Say that I was modelling the number of upvotes for a Facebook post under two treatments. What would be a more ideal distribution for modelling these counts: negative-binomial or a Poisson?
What would be the trade-offs here?
Solved – Negative-Binomial vs Poisson for Count Data
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As your question is related to a Poisson and negative binomial distribution, I assume you are referring to count data, for example a number of goals made during a game, and you also registered the number of training hours of the last year.
Assume you want to model this data using the training hours as explanatory variable, as you assume the player that trains a lot makes more goals. You would assume that this explanatory variable would explain some of the variation in goals made during a game.
However, the Poisson model has as an assumption that the mean of the observed data is equal to the variance. Very often, you end up with a greater variance than demanded by Poisson, and you have to deal with over dispersion (including other predictor variables will further reduce the variance). Negative binomial models can be used in that case.
But even more important is that the zeros in the data have a different "data generating process" than the other counts, as maybe 20% of the kids did not even play during the game and had therefore not even the chance of making a goal. This also leads to lots of unexplained variation, which makes Poisson even less useful. In that case you model the zeros differently, with either zero inflated models or hurdle models.
You can find this summary better explained in this online class: http://www.youtube.com/watch?v=T6cv8n9xBGQ
Multinomial and Poisson are very different. Multinomial regression should be used when your response is categorical with more than 2 categories. Poisson regression is used when your response is a count of an incident.
Classic Poisson regression: response is # of accidents at an intersection in a day. This can be modeled using Poisson regression since there is no obvious upper bound to the # of accidents. Also note that the numbers actually mean something and have quantitative value.
Classic example of multinomial regression: response is dietary habits of an individual, Non-vegetarian, Vegetarian or Vegan. Note that the variable in qualitative and not necessarily quantitative. And there are restrictions to what values the response can take (1 of three).
There are almost no obvious scenarios that I can think of where these two methods can be used interchangeably. For count data, if you want other options, you might look at Negative-Binomial Regression.
If the goal is to model the effect of the treatments, where the effect can be categorical (got well, stayed same, got better) a multinomial (or logistic) regression should be used with the treatment as a predictor.
Best Answer
For $Y \sim$ Poisson$(\lambda)$ we have $\text{E}(Y) = \lambda$ and $\text{Var}(Y) = \lambda$, so the variance should at least be well-approximated by the mean. Often in real applications we find that the variance for count data is larger than the mean.
If on the other hand $Y \sim$ negative binomial$(r, p)$ we have $\text{E}(Y) = rp / (1 - p)$ which is less than $\text{Var}(Y) = rp / (1 - p)^2$, so the negative binomial is sometimes used to account for what's called "overdispersion." The only real downside is that the model is slightly less parsimonious, but that arguably isn't much of a downside.