Suppose we have an outcome variable $y_{ji}$ which is a count of behaviors performed by group $j$ in round $i$, for $j = 1,…,n$ and $i = 1,…,8$. The outcome $y_{ji}$ counts are non-independent within each group $j$. Possible values for $y$ are bounded between 0 and 12, and because the behaviors appear to follow an almost "all or nothing" kind of pattern, the outcome is a u-shaped distribution:
My question is: how should we set up the beta-binomial model when the data has a multilevel / random-effects component?
I was planning on modelling the outcome data $y_{ji}$ with a Beta-binomial($m$, $\alpha$, $\beta$) distribution, where $y_{ji}$ are binomial observations with a number of trials $m = 12$, and where the probabilities $\pi_1,…,\pi_n$ follow a Beta($\alpha$, $\beta$) distribution. This way the U-shaped outcome distribution could be modeled with $0 < \alpha < 1$ and $0 < \beta < 1$.
In a linear random-effects model I would model a random intercept (or more) varying by group, and this would account for the non-independent error terms within each group's 8 rounds of behavior. Is it possible to do that with a beta-binomial model?
Here is what I've tried so far:
hglm
A hierarchical / random-effects GLMM package in R which says it can model beta-binomial by modelling the fixed-effects portion with binomial(link=logit), and the "random-effects" portion with Beta(link=logit).
question: If I understand correctly, the random-effects portion is modelling the dispersion parameter (they call it $\alpha$…). Is this "random-effect" the same as the random effect in a HLM / MLM / linear random-effects model, in that it would account for the within-group non-independence?
aod
Another package that has an explicit betabin function that models the beta-binomial mean parameter $\mu$ as a function of any number of explanatory variables. The dispersion parameter $\phi$ can only be modeled with one explanatory variable.
question: is $\phi$ being treated as the "random effect"? If so, I have the same question as above.
glmmADMB
Dr. Bolker is still working on implementing the beta-binomial, so I can't use this.
bayesian
Perhaps I should just model it myself?
Thank you all for any help (pointers to papers or books would be appreciated).
Best Answer
In linear random effects models, the additional source of variability due to the random effect results in an additive increase in the total variance. In the beta-binomial model the additional variability is accounted for by a multiplicative overdispersion factor $\phi$. The random effect here is modeled implicitly.
\begin{align} & y_{ji}|\pi_{j} \sim \textit{Ber}(\pi_{j}) , \quad \pi_j \sim Beta(\mu,\rho) \nonumber \\ & E(\pi_{j}) = \mu \quad \textbf{Var}(\pi_{j}) = \rho\mu(1-\mu) \end{align}
$\mu$ and $\rho$ are the mean and intra-class correlation and can be reparametrized in terms of $\alpha$, $\beta$ of the Beta distribution.
Together the mean and variance of the marginal proportions $y_{j.}$ look like
\begin{align} E(y_{j.}) &= m\pi_{j} & \\ \text{Var}(y_{j .}) &= m\mu(1-\mu) + m(m-1)\rho\mu(1-\mu) & \\ & = m\mu(1-\mu)\phi & \nonumber \end{align}
The effect of assuming that $\pi_j$ is beta distributed is an overdispersion $\phi = (1+(m-1)\rho)$.
If your goal is inference on the mean proportions, method of moments estimates for $\mu$ and $\rho$ are quite effective, see [1]. These are particularly good estimates, if the number of observations per group are balanced.
References
[1] Kleinman, Joel C. "Proportions with extraneous variance: single and independent samples." Journal of the American Statistical Association 68.341 (1973): 46-54.