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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.

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  • $\begingroup$ The author has his preprint here $\endgroup$
    – Glen_b
    Nov 4, 2014 at 1:21
  • $\begingroup$ it may also be worth people looking at the comment by Antonio Martín Andrés the following year (there's a link on the pubmed page for the Campbell paper), since it raises a number of issues. One can always use simulation from the relevant model for one's situation instead. $\endgroup$
    – Glen_b
    Nov 4, 2014 at 1:30
  • $\begingroup$ There's also this followup by Campbell. [See also this and perhaps this.] $\endgroup$
    – Glen_b
    Nov 4, 2014 at 2:09
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    $\begingroup$ Having read the paper (&some of the other material), and A. Martín Andrés' comment, my guess is that - if you're looking for a basis akin to that used by Campbell, unless there's further research I haven't seen, it's probably an open question (unless someone wants to undertake a much larger study than Campbell did just to answer the question). The justification that the $N-1$ adjustment corresponds to an unbiased estimate of the variance of the proportion should still carry over to larger tables, but there's no clear reason to expect the type I error rate will also correspond in rxc tables. $\endgroup$
    – Glen_b
    Nov 4, 2014 at 4:16

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