# Poisson and Negative Binomial Models - Are real counts really necessary?

The Poisson model is probably the first that pops into mind when trying to model count data, which is generally described as: non-negative integer data. The Poisson distribution is defined as modeling the number of counts which occur in a specific time interval.

My question is: What if I model non-negative integer data with a Poisson GLM for which the underlying process is not a count within a specific time interval?

More specifically, what if I want to model successive counts? For example, what if I want to model the number of successive wins for a football club? Or the number of webpages someone visits on a website? The length of the time interval is then determined based on when the (successive) count ends and different for each observation in a dataset, right? Not entirely the same as for the underlying Poisson process I guess... Therefore I hope someone can give some insight into the following subquestions arising from my initial question.

• What if still the Poisson GLM is used in these situations, what will be the consequences?
• If there are consequences do these also hold for the Negative Binomial GLM as this can be defined as a Gamma - Poisson mixture?
• Is a more flexible method such as quasi-poisson more suitable in these cases?

Edit: I was triggered by this post, where the answer mentions that real counts are not necessary, but the linked webpage provides no information on this issue, nor can I find anything while searching the internet.

• The link in your edit doesn't work. – Sycorax says Reinstate Monica Jan 24 '17 at 20:13
• These questions are premised on conflating a definition with a characterization. The Poisson distribution is defined in terms of its mathematical properties. It often applies to counts within time intervals--but that certainly does not limit its scope of application! This conflation has given rise to a jumble of interrelated questions about when this distribution might be applicable, about GLMs, about other distributions, about wins, Webpage visits, and so on. That looks like it will be too broad to answer on this site. Could you narrow your question? – whuber Jan 24 '17 at 20:21
• @whuber Thanks for your comment. I'm not exactly sure how to restrict the questions in my post to an appropriate size, if that is what you mean, however I'll try to be more succinct. The reason for this post is that I had a discussion with someone who argued that for the examples I gave, the Poisson distribution assumptions such as event independence do not hold. While it is assumed that the response variable follows the Poisson distribution. I could not however find any consequences of these assumptions not holding. – Andrew Jan 24 '17 at 20:40
• Did you read the article that Nick Cox was referencing? It's accessible, non-technical, and may answer your questions. Visit blog.stata.com/2011/08/22/…. – whuber Jan 24 '17 at 21:22
• One big problem with modelling cumulative values is that the assumption of independence is obviously violated. You can't just ignore that issue (i.e. that $\text{Cov}(X_1+X_2+...+X_t, X_1+X_2+...+X_{t+k})$ $=$ $\text{Var}(X_1+X_2+...+X_t)+\text{Cov}(X_1+X_2+...+X_t,X_{t+1}+...+X_{t+k})$ will usually be large and positive even when the $X$'s are nearly independent -- though I'm astounded by how often people - including a lot of academics of my experience - manage do so anyway) – Glen_b -Reinstate Monica Jan 25 '17 at 1:19