Extended Bradley-Terry Model

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 btm
summary(object, file=NULL, digits=4,...)

# S3 method for 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