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I learned about the zero-inflated negative binomial distribution a few months ago when I was trying to do regression on some discrete data. I have a different data set now, and it seems to be very similar except that the value 1 seems to be over-represented (as opposed to 0). Is there such a thing as a one-inflated negative binomial distribution? How could I model these data?

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    $\begingroup$ Before you start modeling a special term for 1, make sure that there is a logical reason for the 1s to be over-represented. When you figure out why are there too many 1s, you might find the most appropriate model. $\endgroup$ – Aniko Apr 29 '11 at 19:55
  • $\begingroup$ @Aniko makes a good point, you ought to be sure that there really are excess ones beyond what you can chalk up to sampling variability or you risk overfitting. And also that there are enough excess ones that you need the more flexible model. $\endgroup$ – JMS Apr 30 '11 at 15:51
  • $\begingroup$ @Aniko The 1 value is significantly over-represented because it is the minimum value representing the simplest case for the data I am measuring. I guess I could transform the data (x' = x-1) and then use the zero-inflated model, but I wasn't sure what effect that would have and if the interpretation of the results would be the same. $\endgroup$ – Daniel Standage May 2 '11 at 16:51
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    $\begingroup$ Then it means that you have no 0's? Subtracting 1 from the data sounds like a good idea, but you could also consider the positive negative binomial distribution. It all depends on the actual meaning of the data points. $\endgroup$ – Aniko May 2 '11 at 18:20
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Sure. You can write

$Pr(X=x) = p*f(x) + (1-p)1(x=1)$

where $f$ is the NB pmf, $1()$ is just an indicator function and $p$ is some probability. Of course in general the $x=1$ could be $x=k$ for any integer $k$, and $f$ could be any pmf.

In principle it shouldn't be harder to fit than a ZI model, but an off-the-shelf model-fitting solution may not exist.

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