I am using
dispersiontest(fit, trafo=2) from the AER package in R to see if my data is overdispersed and what the dispersion parameter $\alpha$ is. Since I use
trafo=2 I assume that $Var(y) = \mu + \alpha \cdot \mu^2$.
However, I get a dispersion parameter of about $0.4$, which is not at all in line with
mean(y) = 68 and
var(y) = 8124, because obivously $68 + 0.4\cdot68^2 = 1917$. If I estimate $\alpha$ "by hand", i.e. just solving the above equation for $\alpha$, I get a value of $51$.
Could somebody explain why
dispersiontest estimates $\alpha$ so small?