Create a dispersion parameter initial object that specifies
initial and/or known values to use while estimating the dispersion parameter
with spglm() or spgautor().
Arguments
- family
The generalized linear model family describing the distribution of the response variable to be used.
"poisson","nbinomial","binomial","beta","Gamma", and"inverse.gaussian".- dispersion
The value of the dispersion parameter.
- known
A character vector indicating whether the dispersion parameter is to be assumed known. The value
"dispersion"or"given"is assumes the dispersion parameter is known.
Value
A list with two elements: initial and is_known.
initial is a named numeric vector indicating the dispersion parameters
with a specified initial and/or known value. is_known is a named
numeric vector indicating whether the dispersion parameters in
initial is known or not. The class of the list
matches the value given to the family argument.
Details
The dispersion_initial list is later passed to spglm() or spgautor().
The variance function of an individual \(y\) (given \(\mu\)) for each generalized linear model family is given below:
family: \(Var(y)\)
poisson: \(\mu \phi\)
nbinomial: \(\mu + \mu^2 / \phi\)
binomial: \(n \mu (1 - \mu) \phi\)
beta: \(\mu (1 - \mu) / (1 + \phi)\)
Gamma: \(\mu^2 / \phi\)
inverse.gaussian: \(\mu^2 / \phi\)
The parameter \(\phi\) is a dispersion parameter that influences \(Var(y)\).
For the poisson and binomial families, \(\phi\) is always
one. Note that this inverse Gaussian parameterization is different than a
standard inverse Gaussian parameterization, which has variance \(\mu^3 / \lambda\).
Setting \(\phi = \lambda / \mu\) yields our parameterization, which is
preferred for computational stability. Also note that the dispersion parameter
is often defined in the literature as \(V(\mu) \phi\), where \(V(\mu)\) is the variance
function of the mean. We do not use this parameterization, which is important
to recognize while interpreting dispersion parameter estimates using spglm() or spgautor().
For more on generalized linear model constructions, see McCullagh and
Nelder (1989).