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I am doing a chi-squared test to compare two treatment groups for any baseline differences eg gender, ethnicity etc. Most of these are 2x2 but ethnicity has at least 5 categories.

In several instances, the assumptions are not met, i.e. cells have counts smaller than 5 in them.

Therefore which test would I use rather than a chi-squared? Please remember that some of the tables are 2x3 or even 2x5.

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Recall that the assumption is that each cell has an expected count of at least 5. (The expected count in each cell is $\frac{\text{row total}\times\text{column total}}{\text{table total}}$.)

If this doesn't hold for each cell, then I might try combining cells with expected counts of less than 5 until you reach the appropriate threshold. For example, if your table of ethnicities includes White, Black, Asian, Latino, and Other and only White and Black have expected counts of above 5, then you could collapse the Asian, Latino, and Other categories together. This is a way to ensure expected counts get above 5 in each cell and then you can conduct your analysis as you initially intended with minimal loss of information.

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