GenDavidson: Specify a Generalised Davidson Term in a gnm Model Formula
In BradleyTerry2: Bradley-Terry Models

Description Usage Arguments Details Value Author(s) References See Also Examples

GenDavidson is a function of class "nonlin" to specify a generalised Davidson term in the formula argument to gnm::gnm(), providing a model for paired comparison data where ties are a possible outcome.

GenDavidson(
  win,
  tie,
  loss,
  player1,
  player2,
  home.adv = NULL,
  tie.max = ~1,
  tie.mode = NULL,
  tie.scale = NULL,
  at.home1 = NULL,
  at.home2 = NULL
)

`win`	a logical vector: `TRUE` if player1 wins, `FALSE` otherwise.
`tie`	a logical vector: `TRUE` if the outcome is a tie, `FALSE` otherwise.
`loss`	a logical vector: `TRUE` if player1 loses, `FALSE` otherwise.
`player1`	an ID factor specifying the first player in each contest, with the same set of levels as `player2`.
`player2`	an ID factor specifying the second player in each contest, with the same set of levels as `player2`.
`home.adv`	a formula for the parameter corresponding to the home advantage effect. If `NULL`, no home advantage effect is estimated.
`tie.max`	a formula for the parameter corresponding to the maximum tie probability.
`tie.mode`	a formula for the parameter corresponding to the location of maximum tie probability, in terms of the probability that `player1` wins, given the outcome is not a draw.
`tie.scale`	a formula for the parameter corresponding to the scale of dependence of the tie probability on the probability that `player1` wins, given the outcome is not a draw.
`at.home1`	a logical vector: `TRUE` if `player1` is at home, `FALSE` otherwise.
`at.home2`	a logical vector: `TRUE` if `player2` is at home, `FALSE` otherwise.

GenDavidson specifies a generalisation of the Davidson model (1970) for paired comparisons where a tie is a possible outcome. It is designed for modelling trinomial counts corresponding to the win/draw/loss outcome for each contest, which are assumed Poisson conditional on the total count for each match. Since this total must be one, the expected counts are equivalently the probabilities for each possible outcome, which are modelled on the log scale:

log(p(i beats j)_k) = theta_{ijk} + log(mu * alpha_i)

log(p(draw)_k) = theta_{ijk} + log(delta) + c + sigma * (pi * log(mu * alpha_i) + (1 - pi) * log(alpha_j)) + (1 - sigma) * log(mu * alpha_i + alpha_j)

log(p(j beats i)_k) = theta_{ijk} + log(alpha_j)

Here theta_{ijk} is a structural parameter to fix the trinomial totals; mu is the home advantage parameter; alpha_i and alpha_j are the abilities of players i and j respectively; c is a function of the parameters such that plogis(delta) is the maximum probability of a tie, sigma scales the dependence of the probability of a tie on the relative abilities and pi allows for asymmetry in this dependence.

For parameters that must be positive (alpha, sigma, mu), the log is estimated, while for parameters that must be between zero and one (δ, π), the logit is estimated, as illustrated in the example.

A list with the anticipated components of a "nonlin" function:

`predictors`	the formulae for the different parameters and the ID factors for player 1 and player 2.
`variables`	the outcome variables and the “at home” variables, if specified.
`common`	an index to specify that common effects are to be estimated for the players.
`term`	a function to create a deparsed mathematical expression of the term, given labels for the predictors.
`start`	a function to generate starting values for the parameters.

Heather Turner

Davidson, R. R. (1970). On extending the Bradley-Terry model to accommodate ties in paired comparison experiments. Journal of the American Statistical Association, 65, 317–328.

football(), plotProportions()

### example requires gnm
if (require(gnm)) {
    ### convert to trinomial counts
    football.tri <- expandCategorical(football, "result", idvar = "match")
    head(football.tri)

    ### add variable to indicate whether team playing at home
    football.tri$at.home <- !logical(nrow(football.tri))

    ### fit shifted & scaled Davidson model
    ###  - subset to first and last season for illustration
    shifScalDav <- gnm(count ~
        GenDavidson(result == 1, result == 0, result == -1,
                    home:season, away:season, home.adv = ~1,
                    tie.max = ~1, tie.scale = ~1, tie.mode = ~1,
                    at.home1 = at.home,
                    at.home2 = !at.home) - 1,
        eliminate = match, family = poisson, data = football.tri,
        subset = season %in% c("2008-9", "2012-13"))

    ### look at coefs
    coef <- coef(shifScalDav)
    ## home advantage
    exp(coef["home.adv"])
    ## max p(tie)
    plogis(coef["tie.max"])
    ## mode p(tie)
    plogis(coef["tie.mode"])
    ## scale relative to Davidson of dependence of p(tie) on p(win|not a draw)
    exp(coef["tie.scale"])

    ### check model fit
    alpha <- names(coef[-(1:4)])
    plotProportions(result == 1, result == 0, result == -1,
                    home:season, away:season,
                    abilities = coef[alpha], home.adv = coef["home.adv"],
                    tie.max = coef["tie.max"], tie.scale = coef["tie.scale"],
                    tie.mode = coef["tie.mode"],
                    at.home1 = at.home, at.home2 = !at.home,
                    data = football.tri, subset = count == 1)
}

### analyse all five seasons
### - takes a little while to run, particularly likelihood ratio tests
## Not run: 
### fit Davidson model
Dav <- gnm(count ~ GenDavidson(result == 1, result == 0, result == -1,
                               home:season, away:season, home.adv = ~1,
                               tie.max = ~1,
                               at.home1 = at.home,
                               at.home2 = !at.home) - 1,
           eliminate = match, family = poisson, data = football.tri)

### fit scaled Davidson model
scalDav <- gnm(count ~ GenDavidson(result == 1, result == 0, result == -1,
                                  home:season, away:season, home.adv = ~1,
                                  tie.max = ~1, tie.scale = ~1,
                                  at.home1 = at.home,
                                  at.home2 = !at.home) - 1,
               eliminate = match, family = poisson, data = football.tri)

### fit shifted & scaled Davidson model
shifScalDav <- gnm(count ~
    GenDavidson(result == 1, result == 0, result == -1,
                home:season, away:season, home.adv = ~1,
                tie.max = ~1, tie.scale = ~1, tie.mode = ~1,
                at.home1 = at.home,
                at.home2 = !at.home) - 1,
    eliminate = match, family = poisson, data = football.tri)

### compare models
anova(Dav, scalDav, shifScalDav, test = "Chisq")

### diagnostic plots
main <- c("Davidson", "Scaled Davidson", "Shifted & Scaled Davidson")
mod <- list(Dav, scalDav, shifScalDav)
names(mod) <- main

## use football.tri data so that at.home can be found,
## but restrict to actual match results
par(mfrow = c(2,2))
for (i in 1:3) {
    coef <- parameters(mod[[i]])
    plotProportions(result == 1, result == 0, result == -1,
                    home:season, away:season,
                    abilities = coef[alpha],
                    home.adv = coef["home.adv"],
                    tie.max = coef["tie.max"],
                    tie.scale = coef["tie.scale"],
                    tie.mode = coef["tie.mode"],
                    at.home1 = at.home,
                    at.home2 = !at.home,
                    main = main[i],
                    data = football.tri, subset = count == 1)
}

## End(Not run)