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I have a question about the analysis of a unequal sample size repeated measures data with categorical response variable.

This experiment looks at 5 young and aged persons and for each of the individual we have a few data-points measuring their responses to a stimulus. The responses are categorical with four factors say A,B,C,D. So, for young person 1 we have $n^1_1$ observations, say (A,B,B,B,A,C,C, ...) and for old person 1 we have $n^2_1$ observations, say (C,D,D,D,C, ...,D). (Please note that $n_i^1$ and $n^2_i$'s might be very different). My goal is to compare the proportions of (A,B,C,D)'s among young people and older people. I am worried that an usual $\chi^2$ test might not be appropriate as the within-subject responses aren't independent.

Can anyone suggest what would be the best way to go about testing the equality of proportions?

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