Which distribution for a bounded count?

[proctice]

My outcome is a count of the number of 5 different activities a person engaged in.  It ranges from 0 to 5.  I'm thinking of a binomial model with genmod or a cumulative logit model with proc logistic.   For the former, I'm not really certain that these are trials and if they are, they probably aren't independent trials.  So, I'm leaning towards the ordinal outcome.  Are there others that I should consider?  Thank you! 

[ballardw]

You might describe the question of interest that the analysis is supposed to answer.

There would be different approaches to "mean number of activities", "maximum/minimum number of activities" or "number of activities in relationship to circumstances x, y and z (and whether x, y and z are categorical, nominal ordinal or contimuous)

[Reeza]

Multinomial logistic regression is a good thought, but my head also runs to Market Basket Analysis if you have EM. If you don't, I would still consider it as an analysis methodology. But it depends on what you're trying to do overall. 

[Rick_SAS · Accepted solution]

I agree with ballardw. A binomial model would be suitable if all the activities are equivalent and you are interested only in the question "did the subjects participate in 0, 1, ..., 5 activities." If the activities are hierachical or cumulative (walking, walking and jogging, walking and jogging and weightlifting,....) then an ordinal model might be better.  Basically we need more information.

[proctice]

It would be something like how many of the following activities did you participate in this week (running, walking, skipping, jumping, jogging)?  

[StatDave]

Yes. Such a count response could be modeled as I suggested.

[proctice]

As a poisson or negative binomial?  This distribution does not look like a poisson distribution at all.  Poisson distributions are not usually bounded at the upper end, they usually trail off the upper end. They usually look like left skewed normal distributions.  My distribution is more u shaped with lots of zeros and fives.   Notice that none of the other posters recommended this.

[StatDave]

You say your response is a count: 0, 1,...,5. A count response is typically modeled using a Poisson or negative binomial model. This can be done in PROC GENMOD.

[proctice]

Would love a second opinion on Poisson for a situation when it is impossible to have a count higher than 5. 

[Reeza]

You may get better answers on stats.stackexchange.com (CV =Cross Validated). There are statisticians on here, but the very specific questions obviously have a smaller subset of people who can answer them. By comparison, CV is solely for statistical methodology question. 

[sld]

The Poisson distribution is bounded by zero and has no upper bound. So if your count can be no greater than 5, the Poisson quite likely is not appropriate.

@Rick_SAS recommended the binomial distribution in a previous response in this thread. I agree with his suggestion. The binomial will produce an analysis of the proportion of activities engaged in: 0/5, 1/5, 2/5, 3/5, 4/5, 5/5.