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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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  • The author has his preprint [here](http://www.iancampbell.co.uk/twobytwo/files/campbell_twobytwo_preprint.pdf) – Glen_b Nov 04 '14 at 01:21
  • 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](http://www.ncbi.nlm.nih.gov/pubmed/17315184) 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. – Glen_b Nov 04 '14 at 01:30
  • There's also [this](http://www.iancampbell.co.uk/twobytwo/results.htm) followup by Campbell. \[See also [this](http://www.iancampbell.co.uk/twobytwo/n-1_theory.htm) and perhaps [this](http://www.iancampbell.co.uk/twobytwo/sawtooth_appearance.htm).\] – Glen_b Nov 04 '14 at 02:09
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    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. – Glen_b Nov 04 '14 at 04:16

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