R: Wedderburn Quasi-likelihood Family

wedderburn {gnm} | R Documentation

## Wedderburn Quasi-likelihood Family

### Description

Creates a `family` object for use with `glm`, `gnm`, etc., for the variance function [μ(1-μ)]^2 introduced by Wedderburn (1974) for response values in [0,1].

### Usage

    wedderburn(link = "logit")

### Arguments

`link` | The name of a link function. Allowed are "logit", "probit" and "cloglog".

### Value

An object of class `family`.

### Note

The reported deviance involves an arbitrary constant (see McCullagh and Nelder, 1989, p330); for estimating dispersion, use the Pearson chi-squared statistic instead.

### Author(s)

Modification of `binomial` by the R Core Team. Adapted for the Wedderburn quasi-likelihood family by David Firth.

### 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.

### See Also

`glm`, `gnm`, `family`

### Examples

    set.seed(1)
    ###  Use data from Wedderburn (1974), see ?barley

    ###  Fit Wedderburn's logit model with variance proportional to the
    ###  square of mu(1-mu)
    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(LETTERS[1:9], as.character(1:9), "X"))

    ###  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)

[Package gnm version 1.1-2 Index]