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I know that a negative binomial model is often use to solve the problem of overdispersion in count data (poisson regression). Now, someone said that a beta binomial model can also be used to solve the problem of overdispersion in binomial data (e.g. if you use logistic regression). Can someone explain why this is?

Also, I know that zero-inflation is a problem when you have count data and want to fit a poisson model. Can zero-inflation be a problem when you want to fit a logistic model for binary data? And what if I fit a beta binomial model to solve the problem of overdispersion? I notice that there exist a zero inflated beta binomial model, but not a zero inflated logistic model?

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In general, random effects models can be used to explain variation in outcomes beyond what the standard distribution in a generalized linear model assumes by assuming that units vary in a random manner (not explained by available covariates; this results in overdispersion = variation beyond the "base"-distribution). A beta-binomial distribution or logistic regresion with a random effect (either normally distributed on the logit scale or beta distributed on the probability scale) are examples of such models.

Quite often overdispersion also results in an excess of zeros (e.g. in the Poisson case or in the binomial case). Sometimes people deal with this by going to a zero-inflated model (e.g. zero-inflated Poisson model) and sometimes by going to a random effects model (e.g. Poisson with random effect on log-rate, Poisson with gamma distributed random effect on rate = negative binomial, binomial with normally distributed random effect on logit etc.). Occasionally, people even do both (e.g. zero-inflated negative binomial model).

< start of personal opinion > Personally, I have not run into many problems that truly required a zero-inflated model (i.e. a random effects model explained the data just as well and was in some sense more logical). Often people in medicine seem (to me) to get carried away with interpreting excess zeros as people being healed/not susceptible to a disease/event, when this can often just as easily be explained by the risk of events/disease varying a bit between patients. < /end of personal opinion >

In the case of a binomial distribution $Y_i \sim \text{Bin}(n, \pi)$, adding zero inflation (e.g. average proportion of $\xi$ being zeros) is actually just the same thing as $Y_i \sim \text{Bin}(n, \pi(1-\xi))$. I.e. you stay within the binomial distribution family and you just end up estimating $\pi' := \pi(1-\xi)$.

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