R: Extended Bradley-Terry Model

btm {sirt}

R Documentation

Extended Bradley-Terry Model

Description

The function btm estimates an extended Bradley-Terry model (Hunter, 2004; see Details). Parameter estimation uses a bias corrected joint maximum likelihood estimation method based on \varepsilon-adjustment (see Bertoli-Barsotti, Lando & Punzo, 2014). See Details for the algorithm.

The function btm_sim simulated data from the extended Bradley-Terry model.

Usage

btm(data, judge=NULL, ignore.ties=FALSE, fix.eta=NULL, fix.delta=NULL, fix.theta=NULL,
       maxiter=100, conv=1e-04, eps=0.3, wgt.ties=.5)

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

## S3 method for class 'btm'
predict(object, data=NULL, ...)

btm_sim(theta, eta=0, delta=-99, repeated=FALSE)

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).
`judge`	Optional vector of judge identifiers (if multiple judges are available)
`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, Lando & Punzo, 2014) which reduces bias in ability estimates. In case of `\varepsilon=0`, persons with extreme scores are removed from the pairwise comparison.
`wgt.ties`	Weighting parameter for ties, see formula in Details. The default is .5
`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.
`theta`	Vector of abilities
`eta`	Value of `\eta` parameter
`delta`	Value of `\delta` parameter
`repeated`	Logical indicating whether repeated ratings of dyads (for home advantage effect) should be simulated

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 + w_T ( \eta + \theta_i +\theta_j ) )

The parameters \theta_i denote the abilities, \delta is the tendency of the occurrence of ties and \eta is the home-advantage effect. The weighting parameter w_T governs the importance of ties and can be chosen in the argument wgt.ties.

A joint maximum likelihood (JML) estimation is applied for simulataneous estimation of \eta, \delta and all \theta_i parameters. In the Rasch model, it was shown that JML can result in biased parameter estimates. The \varepsilon-adjustment approach has been proposed to reduce the bias in parameter estimates (Bertoli-Bersotti, Lando & Punzo, 2014). This estimation approach is adapted to the Bradley-Terry model in the btm function. To this end, the likelihood function is modified for the purpose of bias reduction. It can be easily shown that there exist sufficient statistics for \eta, \delta and all \theta_i parameters. In the \varepsilon-adjustment approach, the sufficient statistic for the \theta_i parameter is modified. In JML estimation of the Bradley-Terry model, S_i=\sum_{j \ne i} ( x_{ij} + x_{ji} ) is a sufficient statistic for \theta_i. Let M_i the maximum score for person i which is the number of x_{ij} terms appearing in S_i. In the \varepsilon-adjustment approach, the sufficient statistic S_i is modified to

S_{i, \varepsilon}=\varepsilon + \frac{M_i - 2 \varepsilon}{M_i} S_i

and S_{i, \varepsilon} instead of S_{i} is used in JML estimation. Hence, original scores S_i are linearly transformed for all persons i.

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
`fit_judges`	Fit statistics (outfit and infit) for judges if `judge` is provided. In addition, average agreement of the rating with the mode of the ratings is calculated for each judge (at least three ratings per dyad has to be available for computing the agreement).
`residuals`	Unstandardized and standardized residuals for each observation

References

Bertoli-Barsotti, L., Lando, T., & Punzo, A. (2014). Estimating a Rasch Model via fuzzy empirical probability functions. In D. Vicari, A. Okada, G. Ragozini & C. Weihs (Eds.). Analysis and Modeling of Complex Data in Behavioral and Social Sciences. Springer. doi:10.1007/978-3-319-06692-9_4

Hunter, D. R. (2004). MM algorithms for generalized Bradley-Terry models. Annals of Statistics, 32, 384-406. doi: 10.1214/aos/1079120141

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 <- sirt::btm( dat)
summary(mod1)

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

#*** Model 3: Estimation with ties, no epsilon adjustment, weight for ties of .333 which
#    corresponds to the rule of 3 points for a victory and 1 point of a draw in football
mod3 <- sirt::btm( dat, eps=0, wgt.ties=1/3)
summary(mod3)

#*** Model 4: Some fixed abilities
fix.theta <- c("Anhalt Dessau"=-1 )
mod4 <- sirt::btm( dat, eps=0, fix.theta=fix.theta)
summary(mod4)

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

#*** Model 6: Ignoring ties, no home advantage effect (JML approach and eps=0)
mod6 <- sirt::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 <- sirt::btm( results, ignore.ties=TRUE, fix.eta=0, eps=0 )
summary(mod1)

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

#############################################################################
# EXAMPLE 3: Simulated data from the Bradley-Terry model
#############################################################################

set.seed(9098)
N <- 22
theta <- seq(2,-2, len=N)

#** simulate and estimate data without repeated dyads
dat1 <- sirt::btm_sim(theta=theta)
mod1 <- sirt::btm( dat1, ignore.ties=TRUE, fix.delta=-99, fix.eta=0)
summary(mod1)

#*** simulate data with home advantage effect and ties
dat2 <- sirt::btm_sim(theta=theta, eta=.8, delta=-.6, repeated=TRUE)
mod2 <- sirt::btm(dat2)
summary(mod2)

#############################################################################
# EXAMPLE 4: Estimating the Bradley-Terry model with multiple judges
#############################################################################

#*** simulating data with multiple judges
set.seed(987)
N <- 26  # number of objects to be rated
theta <- seq(2,-2, len=N)
s1 <- stats::sd(theta)
dat <- NULL
# judge discriminations which define tendency to provide reliable ratings
discrim <- c( rep(.9,10), rep(.5,2), rep(0,2) )
#=> last four raters provide less reliable ratings

RR <- length(discrim)
for (rr in 1:RR){
    theta1 <- discrim[rr]*theta + stats::rnorm(N, mean=0, sd=s1*sqrt(1-discrim[rr]))
    dat1 <- sirt::btm_sim(theta1)
    dat1$judge <- rr
    dat <- rbind(dat, dat1)
}

#** estimate the Bradley-Terry model and compute judge-specific fit statistics
mod <- sirt::btm( dat[,1:3], judge=paste0("J",100+dat[,4]), fix.eta=0, ignore.ties=TRUE)
summary(mod)

## End(Not run)

[Package sirt version 4.1-15 Index]