Interpretation of odds ratio in a multivariable logistic model when dealing with categorical independent variables (more than 2 levels)

Question

I have doubts concerning the correct interpretation of the odds ratio when I am dealing with categorical variables. I try to explain it better:
Suppose having a multivariable logistic model (glm) with gender as one of the possible predictor. Then, I compute the odds ratio with the corresponding confidence intervals and then I can get the interpretation regarding my variable gender. For example, I can conclude that "Fixing the other variables, males are more likely to ... than females".
Now, suppose instead of a binary variable as gender, there is a variable with 4 levels (and none of them can be used as a reference one, because that has no meaning for my particular model). For example, color with 4 different values (yellow, blue, green,red).
My question is: what is the correct interpretation of the concept of odds ratio in this case, when I cannot compare the 4 different levels ? Is there a way to compute the odds for all the possible levels?

Hope it is clear enough.

Answer 1

To retain all levels of your categorical variable in the model output, you would need to drop the intercept from the model (i.e. y ~ 0 + color).

However, it rarely makes sense to fit a logistic regression without an intercept, because it implies that $P(Y=1 |\mathbf{x}=0)=0.5$ (which is often not the case).

Take a look a these questions & answers: 1, 2, 3.

This page might also be helpful.