Bradley Terry Model
Fits a Bradley Terry model (intercept-only model) by maximum likelihood estimation.
brat(refgp = "last", refvalue = 1, ialpha = 1)
refgp |
Integer whose value must be from the set {1,...,M+1}, where there are M+1 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. |
The Bradley Terry model involves M+1 competitors
who either win or lose against each other (no draws/ties
allowed in this implementation–see bratt
if there are ties). The probability that Competitor
i beats Competitor j is alpha_i / (alpha_i + alpha_j),
where all the alphas are positive.
Loosely, the alphas can be thought of as
the competitors' ‘abilities’. 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+1
by M+1 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 M linear predictors. The log link ensures that the alphas are positive.
The Bradley Terry model can be fitted by logistic regression, but this approach is not taken here. 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.
Presently, the residuals are wrong, and the prior weights
are not handled correctly. Ideally, the total number of
counts should be the prior weights, after the response has
been converted to proportions. This would make it similar
to family functions such as multinomial
and binomialff.
The function Brat is useful for coercing
a M+1 by M+1 matrix of counts into a one-row
matrix suitable for brat. 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
1-row matrix with M(M+1) columns.
Only an intercept model is recommended with brat.
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
Agresti, A. (2013). Categorical Data Analysis, 3rd ed. Hoboken, NJ, USA: Wiley.
Stigler, S. (1994). Citation patterns in the journals of statistics and probability. Statistical Science, 9, 94–108.
The BradleyTerry2 package has more comprehensive capabilities than this function.
# 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)
fit <- vglm(Brat(mat) ~ 1, brat(refgp = 1), trace = TRUE)
fit <- vglm(Brat(mat) ~ 1, brat(refgp = 1), trace = TRUE, crit = "coef")
summary(fit)
c(0, coef(fit)) # Log-abilities (in order of "journal")
c(1, Coef(fit)) # Abilities (in order of "journal")
fitted(fit) # Probabilities of winning in awkward form
(check <- InverseBrat(fitted(fit))) # Probabilities of winning
check + t(check) # Should be 1's in the off-diagonalsPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.