 6 player 7 If player 4 has a covariate value missing, all data is lost! Omitting a case (here a contest) is equivalent to including a nuisance parameter for that case. But really a case should be a player (here, a lizard), not a contest. so that the linear predictor for contest outcomes is p β r(x ir x jr ) + U i U j normally λ i p β rx jr U j if i has missing data p β rx ir + U i λ j if j has missing data λ i λ j if both i and j have missing data In the extreme example above, this would result in the estimation of a single extra parameter λ 4 for player 4, rather than complete non-estimability of all the β parameters.

5 Some results Use of standard model-search methods (forwards, backwards, stepwise, based on either significance tests or AIC) points to the following model, as specified in the BradleyTerry2 package for R : > library(bradleyterry2) > BTm(result, winner, loser, ~ throat.pc1[..] + throat.pc3[..] + head.length[..] + SVL[..] + (1..), data = list(contests, predictors)) The random term U i U j turns out to have substantial variance: ˆσ U = 1.1 (st. error 0.3) Main interest is in the coefficients for covariates: Fixed Effects: Estimate Std. Error z value Pr(> z )..lizard e e e lizard e e throat.pc1[..] e e throat.pc3[..] 3.735e e head.length[..] e e SVL[..] 1.722e e same story (thankfully!) as in the published 2006 paper, which ignored the error terms. (The standard errors are a bit larger here, more realistic.) Main substantive conclusions: Normal Q Q plot of contest residuals Clear evidence for throat colour as a predictor of fight-winning ability. Overall brightness (PC1) and UV intensity (PC3) of the throat are clearly significant predictors. PC3 has by far the largest effect: the standard deviation of PC3throat is 2.34, so in a contest between lizards at ±2 standard deviations the odds are estimated as exp( ) 30 in favour of the lizard with greater UV reflectance on the throat. Diagnostics? Based on lizard residuals. Sample Quantiles Theoretical Quantiles Normal Q Q plot of lizard residuals Sample Quantiles 3 Theoretical Quantiles