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Are there any other methods for an $m\times n$ contingency table with $m$ or $n$ greater than 2 for use with small samples ($np<5$) other than Fisher's exact test?

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Smaller expecteds than that aren't necessarily a problem for the Pearson chi-square or for the G-test. The rule you mention is more than 50 years old and is pretty conservative, but in any case, the actual discrete distribution for them can be simulated and exact p-values computed for them as well.

What's the specific situation you're in?

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