Suitable regression model for limited, discrete dependent variable I have data where the dependent variable is discrete and lies between 20 and 40 (possible values are 20, 20.5, 21, 21.5, ..., 39, 39.5, 40). The variable measures some results from a game which can be between 20 (lowest achievable value) and 40 (highest). 
After some hours of research on the web, I could not find a regression model that ideally fits those characteristics of the described dependend variable. 
Given what I found, maybe a multinomial logistic regression fits best although my dependent variable is not nomial. 
I would be very thankful if you could propose me a regression model that fits my problem best according to your consideration.
Thanks a lot!!! :)
 A: As long as the range of achieved scores isn't too narrow, you might treat your variable as effectively continuous, but with bounds. The bounds will impact linearity (a relationship can't just blast through a bound, so it must have a curve or bend) and constant variance assumptions (as the mean approaches a bound more closely, the variance will tend to decrease).
So you wouldn't use linear regression if there are scores that closely approach either of the bounds, but you might use nonlinear regression or GLMs of appropriate kinds, for example.
I see two reasonable possibilities that might work for such data - beta regression and quasi-binomial GLMs. Either might be fitted using a logistic model.
If you want to deal with it in a slightly more "correct" fashion, you might write a model where there's an underlying continuous variable, but where you only observe it in bins (and where the recorded value is the bin-center, say). You could then write a likelihood for the observed data and try to fit the model that way; this would still require accounting for the boundary issues with mean and variance I mentioned before. (Such an approach suggests considering something like an EM algorithm as a possibility.)
