# Beta-binomial relationship between dispersion and correlation parameters

Context: I have created a beta-binomial model using glmmTMB() from the glmmTMB package and I am now trying to simulate a beta-binomial outcome using pbetabinom() from the VGAM package. The parametrization of the the variance in each package is different. Here on pg. 28 it's shown that the beta-binomial variance in the glmmTMB package is V = $$\mu$$(1-$$\mu$$)(n($$\phi$$+n)/($$\phi$$+1)) and here it's shown for the VGAM package as V = $$\mu$$(1-$$\mu$$)(1+(n-1)$$\rho$$)/n. How can I related $$\phi$$ to $$\rho$$?

The pbetabinom() utilizes the $$\rho$$ parameter, what it is calling a correlation parameter, which is defined between 0 and 1. But this $$\rho$$ is not given in the output of glmmTMB(), rather we get a dispersion parameter, $$\phi$$.

I want to simulate based off my model output, but I am struggling how to relate $$\phi$$ to $$\rho$$. Setting the variances equal, and isolating $$\rho$$, my algebra is getting me to:

$$\rho$$=(n*n($$\phi$$+n)-($$\phi$$+1))/((n-1)($$\phi$$+1)).

I have tried playing around with simplifying this but cannot get 'n' out the equation. I can also see, assuming some positive value for $$\phi$$, that $$\rho$$ would not be bound between 0 and 1, which per the VGAM details it should.

So, my questions are: How can I use the dispersion parameter, $$\phi$$, given by glmmTMB() output to simulate a beta-binomial outcome which requires the correlation parameter, $$\rho$$ as defined by VGAM's pbetabinom()? Where am I going wrong in relating to the two given variance equations?

• You appear to be comparing moments of a quantity in one case to moments of its mean in the other: see en.wikipedia.org/wiki/….
– whuber
Apr 19 at 20:11
• Okay, so I cannot "convert" my estimated dispersion into a meaningful correlation parameter. Do you have any guidance on how I can use the dispersion to simulate beta-binomial data? @whuber
– Reid
Apr 20 at 17:17
• What about using glmmTMB::simulate? Apr 21 at 20:39