Timeline for How big can a chi-square table be?

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Jul 27, 2017 at 23:31	history	edited	Glen_b	CC BY-SA 3.0	added 386 characters in body
Jul 27, 2017 at 23:24	comment	added	Glen_b		Where the expected counts don't vary much the number can often go quite a lot lower than 5 and still keep a reasonable approximation. If there's a lot of variation in expectation the approximation tends to be poorer.
Jul 27, 2017 at 14:41	comment	added	whuber♦		@Stephan quotes the rule correctly. However, many have found that this rule is unnecessarily strict. The $\chi^2$ distribution often works well provided not too many of the cells have expectations less than $5$ (and watch out for cells with expectations of zero!). "Not too many" may be less than $25\%$ or so. I suspect that the larger the table gets, the more you can tolerate low-expectation cells for this analysis. At some point the whole exercise becomes pointless: with a huge table, the model grows too complicated and should be replaced (typically) by a mixed model.
Jul 27, 2017 at 12:42	comment	added	Stephan Kolassa		+1. One common rule of thumb is to have at least 5 expected entries in each cell. With $n=81$ and $6\times 4=24$ cells, we'd expect $81/24=3.4$ entries, so I'd be careful with the $\chi^2$ approximation here. However, the example is still small enough that one can approximate the $p$ value by simulations, e.g., using R's `chisq.test` function with `simulate.p.value=TRUE`.
Jul 27, 2017 at 12:04	history	answered	Glen_b	CC BY-SA 3.0