Offsetting the Tweedie by sampling effort: does this only affect the Poisson component?

[gerverska]

Hi folks,

This is a pretty broad GLM question, but I revisited Foster and Bravington (2013), and I began to wonder about about the Tweedie / compound Poisson-gamma and how it relates to gllvm.

I'm mostly focused on this quote:

Without covariates there is no difference in the distribution defined by the Poisson sum-of-gammas formulation, (λi ,α,βi ), and that defined by standard Tweedie formulation (μi ,φ, p). However, differences do arise when covariates are allowed to affect the parameters. We model the expectations E (Ni ) and E (wij ) separately, rather than the marginal expectation E (yi ) that is the focus of the Tweedie GLM.

They go on to say:

We assume that the offset (e.g. duration of sample) affects the expected number of summands, rather than the expected weight of those summands.

Given the above, I have the following question: can we make the same assumption about gllvm Tweedie models offset by sampling effort?

Lecomte et al. (2013) also note:

...when no covariates are included in either the Poisson or gamma latent components, the CPG model belongs to the Tweedie family, and, in addition, a reviewer has noted that this is still the case when the set of covariates is identical in each of the Poisson and gamma components.

My understanding is that gllvm doesn't allow the Poisson and gamma components to include different covariates (i.e., it uses the standard Tweedie formulation mentioned above), so it's not entirely apparent to me that a sampling effort offset would only affect the Poisson process for gllvm.

Thanks in advance!