## Specify a Generalised Davidson Term in a gnm Model Formula

### Description

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.

### Usage

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


### Arguments

 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.

### Details

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 \textrm{beats} j)_k) = \theta_{ijk} + \log(\mu\alpha_i

\log(p(draw)_k) = \theta_{ijk} + \delta + c +

 \sigma(\pi\log(\mu\alpha_i) - (1 - \pi)log(\alpha_j)) + 

 (1 - \sigma)(\log(\mu\alpha_i + \alpha_j))

\log(p(j \textrm{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 \textrm{expit}(\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_i, \sigma, \mu), the log is estimated, while for parameters that must be between zero and one (\delta, \pi), the logit is estimated, as illustrated in the example.

### Value

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

### References

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()

### Examples


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

### 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,
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)
## 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,
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,
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,
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,
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],
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)