I would choose one race to be a baseline category, and express the other races as ratios from that category. Eg: in the 2016 US Census,
- 61% of people included were Caucasian alone,
- 18% Hispanic,
- 13% African-American,
- 6% Asian, and
- <2% everyone else.
I would represent this as
- Hispanic = 18 / 61 = 0.295 times as likely as Caucasian alone,
- AA = 0.213 times,
- Asian = 0.098 times, and
- Everyone else = 0.033 times.
This removes one of the columns, while still preserving the change from baseline (so that you can explain all five groups). This means you are explaining if the different proportions of the races / ethnicities have an effect on the outcome.