Which measurement level would this be? I have a variable times of collaboration: the possible answers are 0 (meaning no collaboration), 1, 2, or 3.
Would this mean I am dealing with an interval measurement level?
 A: Yes, interval is acceptable if you are willing to assume the difference between every increment is the same. (Though nitpickers will say it should be "ratio" because the 0 is a true zero, representing none.) In some cases, this assumption may be useful, in some other not so. For instance, researchers who collaborate with each other may have a non-linear improvement in some outcome (e.g. time spent on lab meeting, or number of grants, etc.) According to this complexity there can be implication on how to model this particular variable.
Particular attention should be put at the last option. Is it "3" or "3 or more?" The first one may prompt people to think what happens if you have 4 or more collaborations? And the second one will make your variable more of with ordinal nature than interval/ratio, because the difference between "2" and "3 or more" is not 1 anymore.
A: The proper way to treat this is as ordinal, since you clearly have ordered categories, but they are just that, categories. 2 is not twice as high as 1, for example.
In analysis, I often treat ordinal data as interval only the following conditions. I did not get these from any source, it is just based on my experience and understanding:


*

*You have enough levels where you have enough granularity to approximate a continuous scale. If there are a lot of ordinal levels, in a regression model you can have a lot of coefficients to interpret if the ordinal variable is an independent variable.

*You can conceptually think about the levels as approximating a continuous underlying latent distribution

*You have done some testing comparing the treatment as ordinal vs continuous in your analysis, and it is comparable


I know other researchers are far less strict than I, so you have to make your own decision there.
Ordinal data has a lot of natural analysis, for example:


*

*Bivariate analysis using contingency tables using $\chi^2$ tests of independence.

*Bivariate or multivariate analysis with the ordinal variable as the dependent variable, using ordinal logit or probit. Most statistical packages handle this fine. Be sure to assess the "parallel lines" or proportional odds assumption, basically that the move from any given level to the next is proportionally equivalent.

*As an independent variable in a model, where you treat each level as a separate dummy variable, or alternatively as I mentioned above, as a single "continuous" predictor, after you add the dummy variables and test that their coefficients are proportional.
