brat {VGAM}

Bradley Terry Model

Description

Fits a Bradley Terry model (intercept-only model) by maximum likelihood estimation.

Usage

brat(refgp = "last", refvalue = 1, ialpha = 1)

Arguments

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.

Details

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.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm.

Warning

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.

Note

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.

Author(s)

T. W. Yee

References

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.

See Also

bratt, Brat, multinomial, binomialff.

Examples

# 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-diagonals