BTm: 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) when abilities are modelled exactly, or by penalized quasi-likelihood when abilities are modelled by covariates.

Usage

BTm(
  outcome = 1,
  player1,
  player2,
  formula = NULL,
  id = "..",
  separate.ability = NULL,
  refcat = NULL,
  family = "binomial",
  data = NULL,
  weights = NULL,
  subset = NULL,
  na.action = NULL,
  start = NULL,
  etastart = NULL,
  mustart = NULL,
  offset = NULL,
  br = FALSE,
  model = TRUE,
  x = FALSE,
  contrasts = NULL,
  ...
)

Value

An object of class c("BTm", "x"), where "x" is the class of object returned by the model fitting function (e.g. glm). Components are as for objects of class "x", with additionally

id: the id argument.
separate.ability: the separate.ability argument.
refcat: the refcat argument.
player1: a data frame for the first player containing the ID factor and any player-specific contest-level variables.
player2: a data frame corresponding to that for player1.
assign: a numeric vector indicating which coefficients correspond to which terms in the model.
term.labels: labels for the model terms.
random: for models with random effects, the design matrix for the random effects.

Arguments

outcome: the binomial response: either a numeric vector, a factor in which the first level denotes failure and all others success, or a two-column matrix with the columns giving the numbers of successes and failures.
player1: either an ID factor specifying the first player in each contest, or a data.frame containing such a factor and possibly other contest-level variables that are specific to the first player. If given in a data.frame, the ID factor must have the name given in the id argument. If a factor is specified it will be used to create such a data.frame.
player2: an object corresponding to that given in player1 for the second player in each contest, with identical structure -- in particular factors must have identical levels.
formula: a formula with no left-hand-side, specifying the model for player ability. See details for more information.
id: the name of the ID factor.
separate.ability: (if formula does not include the ID factor as a separate term) a character vector giving the names of players whose abilities are to be modelled individually rather than using the specification given by formula.
refcat: (if formula includes the ID factor as a separate term) a character specifying which player to use as a reference, with the first level of the ID factor as the default. Overrides any other contrast specification for the ID factor.
family: a description of the error distribution and link function to be used in the model. Only the binomial family is implemented, with either"logit", "probit" , or "cauchit" link. (See stats::family() for details of family functions.)
data: an optional object providing data required by the model. This may be a single data frame of contest-level data or a list of data frames. Names of data frames are ignored unless they refer to data frames specified by player1 and player2. The rows of data frames that do not contain contest-level data must correspond to the levels of a factor used for indexing, i.e. row 1 corresponds to level 1, etc. Note any rownames are ignored. Objects are searched for first in the data object if provided, then in the environment of formula. If data is a list, the data frames are searched in the order given.
weights: an optional numeric vector of ‘prior weights’.
subset: an optional logical or numeric vector specifying a subset of observations to be used in the fitting process.
na.action: a function which indicates what should happen when any contest-level variables contain NAs. The default is the na.action setting of options. See details for the handling of missing values in other variables.
start: a vector of starting values for the fixed effects.
etastart: a vector of starting values for the linear predictor.
mustart: a vector of starting values for the vector of means.
offset: an optional offset term in the model. A vector of length equal to the number of contests.
br: logical. If TRUE fitting will be by penalized maximum likelihood as in Firth (1992, 1993), using brglm::brglm(), rather than maximum likelihood using glm(), when abilities are modelled exactly or when the abilities are modelled by covariates and the variance of the random effects is estimated as zero.
model: logical: whether or not to return the model frame.
x: logical: whether or not to return the design matrix for the fixed effects.
contrasts: an optional list specifying contrasts for the factors in formula. See the contrasts.arg of model.matrix().
...: other arguments for fitting function (currently either glm(), brglm::brglm(), or glmmPQL())

Author

Heather Turner, David Firth

Details

In each comparison to be modelled there is a 'first player' and a 'second player' and it is assumed that one player wins while the other loses (no allowance is made for tied comparisons).

The countsToBinomial() function is provided to convert a contingency table of wins into a data frame of wins and losses for each pair of players.

The formula argument specifies the model for player ability and applies to both the first player and the second player in each contest. If NULL a separate ability is estimated for each player, equivalent to setting formula = reformulate(id).

Contest-level variables can be specified in the formula in the usual manner, see formula(). Player covariates should be included as variables indexed by id, see examples. Thus player covariates must be ordered according to the levels of the ID factor.

If formula includes player covariates and there are players with missing values over these covariates, then a separate ability will be estimated for those players.

When player abilities are modelled by covariates, then random player effects should be added to the model. These should be specified in the formula using the vertical bar notation of lme4::lmer(), see examples.

When specified, it is assumed that random player effects arise from a \(N(0, \)\( \sigma^2)\) distribution and model parameters, including \(\sigma\), are estimated using PQL (Breslow and Clayton, 1993) as implemented in the glmmPQL() function.

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. Fahrmeir, L., Francis, B. J., Gilchrist, R. and Tutz, G., 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, 12(1), 1--12.

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

Turner, H. and Firth, D. (2012) Bradley-Terry models in R: The BradleyTerry2 package. Journal of Statistical Software, 48(9), 1--21.

Examples

Run this code


########################################################
##  Statistics journal citation data from Stigler (1994)
##  -- see also Agresti (2002, p448)
########################################################

##  Convert frequencies to success/failure data
citations.sf <- countsToBinomial(citations)
names(citations.sf)[1:2] <- c("journal1", "journal2")

##  First fit the "standard" Bradley-Terry model
citeModel <- BTm(cbind(win1, win2), journal1, journal2, data = citations.sf)
BTabilities(citeModel)

##  Now the same thing with a different "reference" journal
citeModel2 <- update(citeModel, refcat = "JASA")
BTabilities(citeModel2)

## Alternatively, use sum-to-zero contrasts for ID factor
citeModel3 <- update(citeModel, contrasts = list(".." = "contr.sum"))
BTabilities(citeModel3)

## Compute probabilities journal i (row) beats journal j (column)
## (same for all contrasts)
alpha <- exp(BTabilities(citeModel3)[,1])
alpha/outer(alpha, alpha, "+")

##################################################################
##  Now an example with an order effect -- see Agresti (2002) p438
##################################################################
data(baseball) # start with baseball data as provided by package

##  Simple Bradley-Terry model, ignoring home advantage:
baseballModel1 <- BTm(cbind(home.wins, away.wins), home.team, away.team,
                      data = baseball, id = "team")

##  Now incorporate the "home advantage" effect
baseball$home.team <- data.frame(team = baseball$home.team, at.home = 1)
baseball$away.team <- data.frame(team = baseball$away.team, at.home = 0)
baseballModel2 <- update(baseballModel1, formula = ~ team + at.home)

##  Compare the fit of these two models:
anova(baseballModel1, baseballModel2)

##
## For a more elaborate example with both player-level and contest-level
## predictor variables, see help(chameleons).
##

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