Bradley Terry Model With Ties
Fits a Bradley Terry model with ties (intercept-only model) by maximum likelihood estimation.
bratt(refgp = "last", refvalue = 1, ialpha = 1, i0 = 0.01)
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 alphas. These are recycled to the appropriate length. |
i0 |
Initial value for alpha_0. If convergence fails, try another positive value. |
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 alpha_i / (alpha_i +
alpha_j + alpha_0), where all the alphas
are positive. The probability that Competitor i
ties with Competitor j is alpha_0 / (alpha_i +
alpha_j + alpha_0). Loosely, the alphas
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 alphas 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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm.
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 NAs.
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.
T. W. Yee
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.
brat,
Brat,
binomialff.
# 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)
fit <- vglm(Brat(mat, ties) ~ 1, bratt(refgp = 1), trace = TRUE, crit = "coef")
summary(fit)
c(0, coef(fit)) # Log-abilities (in order of "journal"); last is log(alpha0)
c(1, Coef(fit)) # Abilities (in order of "journal"); 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)) # Probabilities of a tie
check + t(check) + qprobmat # Should be 1s in the off-diagonalsPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.