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Say that I was modelling the counts for two different treatments. Afterwards, I would fit a particular type of distribution to each treatment population respectively.

The data is count-based. What are the trade-offs from using a multinomial vs. Poisson distribution? From what I understand, both of them could be used effectively here.

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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.

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  • $\begingroup$ I don't think you have count data here. You will need to add more explanation to the question for me to be sure. $\endgroup$
    – Greenparker
    Commented Mar 9, 2016 at 18:26
  • $\begingroup$ I'm just considering a theoretical situation. Say I'm counting the number of upvotes for a post on Facebook. This is a numerical value and I thought multinomial or Poisson could work here. $\endgroup$
    – user46925
    Commented Mar 9, 2016 at 18:27
  • $\begingroup$ Ah, in this situation, this is count data with no reasonable theoretical restriction to an upper value for the number of likes. So Poisson or Negative Binomial is your best bet. $\endgroup$
    – Greenparker
    Commented Mar 9, 2016 at 18:28
  • $\begingroup$ Is the hypergeometic distribution useful for count data like this as well? $\endgroup$
    – user46925
    Commented Mar 9, 2016 at 18:45
  • $\begingroup$ The hypergeometric is not part of the exponential family, so that might not be a viable option with the usual GLM framework. $\endgroup$
    – Greenparker
    Commented Mar 9, 2016 at 22:13

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