bratt {VGAM} | R Documentation |
Fits a Bradley Terry model with ties (intercept-only model) by maximum likelihood estimation.
bratt(refgp = "last", refvalue = 1, init.alpha = 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. |
init.alpha |
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
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.
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") 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) # Add some ties. This is fictitional data. ties = 5 + 0*m ties[2,1] = ties[1,2] = 9 # Now fit the model fit = vglm(Brat(m, ties) ~ 1, bratt(refgp=1), trace=TRUE) fit = vglm(Brat(m, ties) ~ 1, bratt(refgp=1), trace=TRUE, cri="c") 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 1's in the off-diagonals