# Bradley Terry Model With Ties Fits a Bradley Terry model with ties (intercept-only model) by maximum likelihood estimation. ```r bratt(refgp = "last", refvalue = 1, ialpha = 1, i0 = 0.01) ``` ## Arguments - `refgp`: Integer whose value must be from the set {1, ,$M$}, where there are $M$ competitors. The default value indicates the last competitor is used---but don't input a character string, in general. - `refvalue`: Numeric. A positive value for the reference group. - `ialpha`: Initial values for the $alpha$s. These are recycled to the appropriate length. - `i0`: Initial value for $alpha_0$. If convergence fails, try another positive value. ## Details There are several models that extend the ordinary Bradley Terry model to handle ties. This family function implements one of these models. It involves $M$ competitors who either win or lose or tie against each other. (If there are no draws/ties then use `brat`). The probability that Competitor $i$ beats Competitor $j$ is c("$alpha_i / (alpha_i +\n$", "$ alpha_j + alpha_0)$"), where all the $alpha$s are positive. The probability that Competitor $i$ ties with Competitor $j$ is c("$alpha_0 / (alpha_i +\n$", "$ alpha_j + alpha_0)$"). Loosely, the $alpha$s can be thought of as the competitors' `abilities', and $alpha_0$ is an added parameter to model ties. For identifiability, one of the $alpha_i$ is set to a known value `refvalue`, e.g., 1. By default, this function chooses the last competitor to have this reference value. The data can be represented in the form of a $M$ by $M$ matrix of counts, where winners are the rows and losers are the columns. However, this is not the way the data should be inputted (see below). Excluding the reference value/group, this function chooses $log(alpha_j)$ as the first $M-1$ linear predictors. The log link ensures that the $alpha$s are positive. The last linear predictor is $log(alpha_0)$. The Bradley Terry model can be fitted with covariates, e.g., a home advantage variable, but unfortunately, this lies outside the VGLM theoretical framework and therefore cannot be handled with this code. ## Returns An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`. ## References Torsney, B. (2004). Fitting Bradley Terry models using a multiplicative algorithm. In: Antoch, J. (ed.) **Proceedings in Computational Statistics COMPSTAT 2004**, Physica-Verlag: Heidelberg. Pages 513--526. ## Author(s) T. W. Yee ## Note The function `Brat` is useful for coercing a $M$ by $M$ matrix of counts into a one-row matrix suitable for `bratt`. Diagonal elements are skipped, and the usual S order of `c(a.matrix)` of elements is used. There should be no missing values apart from the diagonal elements of the square matrix. The matrix should have winners as the rows, and losers as the columns. In general, the response should be a matrix with $M(M-1)$ columns. Also, a symmetric matrix of ties should be passed into `Brat`. The diagonal of this matrix should be all `NA`s. Only an intercept model is recommended with `bratt`. It doesn't make sense really to include covariates because of the limited VGLM framework. Notationally, note that the `VGAM` family function `brat` has $M+1$ contestants, while `bratt` has $M$ contestants. ## See Also `brat`, `Brat`, `binomialff`. ## Examples ```r # citation statistics: being cited is a 'win'; citing is a 'loss' journal <- c("Biometrika", "Comm.Statist", "JASA", "JRSS-B") mat <- matrix(c( NA, 33, 320, 284, 730, NA, 813, 276, 498, 68, NA, 325, 221, 17, 142, NA), 4, 4) dimnames(mat) <- list(winner = journal, loser = journal) # Add some ties. This is fictitional data. ties <- 5 + 0 * mat ties[2, 1] <- ties[1,2] <- 9 # Now fit the model fit <- vglm(Brat(mat, ties) ~ 1, bratt(refgp = 1), trace = TRUE, crit = "coef") summary(fit) c(0, coef(fit)) # Log-abilities (last is log(alpha0)) c(1, Coef(fit)) # Abilities (last is alpha0) fit@misc$alpha # alpha_1,...,alpha_M fit@misc$alpha0 # alpha_0 fitted(fit) # Probabilities of winning and tying, in awkward form predict(fit) (check <- InverseBrat(fitted(fit))) # Probabilities of winning qprob <- attr(fitted(fit), "probtie") # Probabilities of a tie qprobmat <- InverseBrat(c(qprob), NCo = nrow(ties)) # Pr(tie) check + t(check) + qprobmat # Should be 1s in the off-diagonals ```

bratt function