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I am working on zero-inflated count data models using the pscl package. I am just wondering why there is no development of models for one-inflated count data models! Also why there is no development of bimodal, say zero-and-2-inflated, count data models! Once I generated one-inflated Poisson data and found that neither the glm with family=poisson model nor the negative binomial (glm.nb) model was good enough to fit the data well. If any one can shed some light on my thought, eccentric though it might be, it would be very helpful for me.

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    $\begingroup$ I wrote a grant proposal once to work on software for this, but it got turned down. $\endgroup$ – Peter Flom Jul 3 '14 at 9:34
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    $\begingroup$ I expect there's little to say than there's less call for it so people don't mention it as much, but it's not like it wouldn't ever have been done; it may not have been discussed much. The ideas aren't really introducing much that's substantively different from the common 0-case. $\endgroup$ – Glen_b Aug 25 '16 at 4:03
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A one-inflated Poisson model for a count $Y_i$ is

$$\begin{align}\Pr(Y_i = 1) &= \pi_i +(1-\pi_i)\cdot\mu_i\mathrm{e}^{-\mu_i}\\ \Pr(Y_i = y_i) &= (1-\pi_i)\cdot\frac{\mu_i^{y_i}\mathrm{e}^{-\mu_i}}{y_i!} \qquad \text{when } y_i\neq 1 \end{align}$$

where the Poisson mean $\mu_i$ & Bernoulli probability $\pi_i$ are related to the predictors through appropriate link functions. You can define a similar model to inflate probabilities for any values you choose.

Still, zero has a special (& once controversial) place among the counting numbers—in a sense representing the absence of anything to count. And it's the "nothing" vs "something" distinction, rather than the "one" vs "any other count" distinction that tends to be relevant across a wide range of phenomena we like to model: there's one process that gives a nought, one, two, ... count & another that gives no count at all.

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  • $\begingroup$ Thank you for your response. I agree with how you explained why zero counts have drawn so much interest that other counts. Still, I wanted to learn more about one-inflated count data models. When I tried to write the code in R for one-inflated regression model with covariates then the estimates were wrong. I am sure I have made mistakes in score function and hessian matrix. Can you please recommend me any text/article that may help me to learn more on it? $\endgroup$ – overwhelmed Mar 21 '14 at 20:36
  • $\begingroup$ Don't know of anything on fitting one-inflated Poisson models specifically. As one-inflation is only a slight modification of zero-inflation, reading up on the latter should help. I'd think a few alterations to the zeroinfl code should do it - changing the zero-inflated Poisson likelihood & score to match the model above (or try just changing the likelihood & not passing the score to optim). You can also of course ask here or on SO as appropriate for references or help with things you're stuck on. $\endgroup$ – Scortchi Mar 24 '14 at 12:25
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The R package VGAM has function vglm which can be used to fit all sorts of Poisson-esque models. You can use it to specify a one-inflated model, so something like vglm(Y~X,family=oipospoisson(),data=data). See here for more details.

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