Table 1.

The observation-level variance for the three distributional families: quasi-Poisson, negative binomial and gamma with the three different methods for deriving : the delta method, lognormal approximation and the trigamma function, . when x follows gamma distribution. In the R environment, the function, trigamma can be used to obtain ; also note that ν is known as a shape parameter while κ is as a rate parameter in gamma distribution.

family	distributional parameters	mean (E[y]) variance (var[y])	link function	delta method	lognormal approximation	trigamma function
quasi-Poisson (QP)			log
Poisson (when ω = 1)	λ > 0 ω > 0		square-root	0.25ω	—
negative binomial (NB)			log
	λ > 0 θ > 0		square-root		—
gamma			log
	λ > 0 ν > 0		inverse (reciprocal)		—
gamma (alternative parameterization)			log
	ν > 0 κ > 0		inverse (reciprocal)		—