Jenkyn's Data on Leaf-blotch on Barley
Incidence of R. secalis on the leaves of ten varieties of barley grown at nine sites.
barley
A data frame with 90 observations on the following 3 variables.
the proportion of leaf affected (values in [0,1])
a factor with 9 levels A to I
a factor with 10 levels c(1:9, "X")
This dataset was used in Wedderburn's original paper (1974) on quasi-likelihood.
Originally in an unpublished Aberystwyth PhD thesis by J F Jenkyn.
Gabriel, K R (1998). Generalised bilinear regression. Biometrika 85, 689–700.
McCullagh, P and Nelder, J A (1989) Generalized Linear Models (2nd ed). Chapman and Hall.
Wedderburn, R W M (1974). Quasilikelihood functions, generalized linear models and the Gauss-Newton method. Biometrika 61, 439–47.
set.seed(1)
### Fit Wedderburn's logit model with variance proportional to [mu(1-mu)]^2
logitModel <- glm(y ~ site + variety, family = wedderburn, data = barley)
fit <- fitted(logitModel)
print(sum((barley$y - fit)^2 / (fit * (1-fit))^2))
## Agrees with the chi-squared value reported in McCullagh and Nelder
## (1989, p331), which differs slightly from Wedderburn's reported value.
### Fit the biplot model as in Gabriel (1998, p694)
biplotModel <- gnm(y ~ -1 + instances(Mult(site, variety), 2),
family = wedderburn, data = barley)
barleySVD <- svd(matrix(biplotModel$predictors, 10, 9))
A <- sweep(barleySVD$v, 2, sqrt(barleySVD$d), "*")[, 1:2]
B <- sweep(barleySVD$u, 2, sqrt(barleySVD$d), "*")[, 1:2]
## These are essentially A and B as in Gabriel (1998, p694), from which
## the biplot is made by
plot(rbind(A, B), pch = c(levels(barley$site), levels(barley$variety)))
## Fit the double-additive model as in Gabriel (1998, p697)
variety.binary <- factor(match(barley$variety, c(2,3,6), nomatch = 0) > 0,
labels = c("rest", "2,3,6"))
doubleAdditive <- gnm(y ~ variety + Mult(site, variety.binary),
family = wedderburn, data = barley)
## It is unclear why Gabriel's chi-squared statistics differ slightly
## from the ones produced in these fits. Possibly Gabriel adjusted the
## data somehow prior to fitting?Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.