Campbell (2007) recommended what he called the "N-1" correction for chi-square tests performed on two-by-two contingency tables. He described the correction as:
"the K. Pearson chi-squared test but with N replaced by N − 1".
The purpose of the correction is to reduce the rate of Type I errors, for contingency tables where the minimum expected frequency is at least 1. Otherwise, he recommended using the Fisher-Irwin test. Campbell's recommendation has been encouraged by posters to CrossValidated (example).
My question is whether the 'N-1' correction should be used for r x c tables generally? For example, would one apply it to a chi-square test for a three-by-five table? Or is it's use limited to two-by-two tables?
Campbell, Ian. 2007. Chi-squared and Fisher-Irwin tests of two-by-two tables with small sample recommendations. Statistics in Medicine 26: 3661-3675.