# Fisher's exact test or chi-square test

I have a 2x4 table with nominal data (the columns are simply YES/NO, the rows are four categories)

Category A: 7, 13
Category B: 15, 5
Category C: 15, 5
Category D: 19, 1


I am hoping to test the significance of a couple of the categories to each other (2x2), but also to assess the significance of the whole table (2x4, although I am not really sure how to interpret the meaning of the significance that may result here).

I understand that as my sample sizes are small (as you can see, Category D features "1" under NO, and each category has only 20 people) that I should be using Fisher's Exact Test. Is this correct?

Can Fisher's also be used in a 2x4? And what does it mean if this result is significant?

• – nico Mar 26 '14 at 13:31
• – gung - Reinstate Monica Mar 26 '14 at 14:35
• The question referred to by @gung is answered there, but the best one-word answer is Yes (in my view). – Nick Cox Mar 26 '14 at 14:49

It takes more time to post the question than to try it out. Here is Stata:

. tabi  7  13 \ 15 5 \ 15 5 \ 19 1 , exact

Enumerating sample-space combinations:
stage 4:  enumerations = 1
stage 3:  enumerations = 14
stage 2:  enumerations = 65
stage 1:  enumerations = 0

|          col
row |         1          2 |     Total
-----------+----------------------+----------
1 |         7         13 |        20
2 |        15          5 |        20
3 |        15          5 |        20
4 |        19          1 |        20
-----------+----------------------+----------
Total |        56         24 |        80

Fisher's exact =                 0.000

. ret li

scalars:
r(p_exact) =  .000426720882576
r(c) =  2
r(r) =  4
r(N) =  80


You could report the P-value as 0.0004 or 0.00043, say. So, Fisher's test can be done for tables this size. A standard chi-square test (not shown here) gives a P-value of 0.00042, which every statistical person I know would regard as essentially identical. The tests support the interpretation that is evident from eyeballing the table of an association between row and column variables.

• Thank you Nick, that is most helpful! Please excuse my naivety concerning all things stats. – Kay Mar 26 '14 at 14:58
• In terms of 2x2 testing (eg comparing A to B, and A to C, and B to D, should I be used Fishers or Chi? – Kay Mar 26 '14 at 14:59
• Same answer: Do try it and so how they compare! If you get very different P-values, watch out. Also: watch out, as choosing specific comparisons on looking at the data is widely disparaged as data snooping, if that is what you are doing. – Nick Cox Mar 26 '14 at 15:06
• thank you-- I have found significance in all my relevant tests for both fisher's and chi-squared. However, I do not understand the meaning of the 2x4 output/significance. What post-hoc analysis could I use? – Kay Mar 26 '14 at 15:38
• Sorry, but that's against my statistical religion. The post hoc analysis you most need is to think about what the results mean scientifically. Others might well give different answers. – Nick Cox Mar 26 '14 at 15:45