# Difference between binomial, negative binomial and Poisson regression

I am looking for some information about the difference between binomial, negative binomial and Poisson regression and for which situations are these regression best fitted.

Are there any tests I can perform in SPSS that can tell me which of these regressions is the best for my situation?

Also, how do I run a Poisson or negative binomial in SPSS, since there are no options such as I can see in the regression part?

If you have any useful links I would appreciate it very much.

Only the nature of your data and your question of interest can tell you which of these regressions are best for your situation. So there are no tests that will tell you which one of these methods is the best for you. (Click on the links of the regression methods below to see some worked examples in SPSS.)

Remember that the Poisson distribution assumes that the mean and variance are the same. Sometimes, your data show extra variation that is greater than the mean. This situation is called overdispersion and negative binomial regression is more flexible in that regard than Poisson regression (you could still use Poisson regression in that case but the standard errors could be biased). The negative binomial distribution has one parameter more than the Poisson regression that adjusts the variance independently from the mean. In fact, the Poisson distribution is a special case of the negative binomial distribution.

This is too long to be a comment, so I will make it an answer.

The distinction between binomial on the whole hand and Poisson and negative binomial on the other is in the nature of the data; tests are irrelevant.

There are widespread myths about the requirements for Poisson regression. Variance equal to mean is characteristic of a Poisson, but Poisson regression does not require that of the response, nor that the marginal distribution of the response be Poisson, any more than classical regression requires it to be normal (Gaussian).

Having dubious standard errors is not fatal, not least because you can get better estimates of standard errors in decent implementations of Poisson regression.

Nor does Poisson absolutely require the response to be counted. It often works well with non-negative continuous variables. For more on the underestimation (pun intended) of Poisson, see

http://blog.stata.com/tag/poisson-regression/

and its references. The Stata content of that blog entry should not stop it being of interest and use to people who don't use Stata.

It's difficult to advise well on the choice between Poisson and negative binomial regression. See if Poisson regression does a good job; otherwise consider the greater complication of negative binomial regression.

I can't advise on using SPSS. It wouldn't surprise me if you needed to use other software for flexible implementation of Poisson or negative binomial regression.

• Re the myths about requirements: saying "Poisson regression" to mean "using the same score function as for the Poisson GLM in an estimating-equation approach to get point estimates for coefficients, & sandwich estimators for their standard errors" is most likely at the root of any confusion. After all, OLS doesn't get called Gaussian regression. Unfortunately "quasi-Poisson regression with robust standard errors" is the most concise name I can think of. – Scortchi - Reinstate Monica Oct 10 '13 at 16:41
• Agreed. Anybody who reads my papers is likely to note much emphasis on the power of names for good or ill; it is good to get back some of my advice. – Nick Cox Oct 10 '13 at 17:31

In SPSS Statistics, the GENLIN command handles Poisson, negative binomial and a bunch of others. (Analyze > Generalized Linear Models). It is part of the Advanced Statistics option.

Poisson/Negative binomial can also be used with a binary outcome with offset equal to one. Of course it necessitates that the data be from a prospective design (cohort, rct, etc). Poisson or NB regression gives the more appropriate effect measure (IRR) versus odds ratio from logistic regression.

NB regression is "safer" to run than Poisson regression because even if the overdispersion parameter (alpha in Stata) is not statistically significant, the results will be exactly the same as its Poisson regression form.