barley: Jenkyn's Data on Leaf-blotch on Barley

In gnm: Generalized Nonlinear Models

Jenkyn's Data on Leaf-blotch on Barley

Description

Incidence of R. secalis on the leaves of ten varieties of barley grown at nine sites.

Usage

barley

Format

A data frame with 90 observations on the following 3 variables.

y

the proportion of leaf affected (values in [0,1])

site

a factor with 9 levels A to I

variety

a factor with 10 levels c(1:9, "X")

Note

This dataset was used in Wedderburn's original paper (1974) on quasi-likelihood.

Source

Originally in an unpublished Aberystwyth PhD thesis by J F Jenkyn.

References

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.

Examples

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?

gnm documentation built on Sept. 16, 2023, 5:06 p.m.