# 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, 2014 at 13:31
• Mar 26, 2014 at 14:35
• The question referred to by @gung is answered there, but the best one-word answer is Yes (in my view). Mar 26, 2014 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, 2014 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, 2014 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. Mar 26, 2014 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, 2014 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. Mar 26, 2014 at 15:45