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Last data update: 2014.03.03

R: Extended Bradley-Terry Model

btm	R Documentation

Extended Bradley-Terry Model

Description

Estimates an extended Bradley-Terry model (Hunter, 2004; see Details).

Usage

btm(data, ignore.ties = FALSE, fix.eta = NULL, fix.delta = NULL, fix.theta = NULL, 
       maxiter = 100, conv = 1e-04, eps = 0.3)

## S3 method for class 'btm'
summary(object, file=NULL, digits=4,...)

Arguments

`data`	Data frame with three columns. The first two columns contain labels from the units in the pair comparison. The third column contains the result of the comparison. "1" means that the first units wins, "0" means that the second unit wins and "0.5" means a draw (a tie).
`ignore.ties`	Logical indicating whether ties should be ignored.
`fix.eta`	Numeric value for a fixed η value
`fix.delta`	Numeric value for a fixed δ value
`fix.theta`	A vector with entries for fixed theta values.
`maxiter`	Maximum number of iterations
`conv`	Convergence criterion
`eps`	The varepsilon parameter for the varepsilon-adjustment method (see Bertoli-Barsotti & Punzo, 2012) which reduces bias in ability estimates. In case of varepsilon=0, persons with extreme scores are removed from the pairwise comparison.
`object`	Object of class `btm`
`file`	Optional file name for sinking the summary into
`digits`	Number of digits after decimal to print
`...`	Further arguments to be passed.

Details

The extended Bradley-Terry model for the comparison of individuals i and j is defined as

P(X_{ij} = 1 ) propto exp( η + θ_i )

P(X_{ij} = 0 ) propto exp( θ_j )

P(X_{ij} = 0.5) propto exp( δ + ( η + θ_i +θ_j )/2 )

The parameters θ_i denote the abilities, δ is the tendency of the occurence of ties and η is the home-advantage effect.

Value

List with following entries

`pars`	Parameter summary for η and δ
`effects`	Parameter estimates for θ and outfit and infit statistics
`summary.effects`	Summary of θ parameter estimates
`mle.rel`	MLE reliability, also known as separation reliability
`sepG`	Separation index G
`probs`	Estimated probabilities
`data`	Used dataset with integer identifiers

Author(s)

Alexander Robitzsch

References

Bertoli-Barsotti, L., & Punzo, A. (2012). Comparison of two bias reduction techniques for the Rasch model. Electronic Journal of Applied Statistical Analysis, 5, 360-366.

Hunter, D. R. (2004). MM algorithms for generalized Bradley-Terry models. Annals of Statistics, 32, 384-406.

Examples

#############################################################################
# EXAMPLE 1: Bradley-Terry model | data.pw01
#############################################################################

data(data.pw01)

dat <- data.pw01
dat <- dat[ , c("home_team" , "away_team" , "result") ]

# recode results according to needed input
dat$result[ dat$result == 0 ] <- 1/2   # code for ties
dat$result[ dat$result == 2 ] <- 0     # code for victory of away team

#********************
# Model 1: Estimation with ties and home advantage
mod1 <- btm( dat)
summary(mod1)

## Not run: 
#********************
# Model 2: Estimation with ties, no epsilon adjustment
mod2 <- btm( dat , eps=0 , fix.eta=0)
summary(mod2)

#********************
# Model 3: Some fixed abilities
fix.theta <- c("Anhalt Dessau" = -1 )
mod3 <- btm( dat , eps=0, fix.theta=fix.theta)
summary(mod3)

#********************
# Model 4: Ignoring ties, no home advantage effect
mod4 <- btm( dat , ignore.ties=TRUE , fix.eta = 0)
summary(mod4)

#********************
# Model 5: Ignoring ties, no home advantage effect (JML approach -> eps=0)
mod5 <- btm( dat , ignore.ties=TRUE , fix.eta = 0 , eps=0)
summary(mod5)

#############################################################################
# EXAMPLE 2: Venice chess data 
#############################################################################

# See http://www.rasch.org/rmt/rmt113o.htm
# Linacre, J. M. (1997). Paired Comparisons with Standard Rasch Software.
# Rasch Measurement Transactions, 11:3, 584-585.

# dataset with chess games -> "D" denotes a draw (tie)
chessdata <- scan( what="character")
    1D.0..1...1....1.....1......D.......D........1.........1.......... Browne
    0.1.D..0...1....1.....1......D.......1........D.........1......... Mariotti
    .D0..0..1...D....D.....1......1.......1........1.........D........ Tatai
    ...1D1...D...D....1.....D......D.......D........1.........0....... Hort
    ......010D....D....D.....1......D.......1........1.........D...... Kavalek
    ..........00DDD.....D.....D......D.......1........D.........1..... Damjanovic
    ...............00D0DD......D......1.......1........1.........0.... Gligoric
    .....................000D0DD.......D.......1........D.........1... Radulov
    ............................DD0DDD0D........0........0.........1.. Bobotsov
    ....................................D00D00001.........1.........1. Cosulich
    .............................................0D000D0D10..........1 Westerinen
    .......................................................00D1D010000 Zichichi 

L <- length(chessdata) / 2
games <- matrix( chessdata , nrow=L , ncol=2 , byrow=TRUE )
G <- nchar(games[1,1])
# create matrix with results
results <- matrix( NA , nrow=G , ncol=3 )
for (gg in 1:G){
    games.gg <- substring( games[,1] , gg , gg )
    ind.gg <- which( games.gg != "." )
    results[gg , 1:2 ] <- games[ ind.gg , 2]
    results[gg, 3 ] <- games.gg[ ind.gg[1] ]
            }
results <- as.data.frame(results)            
results[,3] <- paste(results[,3] )
results[ results[,3] == "D" , 3] <- 1/2
results[,3] <- as.numeric( results[,3] )

# fit model ignoring draws
mod1 <- btm( results , ignore.ties=TRUE , fix.eta = 0 , eps=0 )
summary(mod1)

# fit model with draws
mod2 <- btm( results , fix.eta = 0 , eps=0 )
summary(mod2)

## End(Not run)