# Zero-inflation GLMMs: On the use of different sets of explanatory variables in main and ZI formulas

my questions are general in nature so I won't provide any data.

For reference:
I am using the package glmmTMB in R so if my terminology is weird it is because it is a mix of this and other sources I've been using to try and answer this myself. I am using GLMMs to model abundance data in relation to physical attributes of the environment. As is expected the count data is massively zero inflated. I am using zero inflation models, not hurdle-models, as the zeros are likely a mixture of real zero observations and some false zeros due to methodological issues.

The Situation
I am currently using either all of the explanatory variables from the main model:

glmmTMB(formula = RV ~ EV1 + EV2 + EV3, ziformula = ~., ...)


or a single zero inflation parameter applied to all observations:

glmmTMB(formula = RV ~ EV1 + EV2 + EV3, ziformula = ~1, ...)


Some of the variables that are not significant in the main conditional model are significant in the zero inflation conditional model.

My Questions:
If I want to reduce the number of explanatory variables down via a drop-1 process (using AIC), does it make sense to have different sets of explanatory variables in the main and zero inflation model specifications? For example:

glmmTMB(formula = RV ~ EV1 + EV2, ziformula = EV1 + EV3, ...)


Furthermore, if this does make sense, what are the implications, in terms of inference, when trying to interpret the selected model if it has different main and zero-inflation model specifications?

OK, so after consulting with some colleagues, and since no one else attempted an answer, I'll give it my best shot.

1. Don't use AIC in a drop-1 model selection approach. One should probably take the information theory approach to model selection (Burnham and Anderson, 2002), maybe use Likelihood ratio tests to test if random effects are beneficial (Bolker et al, 2009). The best situation would be to take a simulation approach which is currently above my pay-grade for explanation.
2. It can make sense to have different variables in the zero-inflation model and the main effects model. This is entirely dependent on your prior knowledge of the system you are trying to model and is informed by that, not by model selection. Note also the difference between hurdle models (sometimes called Delta-models in primary literature) and zero-inflation. The former assumes all zeros are structural (caused by sampling design, not real zeros), the latter assumes some zeros are structural and some are genuine observations of zero (see the answer to this question and ref's within)
3. The inference made will be dependent on the variables you decided to include in the zi-model, which were decided based on your knowledge of the system you're trying to model.

TLDR; Have a good understanding of the system you are modelling and the questions you are asking before changing the ZI-model independently of the main model - Chur.

PS if you're looking for a good package to work with these types of models I can plug glmmTMB (Brooks et al, 2017)

References above:

Bolker, B.M., Brooks, M.E., Clark, C.J., Geange, S.W., Poulsen, J.R., Stevens, M.H.H., White, J.S.S., 2009. Generalized linear mixed models: a practical guide for ecology and evolution. Trends Ecol. Evol. 24, 127–135. doi:10.1016/j.tree.2008.10.008

Brooks, M.E., Kristensen, K., Benthem, K.J. Van, Magnusson, A., Berg, C.W., Nielsen, A., Skaug, H.J., Mächler, M., Bolker, B.M., 2017. glmmTMB Balances Speed and Flexibility Among Packages for Zero-inflated Generalized Linear Mixed Modeling. R J. 9, 378–400.

Burnham, K.P. and Anderson, D.R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer-Verlag