GenDavidson {BradleyTerry2} | R Documentation |
GenDavidson is a function of class "nonlin"
to specify a
generalised Davidson term in the formula argument to
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: |
tie |
a logical vector: |
loss |
a logical vector: |
player1 |
an ID factor specifying the first player in
each contest, with the same set of levels as |
player2 |
an ID factor specifying the second player in
each contest, with the same set of levels as |
home.adv |
a formula for the paramter corresponding to the
home advantage effect. If |
tie.max |
a formula for the parameter corresponding to the maximum tie probability. |
tie.scale |
a formula for the parameter corresponding to the
scale of dependence of the tie probability on the probability
that |
tie.mode |
a formula for the parameter corresponding to the
location of maximum tie probability, in terms of the probability
that |
at.home1 |
a logical vector: |
at.home2 |
a logical vector: |
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.
### 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)