Produces a generalized linear model family object with any power variance function and any power link. Includes the Gaussian, Poisson, gamma and inverse-Gaussian families as special cases.
tweedie(var.power=0, link.power=1-var.power)index of power variance function
index of power link function. link.power=0 produces a log-link. Defaults to the canonical link, which is 1-var.power.
A family object, which is a list of functions and expressions used by glm and gam in their iteratively reweighted least-squares algorithms.
See family and glm in the R base help for details.
This function provides access to a range of generalized linear model response distributions which are not otherwise provided by R, or any other package for that matter. It is also useful for accessing distribution/link combinations which are disallowed by the R glm function.
Let \(\mu_i = E(y_i)\) be the expectation of the \(i\)th response. We assume that $$\mu_i^q = x_i^Tb, var(y_i) = \phi \mu_i^p$$
where \(x_i\) is a vector of covariates and b is a vector of regression cofficients, for some \(\phi\), \(p\) and \(q\). This family is specified by var.power = p and link.power = q. A value of zero for \(q\) is interpreted as \(\log(\mu_i) = x_i^Tb\).
The variance power \(p\) characterizes the distribution of the responses \(y\). The following are some special cases:
| p | Response distribution |
| 0 | Normal |
| 1 | Poisson |
| (1, 2) | Compound Poisson, non-negative with mass at zero |
| 2 | Gamma |
| 3 | Inverse-Gaussian |
The name Tweedie has been associated with this family by Joergensen (1987) in honour of M. C. K. Tweedie.
Tweedie, M. C. K. (1984). An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. (Eds. J. K. Ghosh and J. Roy), pp. 579-604. Calcutta: Indian Statistical Institute.
Joergensen, B. (1987). Exponential dispersion models. J. R. Statist. Soc. B 49, 127-162.
Smyth, G. K. (1996). Regression modelling of quantity data with exact zeroes. Proceedings of the Second Australia-Japan Workshop on Stochastic Models in Engineering, Technology and Management. Technology Management Centre, University of Queensland, pp. 572-580.
Joergensen, B. (1997). Theory of Dispersion Models, Chapman and Hall, London.
Smyth, G. K., and Verbyla, A. P., (1999). Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10, 695-709.
# NOT RUN {
y <- rgamma(20,shape=5)
x <- 1:20
# Fit a poisson generalized linear model with identity link
glm(y~x,family=tweedie(var.power=1,link.power=1))
# Fit an inverse-Gaussion glm with log-link
glm(y~x,family=tweedie(var.power=3,link.power=0))
# }
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