Attempting to investigate a relationship between one variable and another that has three possible values

I'm looking to investigate the relationship between a series of variables on a recurring basis.

One variable will always be the dependent variable, and will take three possible values: good, bad, or other, which I have coded as 1, -1, and 0, respectively.

I'm generally interested in seeing how a series of additional independent variables interact with this first variable, all of which will range linearly. For instance, one such variable might be called minutes and will range from zero to a few thousand.

Essentially, I want to see how variables like minutes might affect the likelihood of a good outcome. Ordinarily, I would run a regression, but the fact that the dependent variable here takes three possible forms is making that tricky. What's the correct way to run this sort of a series of tests?

There are a variety of analyses that can be done with an ordered categorical response.

One fairly common approach is to extend logistic regression in one of several ways, such as ordered logistic regression (ordered logit, proportional odds):

http://en.wikipedia.org/wiki/Ordered_logit

http://www.stanford.edu/~hastie/Papers/ordered.pdf

This can be fitted using polr in the MASS package that comes with R and many other statistics packages can fit it as well.

[Another is multinomial logistic regression (which is suitable for nominal data but is also at least sometimes used with ordered data).]

Another suitable model is the stereotype model (related to the multinomial logit model):

Anderson J. A. (1984), Regression and Ordered Categorical Variables,
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 46, No. 1, pp. 1-30

The R package gnm can fit the stereotype model, as can a number of other statistical packages.

There are a number of other possibilities.

Sounds like you want to give logistic regression a try. But think carefully about how to treat the dependent variable: is it ordinal (good > other > bad) or multinomial, or perhaps even binomial (good vs the rest).