# Zero-and-one inflated beta regression vs. binomial GLMM?

I appreciate some help with deciding whether I should (and how to) construct a zero-and-one-inflated beta regression model. I want to use R to test the hypothesis that there is a condition * age interaction effect on investment.

investment can be none (i.e. 0), half (i.e. 0.5) or all (i.e. 1). This makes investment non-binomial, and, strictly speaking, inappropriate for a GLMM with family = binomial. It looks like I should use gamlss() to run a beta regression with family = BEINF.

condition is a factor with 3 levels. age is a continuous variable. There is another fixed effect sex, and a random effect groupID

the mu.formula of a gamlss beta regression should look like this: investment ~ condition * age + sex + re(random = ~1 | groupID)

I have three questions:

1. Since my goal is hypothesis testing other than model selection, shall I just use something like drop1 to test the effect of each predictor?

2. If the answer to #1 is yes, how should I specify sigma.formula, nu.formula and tau.formula? (I am aware of the function stepGAICAll.A for selecting the best model, but that's kind of a forward model selection and my goal isn't model selection anyway).

3. Let's forget beta regression for the moment. How justified I am if I just go ahead and use a binomial GLMM via lme4? The two reasons for this include:

(1) investment has only three possible values (0, 0.5 and 1), it's clearly not a binary response, but it's not far from it. It doesn't look like a true frequency either. Even if I stick with zero-and-one-inflated beta regression, I have a hard time understanding what the mu.formula means? Does it mean the probability of investment equals 0.5?

(2) a quote from this paper (Warton & Hui 2011 Ecology pp9):

For non-binomial proportions, there was never any theoretical reason to use the arcsine transform in the first place, and we instead suggest using the logit transform.

• I think I would recommend trying an ordinal model, e.g. clmm from the ordinal package in R ... – Ben Bolker Jul 17 '15 at 21:06