I have data on 44 firms that have all been ranked by an expert. The "best" firm has rank 1, the second best has rank 2, ..., the last one has rank 44. I have a bunch of explanatory variables and would like to explain the rank of the firm on the basis of these variables. My inclination is to use a regression model, but am concerned about the fact that the dependent variable is limited, it can only be a positive discrete number.
I have thought about ordinal regression, but that seems impossible since I would have as many categories as I have observations.
What regression models would be possible? (preferably to be run in R)