# Running a Negative Binomial Regression with Overdispersion

I am not an expert on SPSS but have been developing my skills and believe I should run a negative binomial regression on my data, though I am not certain.

I am testing the number of days people go running in a week (dependent variable) against a selection of controls. Firstly, am I correct in assuming this is a count dependent variable?

Secondly, given the mean for the DV is 2.45 and the standard deviation is 2.3 I assume that this suggests there is over dispersion (given the variance (2.3 ^2) is greater than the mean).

In this light, am I correct to run a negative binomial regression model on my data? Just looking for some clarity!

• Welcome! Your question is not at all about SPSS, and I think you will get better answers using the title "Is Negative Binomial Regression the right choice?" Apr 1 at 14:10
• Your thinking seems sound to me. Confirmatory info is at stats.idre.ucla.edu . Apr 1 at 14:14
• @rolando2 I have edited their question to reflect their more direct question rather than a programming one. Apr 1 at 14:59
• Thank you very much! So just to check even though the standard deviation is below the mean, given the variance is above it, then it suggests that the data is over-dispersed (hence a Negative binomial will be more suitable than Poisson).
– Vito
Apr 1 at 15:29
• No, it doesn't. See also my comment in response to @ShawnHemelstrand 's answer below. What matters are the conditional means and variances given the regressors. As a thought experiment, imagine that, with your regressors, you could predict with 100% accuracy the # of days a person runs; would the conditional distribution be over-dispersed? (No, it would be a point mass.) Apr 1 at 15:35

I am less experienced with NB regression, but can offer my insight if others don't chime in. To your questions:

I am testing the number of days people go running in a week (dependent variable) against a selection of controls. Firstly, am I correct in assuming this is a count dependent variable?

Yes this is by definition a count, and would be suitable for NB regression.

Secondly, given the mean for the DV is 2.45 and the standard deviation is 2.3 I assume that this suggests there is over dispersion (given the variance (2.3 ^2) is greater than the mean).

This could theoretically be the case, but NB regression is anyway supposed to account for these issues, whereas more strict count models like Poisson regression don't consider this (or at least have unrealistic assumptions related to this).

You may appreciate Hilbe's text on NB regression, which discusses count models in general but obviously centers on NB. It is written for R/SAS users, but the text is regardless informative from a purely descriptive writing on the subject. Rolando also referenced the UCLA tutorial which has some nice discussion on the topic too.

#### Edit

As noted in the comments by @jbowman, I hadn't considered the potential right-censoring of the model here given one can only run 7 days a week. For that perhaps a right-truncated negative binomial regression would be more adequate, discussed in Chapter 12 of the above linked book. However the only way I know of fitting this is using the gamlss package in R using the family = trun argument.

• I would not think this was suitable for an NB regression, as there is an upper bound of 7 on the response. Imagine, for example, a conditional mean of 4 days/week once the regressors are taken into account; the conditional std. deviation is at least 2 and that just doesn't work with an NB distribution and an upper bound of 7. Apr 1 at 15:31
• I do recognise that it has an upper bound. Is it still possible to run a negative binomial and treat it as a count variable?
– Vito
Apr 1 at 16:21
• @jbowman that is an interesting and valid insight I didn't consider my first time reading this. Do you think a right-truncated negative binomial model be better suited for this? I believe I remember this also being discussed in the above linked book. Apr 1 at 20:34
• I think it depends on how good the model is. It's quite possible that the data will be underdispersed relative to a Poisson or that a Binomial regression where you are trying to estimate the conditional probability of running on a day in the observed week would work well, too. If the model isn't strongly predictive, a right-truncated NB might well be better suited, though. Apr 1 at 20:44
• The problem I foresee with using a conditional probability model is that it likely doesn't answer the question posed by OP, which would be how many days one is predicted to run in a given week, rather than the probability of running on a given day, if my understanding of your comment is correct. I have anyway edited my answer to reflect the censoring here. Apr 1 at 21:02