Bradley-Terry Models. Stat 557 Heike Hofmann
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1 Bradley-Terry Models Stat 557 Heike Hofmann
2 Outline Definition: Bradley-Terry Fitting the model Extension: Order Effects Extension: Ordinal & Nominal Response Repeated Measures
3 Bradley-Terry Model (1952) Idea: based on pairwise comparisons, find overall ranking e.g. sports teams, wine tasting,,...
4 Example: American League Baseball 1987 Season each team played every other 13 times losing team winning Baltimore Cleveland Boston NY Toronto Detroit Milwaukee Baltimore Cleveland Boston NY Toronto Detroit Milwaukee This table translates into a matrix of = 21 columns: standings.shtml
5 Bradley-Terry Model Let πab = probability that a beats b assume πab+πba = 1 i.e. no ties are allowed (for now) logit model log πab/πba = µa - µb with µ1 = 0 (estimability)
6 Bradley-Terry Model πab = exp(µa)/(exp(µb)+exp(µa)) πab > 0.5, if µa > µb Bradley-Terry Model is quasi-symmetric model
7 ABL - logit model data abl$pair <- fsym(abl$winner, abl$loser) require(plyr) abl.new <- ddply(abl,.(pair), function(x) { dummy <- as.numeric(teams==x$winner[1]) - as.numeric(teams==x$loser[1]) return(c(dummy, x$times)) }) names(abl.new) <- c("pair", as.character(teams), "scorea", "scoreb") abl.tb <- glm(cbind(scorea, scoreb)~milwaukee + Detroit + Toronto + + NY + Boston + Cleveland-1, data=abl.new, family=binomial(link=logit)) summary(abl.tb)
8 ABL - logit model data > head(abl.new) pair Milwaukee Detroit Toronto NY Boston Cleveland Baltimore scorea scoreb 1 Baltimore,Boston Baltimore,Cleveland Baltimore,Detroit Baltimore,Milwaukee Baltimore,NY Baltimore,Toronto
9 glm(formula = cbind(scorea, scoreb) ~ Milwaukee + Detroit + Toronto + NY + Boston + Cleveland - 1, family = binomial(link = logit), data = abl.new) ALB - logit model Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) Milwaukee e-06 *** Detroit e-05 *** Toronto *** NY *** Boston *** Cleveland * --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 21 degrees of freedom Residual deviance: on 15 degrees of freedom AIC: Number of Fisher Scoring iterations: 4
10 ABL - QS model data > head(abl) loser winner times pair 2 Detroit Milwaukee 7 Detroit,Milwaukee 3 Toronto Milwaukee 9 Milwaukee,Toronto 4 NY Milwaukee 7 Milwaukee,NY 5 Boston Milwaukee 7 Boston,Milwaukee 6 Cleveland Milwaukee 9 Cleveland,Milwaukee 7 Baltimore Milwaukee 11 Baltimore,Milwaukee
11 glm(formula = times ~ pair winner, family = poisson(link = log), data = abl) ABL - QS model Coefficients: Estimate Std. Error z value Pr(> z ) pairbaltimore,boston e-12 *** pairbaltimore,cleveland e-14 ***... pairmilwaukee,ny e-11 *** pairmilwaukee,toronto e-11 *** pairny,toronto e-08 *** winnerdetroit winnertoronto winnerny winnerboston winnercleveland ** winnerbaltimore e-06 *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for poisson family taken to be 1) Null deviance: on 42 degrees of freedom Residual deviance: on 15 degrees of freedom AIC: Number of Fisher Scoring iterations: 5
12 ABL - TB model data > library(bradleyterry2) > > data(baseball, package = "BradleyTerry2") > head(baseball) home.team away.team home.wins away.wins 1 Milwaukee Detroit Milwaukee Toronto Milwaukee New York Milwaukee Boston Milwaukee Cleveland Milwaukee Baltimore 6 0
13 BTm(outcome = cbind(home.wins, away.wins), player1 = home.team, player2 = away.team, id = "team", data = baseball) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) teamboston *** teamcleveland * teamdetroit e-05 *** teammilwaukee e-06 *** teamnew York *** teamtoronto *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 42 degrees of freedom Residual deviance: on 36 degrees of freedom AIC: Number of Fisher Scoring iterations: 4 ABL - TB model
