BTm {BradleyTerry}R Documentation

Bradley-Terry model and extensions

Description

Fits Bradley-Terry models for pair comparison data, including models with structured scores, order effect and missing covariate data. Fits by either maximum likelihood or maximum penalized likelihood (with Jeffreys-prior penalty).

Usage

BTm(formula, refcat = NULL, offset = NULL, contrasts = NULL,
data = NULL, subset = NULL, br = FALSE, order.effect =
NULL, ...)

Arguments

formula A model formula for the Bradley-Terry “ability” parameters. Response variable should be a data frame containing contest results: columns include factors named "winner" (by default column 1) and "loser" (by default column 2), and possibly also "Freq", a numeric vector recording the frequency of each contest result (taken to be 1 if omitted). Variables on RHS have length equal to the number of players. The special RHS formula .. specifies the standard Bradley-Terry model with unstructured abilities.
refcat Character. Which is the “reference” player? Only used with .. on the RHS of the formula (otherwise ignored if supplied). Default is the name of the alphanumerically-first player.
offset An optional offset term in the model. A vector of length equal to the number of players.
contrasts As for glm.
data A data frame, in which RHS variables can be found.
subset An optional logical or numeric vector specifying a subset of observations (ie, a subset of rows of the response dataframe) to be used in the fitting process.
br Logical. If TRUE, fitting will be by penalized maximum likelihood as in Firth (1992, 1993), using brglm. Default is fitting by maximum likelihood.
order.effect An optional vector, numeric, indicating an order effect to be estimated in the model (for example, a home advantage effect). Values should be 1 where contest winner has the advantage, -1 where loser has the advantage, and 0 where neither player is advantaged.
... Other arguments for glm or brglm

Details

No allowance is made for tied contests.

Aside from the possibility of an order effect, contest-specific predictors are not catered for by BTm. However, the availability of the model and x components in objects of class BTm allows a model fitted with only player-specific predictors to be manipulated subsequently to include further terms involving contest-specific predictors.

Value

An object of class c("BTm", "glm", "lm"), or of class c("BTm", "brglm", "glm", "lm") if br = TRUE. Components are as for glm or brglm, with additionally

x0 Model matrix for the formula as supplied (rather than the model matrix actually used in the subsidiary call to glm or brglm, which is included as component x if the call includes x = TRUE). One row for each player.
offset0 The supplied offset vector, if any was specified. One element for each player. (The offset vector actually used in the subsidiary call to glm or brglm is included as component offset.)
y0 The data frame of contest winners and losers, containing only those rows actually used in fitting the model.
order.effect The values of order.effect, if specified.
abilities A two-column matrix of estimated abilities, with the ability for refcat set to zero if refcat is specified (otherwise the first player has zero ability). First column is estimated ability, second column is the standard error for that estimate. One row for each player.

Note

Methods specific to the BTm class of models are

Others are inherited from glm or lm.

Author(s)

David Firth

References

Agresti, A (2002) Categorical Data Analysis (2nd ed). New York: Wiley.

Firth, D. (1992) Bias reduction, the Jeffreys prior and GLIM. In Advances in GLIM and Statistical Modelling, Eds. L Fahrmeir, B J Francis, R Gilchrist and G Tutz, pp91–100. New York: Springer.

Firth, D. (1993) Bias reduction of maximum likelihood estimates. Biometrika 80, 27–38.

Firth, D. (2005) Bradley-Terry models in R. Journal of Statistical Software, to appear.

Stigler, S. (1994) Citation patterns in the journals of statistics and probability. Statistical Science 9, 94–108.

See Also

BTresiduals, BTabilities

Examples

##
##  Statistics journal citation data from Stigler (1994)
##  -- see also Agresti (2002, p448)
data(citations)
                 
##  First fit the "standard" Bradley-Terry model
print(citeModel <- BTm(citations ~ ..))

##  Now the same thing with a different "reference" journal
update(citeModel, . ~ ., refcat = "JASA")

##  Is the "citeability" of a journal predicted by its country of origin?
origin <- factor(c("UK", "USA", "USA", "UK"))
print(citeModel2 <- BTm(citations ~ origin))

##  Hmm... not so sure about the origin of "Comm Statist" ...
is.na(origin[2]) <- TRUE
citeModel2 <- update(citeModel2, . ~ .)

##  Now an example with an order effect -- see Agresti (2002) p438
data(baseball)

##  Simple Bradley-Terry model as in Agresti p437
print(baseballModel <- BTm(baseball ~ ..))

##  Introduce order effect as in Agresti p438
update(baseballModel, order.effect = baseball$home.adv)   


[Package BradleyTerry version 0.8-7 Index]