Logit model fails - collinearity between categoricals. How to fix?

#1
Hello all!

I have a logistic regression model where I'm trying to determine the effects of two categorical variables. However, one of the variables can be predicted from the other. For example:

Obs # / y-value / Fruit or Veg? / Type
1 / 1 / Fruit / Apple
2 / 0 / Fruit / Banana
3 / 0 / Vegetable / Carrot
4 / 0 / Fruit / Apple
5 / 0 / Vegetable / Carrot
6 / 1 / Vegetable / Bok choy

What I'd like to do is include both the 'fruit or veg?' and 'type' variables in the model. Conceptually, that would allow me to determine the effect of fruit/vegetable _and_ the effect of each individual fruit/vegetable type. For example, the odds ratio associated with fruit is 0.8, while the odds ratio associated with Apple is 1.3.

Unfortunately, the model won't run because fruit/veg can obviously be predicted by type. Is there anything I can do to include both of these variables in the model?

Thank you!