Feature Selection On Nominal Variables

I have been in a situation where l needed to carry out a statistical test on a one-hot-encoded nominal variable. It turns out that one-hot-encoding such variables dramatically increases the dimensionality of the dataset. The challenge l always face is that l don't really know how to carry out a statistical test of such a variable in its one-hot-encoded form since it introduces more columns for this variable.

I thought of doing univariate tests on each column but l am not sure

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