Which type of regression model for count data should I use for log-series? I am attempting to identify the effects of mealtime habits on fast food consumption. In this case, the dependent variable is number of times per week a respondent reported eating fast food. As this is count data, I'm using a Poisson or Poisson-related distribution. However there are a few issues with this data. First, tests for overdispersion revealed that my dependent variable is indeed overdispersed:
Overdispersion test
data:  model8.poisson
z = 8.2298, p-value < 2.2e-16
alternative hypothesis: true dispersion is greater than 1
sample estimates:
dispersion 
  1.950116

So, a poisson regression is inappropriate.
I also estimated quasipoisson and zero-inflated models, and tested for model fit versus the poisson models. Both the quasi-poisson and zero-inflated were better fits. 
Ord plot shows a positive slope (1.62) and a negative intercept (-1.82) which suggests that I should use a log-series model. I have a rudimentary understanding of poisson quasi-poisson distribution models, and negative binomial models. However, I can't find anything on log-series models.
Are there specialized models for these types of data, or should I use one of the other Poisson-related distributions or models (quasipoisson, zero-inflated, negative binomial)?
If there are specialized models, is there an r package for them?
If there are not, which type of model would best fit these data?
 A: Another natural extension to the Poisson GLM is the negative binomial GLM, which is implemented in R using the glm.nb function in the MASS package.  The negative binomial distribution is a two-parameter distribution which generalises the Poisson distribution, with an additional parameter that controls "overdispersion" in the variance.  The distribution has a natural association with count data through mixtures of Poisson distributions, or through counts of a binary outcome until its opposite.
The quasi-Poisson and negative binomial GLMs are both two-parameter extensions of the Poisson, but they have different functional relationships between the mean and variance.  The former has a variance that is a linear function of the mean, while the latter has a variance that is a quadratic function of the mean.  This means that the distributions weight small and large counts differently.  It is best to fit both these models and see which fits the data better.  Finally, the zero-inflated Poisson model is also a two-parameter extension of the Poisson, but it involves a generalised allowance for more weight at zero, but the shape of the rest of the distribution remains unchanged.  This latter distribution is appropriate where there is a mixture of Poisson counts, plus additional zeros.
A: you can try package ‘COMPoissonReg’, this package includes Zero-inflated CMP regression, has computation of estimates (including pscl for ZIP regression). Here is documentation  https://cran.r-project.org/web/packages/COMPoissonReg/COMPoissonReg.pdf
