Zero-inflated negative binomial

I am trying to understand zero-inflated negative binomial regression. My impression is that if a zero-inflated negative binomial model does not contain any logit part, the model is identical to the one can obtain with just ordinary negative binomial regression. Is this correct?

PS: the logit part I was talking about - well - zero-inflated model assumes that the 0s within the dataset are generated based on two different process: one is negative binomial and the other is, if I remember it correctly, poisson. By "no logit part" I meant what if we take out the effect of the poisson distribution from the zero-inflated model? would it be same as ordinary negative binomial regression?

• i don't quite understand what you mean by the "logit part". you might want to look at this review: jstatsoft.org/v27/i08 Nov 26 '13 at 17:40
• Zero-inflated models are usually defined as two-component mixture models combining a point mass at zero with a count distribution such as negative binomial. Nov 27 '13 at 3:52
• From that review: "the unobserved probability $\pi$ of belonging to the point mass component is modelled by a binomial GLM" (8). I take it @Jin-Dominique has just confused a binomial GLM with logistic regression. Then they're asking, if $\pi = 0$, isn't a ZIM just an ordinary negative binomial regression? In that case, the RHS of eqn 7 collapses to $f_{count}(y; x, \bet)$, so the first-pass answer is "yes." Differences in the way the models are fit in particular implementations might lead to different estimates, perhaps. Dec 3 '17 at 19:48

$$f_{ZINB}(Y;\pi, \mu, \theta) = \pi Z(Y) + (1-\pi)f_{NB}(Y;\mu, \theta)$$
The "logit-part" you refer to models the mixing parameter $\pi$ as a function on covariates, assuming a binomial distribution with succes probability $\pi$.
To answer your question: yes if you set $\pi=0$ you're indeed back at the regular NB