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Zero-inflated negative binomial models have two components: a count component (negative binomial regression part) and a zero component (logistic regression part).

Why not just run two separate models in the data: a negative binomial regression and a logistic regression? What are the advantages of combining these two models within one model?

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Zero-inflated negative binomial models don't assume that all the zeroes come from the Bernoulli process; some may come from the negative binomial process. Toss a coin & write down zero if it's tails. If it's heads then start tossing again & write down the number of tails until you have three, say, heads. There are two different reasons for writing down zero so you can't separate the data into two parts for different models.

Hurdle models on the other hand, can indeed be seen as two separate models—a truncated negative binomial for the non-zero count component & a Bernoulli for the zero component—for which the likelihoods are separately maximized. See here.

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