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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 whydispersiontest estimates $\alpha$ so small?

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The thing is that you don't estimate $\alpha$ by solving for it from the formula that models $Var(y)$ in terms of $\mu$. Instead $\alpha$ is estimated by analyzing how $Var(y)$ changes with $\mu$ locally in your data. It is not a value that comes from the global estimations of $Var(y)$ and $\mu$ but rather from its local estimations.

Measuring and using these local estimations to estimate $\alpha$ is done via regression, as explained in the Details section of the documentation of dispersiontest():

Overdispersion corresponds to alpha > 0 and underdispersion to alpha < 0. The coefficient alpha can be estimated by an auxiliary OLS regression and tested with the corresponding t (or z) statistic which is asymptotically standard normal under the null hypothesis.

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  • $\begingroup$ thank you! I see what you mean, that means $Var(y) = \mu + \alpha \cdot \mu$ is just not generally correct? What exactly does "locally" mean here? For which datapoints is the equation true? $\endgroup$ – msloryg Jun 24 '19 at 10:08
  • $\begingroup$ No, I don't mean that $Var(y) = \mu + \alpha * \mu^2$ is not generally correct. What I mean is that this formula proposes a relationship between $Var(y)$ and $\mu$ as when you propose a relationship between a dependent variable y and a variable x. The relationship is theoretical and is estimated using regression on the observed data points (x,y). Here is the same thing, you locally observe values of $Var(y)$ and $\mu$ ("local" means in a neighbourhood around each x) and estimate the relationship between $Var(y)$ and $\mu$ using regression. $\endgroup$ – mastropi Jun 24 '19 at 10:48

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