# What regression models can choose between more than two categories?

Logistic regression creates a model that predicts one of two possible responses for a given set of predictors. What if I want to choose between three responses?

I tried this by using a logistic regression, except instead of defining my two outcomes as 0 and 1; I defined a third outcome at 0.5. This seemed to be going great, until I realized that this model would never produce an estimate of probability in all three categories. For example, if outcome A = 0; B = 0.5; and C = 1, and if the regression result for a certain set of predictor variables is 0.75, I could interpret that as saying the probability of B is 0.5 an the probability of C is 0.5. However, the observed probability of all three outcomes is > 0 for any set of predictor variables, so this model is not so useful.

What sorts of regression-type models will estimate probabilities for three or more outcomes, the way that logistic regression estimates probabilities for two outcomes?

• why does it have to be a a regression model? From what you're writing, this sounds like a standard multi-class classification problem Dec 12, 2017 at 21:22
• @Rickyfox, typical ML classifiers (ANNs, SVMs, random forests, etc.) can be considered regression models in an abstract sense. Dec 14, 2017 at 16:16

If the outcome categories are not ordered, you would use multinomial logistic regression. This model makes the assumption of the independence of irrelevant alternatives. This means that the odds that event $A$ occurs instead of event $B$ would be the same whether the event $C$ is among the possible outcomes or not. However, this is known to not hold of human decision making (e.g., marketers can nudge consumers towards one product vs. another by including a specially designed third product). Alternative models that do not require this assumption are discussed starting on p. 28 of the R mlogit package's vignette (pdf).