brat {VGAM} | R Documentation |
Fits a Bradley Terry model (intercept-only model) by maximum likelihood estimation.
brat(refgp = "last", refvalue = 1, init.alpha = 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. |
init.alpha |
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. (2002) Categorical Data Analysis, 2nd ed. New York: 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.
bratt
,
Brat
,
multinomial
,
binomialff
.
# citation statistics: being cited is a 'win'; citing is a 'loss' journal = c("Biometrika", "Comm.Statist", "JASA", "JRSS-B") m = matrix(c( NA, 33, 320, 284, 730, NA, 813, 276, 498, 68, NA, 325, 221, 17, 142, NA), 4,4) dimnames(m) = list(winner = journal, loser = journal) fit = vglm(Brat(m) ~ 1, brat(refgp=1), trace=TRUE) fit = vglm(Brat(m) ~ 1, brat(refgp=1), trace=TRUE, cri="c") 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-diagonals