Tables Larger than 2x2
For tables larger than 2x2, the chi-square distribution with the appropriate degrees of freedom provides a good approximation to the sampling distribution of Pearson's chi-square when the null hypothesis is true, and the following conditions are met:
2x2 Tables: The Standard Advice
The standard advice for 2x2 tables dates back to Cochran (1952, 1954) or earlier, and goes something like this:
2x2 Tables: Advice from Campbell (2007)
Campbell (2007) distinguishes between 3 different research designs that give rise to 2x2 tables:
I'll not go into great detail here. In a nutshell, Campbell showed via simulations that for comparative trials and cross-sectional designs, the Fisher-Irwin test and Pearson's chi-square with Yates' correction are both far too conservative (i.e., the actual Type I error rate is well below the nominal alpha). Here is the advice he gives in closing (p. 3674):
As Campbell then notes, "This policy extends the use of the chi-squared test to smaller samples (where the current practice is to use the Fisher–Irwin test), with a resultant increase in the power to detect real differences."
The 'N -1' chi-square
Where Campbell describes replacing N with N -1, he is referring to this formula for Pearson's chi-square:
chi-square = N(ad-bc)^2 / (mnrs)
'N -1' chi-square = Pearson chi-square x (N -1) / N
Equivalence of the 'N-1' Chi-square and the Linear-by-Linear Association Chi-square
This webpage used to include links to two SPSS syntax files that could be used to compute the 'N-1' chi-square and its p-value in SPSS. I have now removed those links, because it turns out that for 2x2 tables, the Linear-by-Linear Assocation chi-square computed by the SPSS CROSSTABS is equivalent to the 'N-1' chi-square. See this document for details, and this Statistics in Medicine article by Frank Busing, Sacha Dubois and me for examples of how to compute the N-1 Chi-square using SAS, Stata or R.
Pearson's Chi-square vs. the Likelihood Ratio Chi-square
The following is from Alan Agresti's book, Categorical Data Analysis.
It is not simple to describe the sample size needed for the chi-squared distribution to approximate well the exact distributions of X^2 and G^2 [also called L^2 by some authors]. For a fixed number of cells, X^2 usually converges more quickly than G^2. The chi-squared approximation is usually poor for G^2 when n/IJ < 5 [where n = the grand total and IJ = rc = the number of cells in the table]. When I or J [i.e., r or c] is large, it can be decent for X^2 for n/IJ as small as 1, if the table does not contain both very small and moderately large expected frequencies. (Agresti, 1990, p. 49)
* For matched pairs of subjects, or 2 observations per person, McNemar's chi-square (aka. McNemar's Change Test) may be appropriate. In essence, it is a chi-square goodness of fit test on the two discordant cells, with a null hypothesis stating that 50% of the changes (or disagreements) go in each direction.
** Fisher's exact test is a "conditional" test--it is conditional on the observed marginal totals (i.e., row and column totals). Unconditional exact tests are also available. For example:
Agresti, A. (1990). Categorical Data Analysis. New York: Wiley.
Busing FM, Weaver B, Dubois S. 2 × 2 Tables: a note on Campbell's recommendation. Stat Med. 2016;35(8):1354-8. doi: 10.1002/sim.6808. Epub 2015 Nov 17.
Campbell, I. Chi-squared and Fisher-Irwin tests of two-by-two tables with small sample recommendations. Stat Med. 2007; 26:3661-3675. See also www.iancampbell.co.uk/twobytwo/background.htm.
Cochran WG. The [chi-squared] test of goodness of fit. Annals of Mathematical Statistics 1952; 25:315–345.
Cochran WG. Some methods for strengthening the common [chi-squared] tests. Biometrics 1954; 10:417–451.
Yates, D., Moore, Moore, D., McCabe, G. (1999). The Practice of Statistics (1st Ed.). New York: W.H. Freeman.