Reporting betareg outcome - how to compare non-nested models? I am looking for advice how to gain and report results using beta regression for an ANCOVA-like model.
My model is as follows:
Model <- betareg(plants ~ rain * cows)

The questions I want to answer is whether rain, cows, or interaction between these two factors have an influence to plant relative biomass (expressed as % in a range of (0;1)). If this was a GLM, I would use anova(Model) and report results from the likelihood ratio test using Chisq. Since ANOVA is not suitable for betareg, I read that I could use lrtest or wald test and compare it manually, like this: 
 m1 <- betareg(plants~ 1 ,data = mydata)
 m2 <- betareg(plants~ cows,data = mydata)
 m3 <- betareg(plants~ rain ,data = mydata)
 m4 <- betareg(plants~ rain:cows,data = mydata)
 lrtest(m1,m2,m3,m4)

However, as I understand this is not correct, as my models are not fully nested (in particular m2 and m3). I am wondering if there is an alternative to compare these models and determine which factors could have significantly affected plant biomass?
P.S. This question is closely related to
https://stackoverflow.com/questions/44183329/anova-like-object-for-betaregression however, the answer did not help in my case.
 A: The situation is completely analogous to (generalized) linear models where you could do forward or backward selection via the so-called "encompassing" model (cows + rain) in your case to the interaction model. Only the two non-nested models with one main effect cannot be compared directly with a LR or Wald test but both can be compared against the encompassing model. Or you adopt an information criterion (AIC or BIC, ...) instead. So you could do
m0   <- betareg(plants ~ 1)
mC   <- betareg(plants ~ cows)
mR   <- betareg(plants ~ rain)
mCR  <- betareg(plants ~ cows + rain)
mCxR <- betareg(plates ~ cows * rain)

Then you can test both paths:
lrtest(m0, mC, mCR, mCxR)
lrtest(m0, mR, mCR, mCxR)

and use these to do either forward or backward selection. Or you could do:
BIC(m0, mC, mR, mCR, mCxR)

or something like that.
Further comments:


*

*I guess that plants ~ cows * rain and plants ~ cows:rain is really of interest to you.

*You might want to consider models where also the precision (phi) depends on cows and/or rain. Of course, this creates even more potential models to select from.
