# Test for independence in 3x2x2 table with small expected cell values

I have three unordered categorical variables. I have made a 3x2x2 table, but I'm uncertain of how to test for independence. The usual manner is a chi-squared statistic, but I think this is inappropriate when the expected values of the cells are small. I have one cell that is 0, so this is an issue for me. In a 2x2 table, I think that the Fisher-Irwin exact test would work, but I can't find anything covering a table with three dimensions. Any ideas?

• The actual value in a cell is not its expected value--but you seem to equate them. Could you clarify this by describing the typical expected values in the cells? Even giving the total count would help, because we could divide that by $3\times 2\times 2 = 12$ to estimate an average expected value. – whuber Mar 21 '15 at 17:13
• Sure - I should've included that in the original post, sorry. Total N is about 140 (so about 11.6 avg Exp Val). – Alex Mar 22 '15 at 3:03
• Thank you. Since most rules of thumb for applying the $\chi^2$ distribution deem "small" to be less than $5$ (maybe some really conservative ones use $10$ as a threshold), your average value of $11.6$ doesn't look small. You therefore might not have any problem at all unless two or more of the marginal totals in this table are very imbalanced. Perhaps you could display the entire table (since it's only 12 numbers)? – whuber Mar 22 '15 at 17:40