# 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.

Thank you very much for your comments and suggestions in advance!

• 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
• @BenBolker Thank you for the quick suggestion!! That's a nice alternative! – kctam Jul 18 '15 at 12:32