14 ABL Three different solutions: same fits different residual/null deviances different degrees of freedom???
15 Terry Bradley Model Assume, X has J categories (number of teams) There are a total of J(J-1)/2 pairs of categories (J-1) parameters are fit degrees of freedom: (J-1)(J-2)/2
16 ABL For 7 teams we have 21 pairs of teams we fit 6 parameters resulting in 15 degrees of freedom
17 ABL logit has correct deviance and degrees of freedom BTm uses extended data set (comes with package, regards home/away teams) loglinear model computes deviances and degrees of freedom differently, residual deviance and degrees of freedom as with logit model (i.e. correct)
18 Home Advantage most sports show a home advantage 1987 season Away Team Home Team Milwaukee Detroit Toronto New York Boston Cleveland Baltimore Milwaukee Detroit Toronto New York Boston Cleveland Baltimore NY vs Boston lost 4 2 at Boston, and won 4 3 at New York
19 Bradley Terry with Order Effects assume that first team plays at home let πab be the probability that team a beats team b when team a goes first logit model log πab/πba = µ + µa - µb if µ significantly > 0 there is a home advantage
20 ABL TerryBradley2 package baseball$home.team <- data.frame(team = baseball$home.team, at.home = 1) baseball$away.team <- data.frame(team = baseball$away.team, at.home = 0) baseballmodel2 <- update(baseballmodel1, formula = ~ team + at.home) summary(baseballmodel2) > anova(baseballmodel1, baseballmodel2) Analysis of Deviance Table Response: cbind(home.wins, away.wins) Model 1: ~team Model 2: ~team + at.home Resid. Df Resid. Dev Df Deviance
21 BTm(outcome = cbind(home.wins, away.wins), player1 = home.team, player2 = away.team, formula = ~team + at.home, id = "team", data = baseball) Coefficients: Estimate Std. Error z value Pr(> z ) teamboston *** teamcleveland * teamdetroit e-05 *** teammilwaukee e-06 *** teamnew York *** teamtoronto e-05 *** at.home * --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 42 degrees of freedom Residual deviance: on 35 degrees of freedom AIC: Number of Fisher Scoring iterations: 4 ABL
22 ABL - logit with order response<-cbind( c(4,4,4,6,4,6, 3,4,4,6,6,4, 2,4,2,4,4,6, 3,5,2,4,4,6, 5,2,3,4,5,6, 2,3,3,4,4,2, 2,1,1,2,1,3), c(3,2,3,1,2,0, 3,2,3,0,1,3, 5,3,4,3,2,0, 3,1,5,3,2,1, 1,5,3,2,2,0, 5,3,4,3,2,4, 5,5,6,4,6,4)) # 42 pair sets xabl <- expand.grid(teamb=teams, teama=teams) idx <- with(xabl, which(teama==teamb)) xabl <- xabl[-idx,] X <- matrix(0, nrow=nrow(xabl), ncol=length(teams)) for (i in 1:nrow(X)) { X[i,as.numeric(xabl$teamA)[i]] <- 1 X[i,as.numeric(xabl$teamB)[i]] <- -1 } X <- data.frame(x) names(x) <- as.character(teams)
23 ABL - home advantage teamb teama scorea scoreb Milwaukee Detroit Toronto NY Boston Cleveland Baltimore 2 Detroit Milwaukee Toronto Milwaukee NY Milwaukee Boston Milwaukee Cleveland Milwaukee Baltimore Milwaukee fit.bto<-glm(cbind(scorea, scoreb)~1+milwaukee + Detroit + Toronto + NY + Boston + Cleveland, family=binomial(link=logit), data=xabl)
24 glm(formula = cbind(scorea, scoreb) ~ 1 + Milwaukee + Detroit + Toronto + NY + Boston + Cleveland, family = binomial(link = logit), data = xabl) ABL - home advantage Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) * Milwaukee e-06 *** Detroit e-05 *** Toronto e-05 *** NY *** Boston *** Cleveland * --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 41 degrees of freedom Residual deviance: on 35 degrees of freedom AIC: Number of Fisher Scoring iterations: 4
25 Bradley Terry - Extensions Ordinal Response: cumulative logit model logit P(Y j) = µj + µa - µb e.g. loss, tie, win Nominal Response: baseline categorical model log P(Y = j)/p(y = J) = µj + µaj - µbj
26 Repeated Measures Models Extension of matched pairs data Multiple (T 3) measurements observed for same individual, e.g. individuals weekly progress Measurements for cluster of individuals (T 3), e.g. one litter, teeth at dentist s visit,...
27 Example: Drug Comparisons Cross-over effect of drugs A, B, C Interested in marginal distributions P(A=S), P(B=S), P(C=S) A B C count S S S 6 S S F 16 S F S 2 S F F 4 F S S 2 F S F 4 F F S 6 F F F 6 Total 46
28 Multiple Binary Response Yt binary response for time points t=1,..., T logit model logit P (Y t = 1) = α + β t estimability: β T = 0 (or α = 0) = = 0 we ob Marginal ithhomogeneity constraint T = 0 (or β 1 = β 2 =... = β T = 0 ( = 1).
29 Drugs Crossover > head(drugs.m) count id variable value A Y A Y A Y A Y A N A N
30 glm(formula = value ~ variable - 1, family = binomial(link = logit), data = drugsm, weights = count) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) variablea variableb variablec * --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 24 degrees of freedom Residual deviance: on 21 degrees of freedom AIC: Number of Fisher Scoring iterations: 4 Drugs Crossover
31 Marginal Homogeneity > anova(drugs.null, drugs.mh, test="chisq") Analysis of Deviance Table Model 1: value ~ 1 Model 2: value ~ variable - 1 Resid. Df Resid. Dev Df Deviance P(> Chi ) * --- Signif. codes: 0 *** ** 0.01 *
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