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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)
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  • $\begingroup$ @MartijnWeterings as for my toy data set, this will require some forensics on my home computer to see what I was working on 5.5 years ago! The online example, luckily, has been crawled by internet archive link $\endgroup$ – user9437 Nov 17 '17 at 15:28
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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

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