# Using poisson regression for continuous data?

Can the poisson distribution be used to analyze continuous data as well as discrete data?

I have a few data sets where response variables are continuous, but resemble a poisson distribution rather than a normal distribution. However, the poisson distribution is a discrete distribution and is usually concerned with numbers or counts.

• How do your empirical distributions differ from Gamma variates, then? – whuber Feb 10 '11 at 15:15
• I have used the gamma distribution for these data. If you use the gamma distribution with a log link you get almost the exact same result you get from an over-dispersed poisson model.However, in most of the statistical packages I am familiar with poisson regression is simpler and much more flexible. – user3136 Feb 10 '11 at 16:50
• Wouldn't there be other distributions that are better, e.g. whuber's suggestion of gamma? – Peter Flom Feb 10 '11 at 22:49
• @PeterFlom - I wonder if this issue comes up a lot because the glmnet package in R doesn't support either the Gamma family or Gaussian family with a log link function. However, because glmnet is used as a predictive modeling package (hence users are only interested in model coefficients, not coeff. stnd errors) and since the Poisson dbn produces consistent coeff. estimates for models of the form ln[E(y)]=beta0 + beta*X with continuous responses regardless of the distribution, I'm guessing the authors of glmnet didn't bother including these additional families. – RobertF Jun 12 '15 at 16:40

## 2 Answers

The key assumption of a generalized linear model that's relevant here is the relationship between the variance and mean of the response, given the values of the predictors. When you specify a Poisson distribution, what this implies is that you are assuming the conditional variance is equal to the conditional mean.* The actual shape of the distribution doesn't matter as much: it could be Poisson, or gamma, or normal, or anything else as long as that mean-variance relationship holds.

* You can relax the assumption that the variance equals the mean to one of proportionality, and still usually get good results.

If you're talking about using a Poisson response in a generalized linear model, then yes, if you are willing to make the assumption that the variance of each observation is equal to its mean.

If you don't want to do that, another alternative may be to transform the response (e.g. take logs).

• I think, in addition to your point, even if @user3136 is not willing to make the assumption of mean = variance, he/she can use the quasipoisson family in glm . – suncoolsu Feb 11 '11 at 3:45
• But my problem is why would you want to transform continous data to discrete. It is loosing information essentially. Also when a simple log transform would have worked, why discretize your data? Using glm works, but every result is asymptotics based (which may or may not hold) – suncoolsu Feb 11 '11 at 3:47
• @suncoolsu: 1) quasipoisson makes the assumption of mean proportional to variance. 2) I didn't mean transform to discrete, i meant transform (maintaining continuity) so you could use a different model. – Simon Byrne Feb 11 '11 at 10:58
• yeah -- I understood agree with you. Sorry, I was talking about the question. Quasi-poisson, takes into account of overdisperson right? (if I remember correctly, cf Faraway 2006) – suncoolsu Feb 11 '11 at 13:51
• In this particular instance I was not satisfied that any transformation I tried (log, sqrt, box-cox) gave a good approximation to normality. Incidentally, if I use the normal score transformation method then I can transform most data to almost beautiful normality, but I have not seen this transformation widely used so I assume there's a catch (it is hard to back-transform). – user3136 Feb 11 '11 at 17:10