# Use Poisson or Linear Regression for Longitudinal Data?

I want to fit a regression with repeated measurements

Outcome：number of cigarettes smoked in the jth month (j=1,2,3)

The outcome variable DOES NOT follow a normal distribution. The outcome values have minumum number of 20, median of 80, and maximum number of 230.

I'm wondering which distribution should I assume on the outcome? Poisson distribution or Gaussian distribution?

If I can consider it as continuous outcome, then I can use MANOVA, and do some profile analysis.

Otherwise, I can only Poisson Regression (GLM). But in my knowledge, the Poisson outcome is something like rare evens? However, the outcome values have a maximum of 230+.

Thanks for any suggestions!

• Given the range and mean of the data, it seems quite possible that you would want an overdispersed Poisson model (quasiPoisson), but of course what you really care about in making that determination is the variance-to-mean ratio of the data conditional upon the independent variables. "Rare" events is not a requirement for a Poisson distribution, you can have a Poisson with mean $10^{100}$ if you want. Commented Nov 30, 2019 at 20:54
• Rather than guess what the distribution is, test which one it is. Some programs, like Mathematica, have FindDistribution routines. Next either use an appropriate regression or transform the data to be more conveniently distributed, e.g., normally distributed.
– Carl
Commented Nov 30, 2019 at 21:02
• Another issue to keep in mind is that you don’t care about the distribution of all outcomes pooled together. You care about the distribution at particular values of the predictors.
– Dave
Commented Nov 30, 2019 at 22:35
• Also, you need to account for the longitudinal nature of the data; one way is using a multilevel model. Commented Dec 1, 2019 at 10:10

You make a common mistake: the assumptions about the distribution in a regression model are not wrt the response (what you call outcome), but wrt the predictions the fitted model, i.e. you expect residuals to scatter around the model predictions according to the assumed distribution.

It therefore doesn't make sense to select the distribution based on the response. You have to fit a model, and check residuals (e.g. using DHARMa), or compare regression models with different distributions (e.g. using AIC or LRTs).

In your case, I would start with a negative Binomial (this is basically Poisson with a dispersion parameter, which is nearly always better than a pure Poisson). Poisson btw. does not assume rare events, it's only more important if you have rare events. For large means, the Poisson becomes approximately the normal.

Your response variable is a count variable, so it is best to use a regression model that is appropriate for count data. Poisson regression is a bad model since it does not include a free scale parameter. A better starting point would be a negative binomial GLM, which handles regression problems with a count variable as the response, and also has a free scale parameter to fit the variance well. The distribution has a natural association with count data through mixtures of Poisson distributions, or through counts of a binary outcome until its opposite. Since your data is longitudinal, with repeated measures for each individual, you might also wish to use a mixed effects model, to allow correlation in the longitudinal outcomes for individuals.

You can implement a negative binomial linear mixed model using glmer.nb in the lme4 package in R. (Here are some notes on implementing these kinds of models.) I have used this model for longitudinal count data in previous projects, with good results. Depending on the idiosyncrasies of your data set, you might need to make some adjustments to this model, but I would regard it as a good starting point for regression on a longitudinal data set with a count variable as the response.

• I have a similar question, but the "notes" link is just the home page of Tufts Engineering school :(
– Nate
Commented Oct 22, 2022 at 20:24