The assumptions behind ordinal regression are similar to the ones for logistic regression.
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Checklist
Ordinal outcome
The y-variable has to be ordinal.
Independence of errors
Data should be independent, i.e. not derived from any dependent samples design, e.g. before-after measurements/paired samples.
Correct model specification
Your model should be correctly specified. This means that the x-variables that are included should be meaningful and contribute to the model. No important (confounding) variables should be omitted (often referred to as omitted variable bias).
No multicollinearity
Multicollinearity may occur when two or more x-variables that are included simultaneously in the model are strongly correlated with each another. Actually, this does not violate the assumptions, but is does create greater standard errors which makes it harder to reject the null hypothesis.
Parallel lines/Proportional odds
The relationship between each pair of outcome groups is the same, i.e. the coefficients that describe the relationship between, for example, the lowest versus all higher categories of the outcome variable are the same as those that describe the relationship between the next lowest category and all higher categories, and so on.