Compare different risk scores in the same cohort

i have a small question and hope you can help me.

I want to compare the distribution of a prognostic score (ordinal variable) of two score methods to predict survival outcome that use different variables in a given cohort. So, both scores use a numerical score to assign the subjects to different prognostic groups (poor, intemerdiate, and good prognosis), but the criteria between the scores are different. I suspect that distirbution of said three groups is different.

So in short: One cohort, two ways of distribution of subjects that each result in a score, and patients are assigned one of three prognostic groups. How to compare the distribution of the subjects into the score groups (score is ordinal, not really quantitative).

My question is: Can is just use Chi²test to test to do this? It is the same cohort, but technically i would guess i just try to find imbalances between two groups (one prognostic score vs. the other)? Or do i need a different test in that special circumstance?

I am looking forward to hearing your opinion on this and thank you in advance.

Kind regards,

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