Types of dispersion parameter for binomial data For a model with a binomial proportion as response variable, which is fitted with according to a binomial distribution, a dispersion parameter $\phi$ can be calculated, which is equal to the sum of the squared Pearson residuals divided by the residual degrees of freedom.
I was following along with an example I found online and found that the $\phi$ computed by hand matches the dispersion parameter that one gets by specifying family=quasibinomial() in the glm command (rather than family=binomial()) and asking R for 
summary(fittedModelName)$dispersion
For my toy data set, $\phi = 4.904$ in both cases. When I fit the same model with a beta-binomial distribution (using betabin in the aod package), I get a dispersion parameter estimate which is very different, sc.
Overdispersion coefficients:
                 Estimate Std. Error   z value   Pr(> z)
phi.(Intercept) 1.376e-01  3.145e-02 4.374e+00 6.102e-06

I would like to know how the two types of dispersion parameters differ, and whether there is any way they can be related.
These are the parameters I specified for the betabin function:
betabin(formula = cbind(Rcnt, total - Rcnt) ~ LANG, random = ~1, 
    data = toyDataSet)

 A: The glm dispersion
the glm dispersion is an expression based on the normalized residuals. 
So say you have a model fittedModelName Then you get the same for 


*

*manual calculation
sum( residuals(fittedModelName, "pearson")^2 ) / fittedModelName$df.residual


*automatic    
summary(fittedModelName)$dispersion

note that you can define the quasibinomial model in two ways. You could use:


*

*scaled from 0 to 1
formula = Rcnt/total ~ LANG


*scaled from 0 to n
formula = cbind(Rcnt, total - Rcnt) ~ LANG

which gives different results for the calculated dispersion
The beta-binomial regression by aod's betabin
In the betabin case the reported dispersion is a model parameter. This is explained in the documentation of the function.

The function uses the parameterization .... $\varphi = 1 / (a1 + a2 + 1)$ ... and $\varphi$ is the overdispersion parameter.

You can test this also with the code below:
library(aod)

#generate data

set.seed(1)
alpha = 1
beta = 1
n = 40
x <- rbeta(1000, alpha, beta)
y <- qbinom(runif(1000), n, x)

#modeling

mb <- betabin(cbind(y,n-y)~1, random=~1, data=as.data.frame(list(y=y)))

which shows that the used dispersion is the inverse of the sum $1+\alpha+\beta$ with $\alpha$ and $\beta$ the coefficients of the beta-distribution.
#comparison of dispersion

> mb@param[2]   
phi.(Intercept) 
       0.340159 
> 1/(alpha+beta+1)
[1] 0.3333333

References
The documentation is here:
https://www.rdocumentation.org/packages/aod/versions/1.3/topics/betabin
but I prefer to get these documentation files by typing the function name into the console betabin or ??betabin
