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Suppose we have ordinal data that can assume the following values: $0$, $1$, $2$, $3$ and $4$. Also suppose that there are a lot of data points that fall into category $1$. Can we use zero inflated Poisson regression to model this?

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I think if the data are count in nature then you can use zero inflated Poisson regression. The use of mixture model for ordinal data with zero inflation case can be found in the work of Kelley and Anderson. This may come very informative for you.

There is an argument whether zero inflated models are really necessary. If you are interested you may check out this discussion.

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You could also use a Zero Inflated Negative Binomial model. In the negative binomial model, the mean does not have to equal the variance as in the Poisson model.

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