With binomially distributed data, it's straightforward to test the null hypothesis of equiprobable responses, $H_0: p=0.5$, but say you want to test the analogue in a Beta-Binomial model fit to over-dispersed data (not under a Bayesian paradigm, but with the whole thing fit via maximum-likelihood estimation).

It makes sense to define a null distribution where $\frac { \alpha }{ \alpha+\beta }=0.5 $, but how would you decide how large or small these values should be? Is it reasonable to simply define $\alpha+\beta$ as the observed sum $\hat\alpha+\hat\beta$ from the fitted model? Or is there reason to expect that the null would imply some different degree of dispersion?

  • $\begingroup$ What makes you think the data are overdispersed? Do you have a count of successes out of a total number of trials (which is >1) for every unit? Do you have an estimate of the dispersion? $\endgroup$ – gung Jan 31 '16 at 16:16
  • $\begingroup$ Well, I mean hypothetically. Assuming that the data is over-dispersed. $\endgroup$ – Joe Jan 31 '16 at 16:33
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    $\begingroup$ If you don't have multiple trials per unit, it's hard to see how you'd know there was overdispersion present. In addition, usually, you use a measure of overdispersion from your data to fit the beta-binomial (sort of analogous to using the SD from your data to run a t-test). $\endgroup$ – gung Jan 31 '16 at 16:50
  • $\begingroup$ Perhaps it was confusing to lead off with the different hypothetical "With Bernoulli-distributed data..." Yes, I'm assuming multiple trials per unit with over-dispersion present. $\endgroup$ – Joe Jan 31 '16 at 19:36

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