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The following plot shows the data I have. enter image description here Each point indicates the number of actions (x-axis) and the number of different categories (y-axis) on which the actions are performed by a given user.

The number of actions is a discrete variable ($1,\dots,N$), whereas the number of different categories is an ordinal variable ($1 < 2 < \cdots < 28$).

Using the number of actions as a predictor, I aim at predicting the number of categories on which the actions are performed.

Intuitively, it looks like that there is an inverse relationship between the number of actions and the number of categories on which the actions are performed -- i.e. the higher-activity users concentrate their actions on one (or a few) page.

However, I would like to prove such a relationship by means of a statistical model (and rule out the possibility that I'm observing such a relationship only because of the data are skewed -- see points below). The very first thing that came into my mind was a ordinal regression (proportional odds).

However, I have some doubts.

The distribution of actions is very skewed (Pareto-like), and a lot of users performed only one action. This is a problem, since a user who performed only one action falls necessarily in the first category.

Do I observe such an inverse relationship because my data are skewed?

How do you think I could successfully model such a relationship?

Any ideas or suggestions are appreciated.

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