The standard Poisson GLM models the (conditional) mean
\(\mathsf{E}[y] = \mu\) which is assumed to be equal to the
variance \(\mathsf{VAR}[y] = \mu\). `dispersiontest`

assesses the hypothesis that this assumption holds (equidispersion) against
the alternative that the variance is of the form:
$$\mathsf{VAR}[y] \quad = \quad \mu \; + \; \alpha \cdot \mathrm{trafo}(\mu).$$
Overdispersion corresponds to \(\alpha > 0\) and underdispersion to
\(\alpha < 0\). The coefficient \(\alpha\) can be estimated
by an auxiliary OLS regression and tested with the corresponding t (or z) statistic
which is asymptotically standard normal under the null hypothesis.

Common specifications of the transformation function \(\mathrm{trafo}\) are
\(\mathrm{trafo}(\mu) = \mu^2\) or \(\mathrm{trafo}(\mu) = \mu\).
The former corresponds to a negative binomial (NB) model with quadratic variance function
(called NB2 by Cameron and Trivedi, 2005), the latter to a NB model with linear variance
function (called NB1 by Cameron and Trivedi, 2005) or quasi-Poisson model with dispersion
parameter, i.e.,
$$\mathsf{VAR}[y] \quad = \quad (1 + \alpha) \cdot \mu = \mathrm{dispersion} \cdot \mu.$$
By default, for `trafo = NULL`

, the latter dispersion formulation is used in
`dispersiontest`

. Otherwise, if `trafo`

is specified, the test is formulated
in terms of the parameter \(\alpha\). The transformation `trafo`

can either
be specified as a function or an integer corresponding to the function `function(x) x^trafo`

,
such that `trafo = 1`

and `trafo = 2`

yield the linear and quadratic formulations
respectively.