sirt (version 1.9-0)

btm: 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

ignore.ties

Logical indicating whether ties should be ignored.

fix.eta

Numeric value for a fixed $\eta$ value

fix.delta

Numeric value for a fixed $\delta$ 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.

Value

• List with following entries
• pars
  Parameter summary for $\eta$ and $\delta$
• effects
  Parameter estimates for $\theta$ and outfit and infit statistics
• summary.effects
  Summary of $\theta$ parameter estimates
• mle.rel
  MLE reliability, also known as separation reliability
• sepG
  Separation index $G$
• probs
  Estimated probabilities
• data
  Used dataset with integer identifiers

Details

The extended Bradley-Terry model for the comparison of individuals $i$ and $j$ is defined as $$P(X_{ij} = 1 ) \propto \exp( \eta + \theta_i )$$ $$P(X_{ij} = 0 ) \propto \exp( \theta_j )$$ $$P(X_{ij} = 0.5) \propto \exp( \delta + ( \eta + \theta_i +\theta_j )/2 )$$ The parameters $\theta_i$ denote the abilities, $\delta$ is the tendency of the occurence of ties and $\eta$ is the home-advantage effect.

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.

See Also

See also the Rpackages BradleyTerry2, psychotools, psychomix and prefmod.

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)

#********************
# 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)