Nothing

```
options(digits = 4) ## only applies to this file
library(BradleyTerry2)
data(flatlizards, package = "BradleyTerry2")
##
## Fit the standard Bradley-Terry model, using the bias-reduced
## maximum likelihood method:
##
result <- rep(1, nrow(flatlizards$contests))
BTmodel <- BTm(result, winner, loser, br = TRUE, data = flatlizards$contests)
summary(BTmodel)
##
## That's fairly useless, though, because of the rather small
## amount of data on each lizard. And really the scientific
## interest is not in the abilities of these particular 77
## lizards, but in the relationship between ability and the
## measured predictor variables.
##
## So next fit (by maximum likelihood) a "structured" B-T model in
## which abilities are determined by a linear predictor.
##
## This reproduces results reported in Table 1 of Whiting et al. (2006):
##
Whiting.model <- BTm(result, winner, loser, ~ throat.PC1[..] + throat.PC3[..] +
head.length[..] + SVL[..], family = binomial,
data = flatlizards)
summary(Whiting.model)
##
## Equivalently, fit the same model using glmmPQL:
##
Whiting.model <- BTm(result, winner, loser, ~ throat.PC1[..] + throat.PC3[..] +
head.length[..] + SVL[..] + (1|..), sigma = 0,
sigma.fixed = TRUE, data = flatlizards)
summary(Whiting.model)
##
## But that analysis assumes that the linear predictor formula for
## abilities is _perfect_, i.e., that there is no error in the linear
## predictor. This will always be unrealistic.
##
## So now fit the same predictor but with a normally distributed error
## term --- a generalized linear mixed model --- by using the BTm
## function instead of glm.
##
Whiting.model2 <- BTm(result, winner, loser, ~ throat.PC1[..] + throat.PC3[..] +
head.length[..] + SVL[..] + (1|..),
data = flatlizards, trace = TRUE)
summary(Whiting.model2)
##
## The estimated coefficients (of throat.PC1, throat.PC3,
## head.length and SVL are not changed substantially by
## the recognition of an error term in the model; but the estimated
## standard errors are larger, as expected. The main conclusions from
## Whiting et al. (2006) are unaffected.
##
## With the normally distributed random error included, it is perhaps
## at least as natural to use probit rather than logit as the link
## function:
##
Whiting.model3 <- BTm(result, winner, loser, ~ throat.PC1[..] + throat.PC3[..] +
head.length[..] + SVL[..] + (1|..),
family = binomial(link = "probit"),
data = flatlizards, trace = TRUE)
summary(Whiting.model3)
BTabilities(Whiting.model3)
residuals(Whiting.model3, "grouped")
## Note the "separate" attribute here, identifying two lizards with
## missing values of at least one predictor variable
##
## Modulo the usual scale change between logit and probit, the results
## are (as expected) very similar to Whiting.model2.
```

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