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

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.

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).