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For count data with excessive number of zeros, there are two choices of models, zero inflated poisson and zero inflated negative binomial.

Q1: How does one make appropriate choice between the two from theoretical viewpoint?

To me, zero inflated negative binomial has an extra parameter for variance part which allows more flexible variance structure as compared to zero inflated poisson.

Q2: How do I know this behavior beforehand?

Q3: Suppose one is compelled to choose between ZIP and ZINB. Is a badly fitted model(ZIP) with significant parameter better than a good fit model(ZINB) with insignificant parameter? The improvement of fitting is dramatic. I am leaning towards ZINB as it explains my data quite well but insignificance is obtained in return.

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  • $\begingroup$ If the zero-inflation parameter is not significant, why use a zero-inflated model? $\endgroup$
    – jbowman
    Jul 12 at 15:54
  • $\begingroup$ @jbowman Should you use ZIP in this case to declare victory? It sounds like that you are saying even if data fits model well without significance, I should throw away that model. I think there is good chance that there is no association between outcome of interest and covariates included in ZINB. $\endgroup$
    – user45765
    Jul 12 at 16:02
  • $\begingroup$ "The data fits model well without significance" sounds like overfitting to me. The implication is (usually) that a simpler model will also fit the data well. In this case, the simpler model would have no zero inflation term. $\endgroup$
    – jbowman
    Jul 12 at 18:46
  • $\begingroup$ @jbowman When I fit the model, I have selected the most simple model. Unfortunately, doing cross validation will drop sample size for inference model building. I also checked either regression without zero inflation and fitting is not well. Reducing the model further will eliminate covariate of interest to test association. Since I am doing inference to test association, I would not be interested in predictive modelling which you are suggesting here. $\endgroup$
    – user45765
    Jul 12 at 19:44
  • $\begingroup$ What exactly do you mean by "insignificant parameter"? A fit shouldn't go from good to bad by removing an insignificant parameter, that's kind of the point of significance. $\endgroup$
    – jbowman
    Jul 12 at 19:58

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