# When do I use the correction for continuity when performing a chi-square analysis on a 2x2 table?

SPSS and other software output a chi-square which is corrected for continuity (also known as Yates correction) and applies only to 2 x 2 tables. A summary of the wikipedia article here is given below.

In statistics, Yates' correction for continuity (or Yates' chi-square test) is used in certain situations when testing for independence in a contingency table. In some cases, Yates' correction may adjust too far, and so its current use is limited.

Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution. This assumption is not quite correct, and introduces some error.

To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for continuity which adjusts the formula for Pearson's chi-square test by subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table (Yates, 1934). This reduces the chi-square value obtained and thus increases its p-value.

The effect of Yates' correction is to prevent overestimation of statistical significance for small data. This formula is chiefly used when at least one cell of the table has an expected count smaller than 5. Unfortunately, Yates' correction may tend to overcorrect. This can result in an overly conservative result that fails to reject the null hypothesis when it should. So it is suggested that Yates' correction is unnecessary even with quite low sample sizes (Sokal and Rohlf, 1981), such as total sample sizes less than or equal to 20.

Note, however, that Ian Campbell (2007) mentions here that the exact test is too conservative for 2x2 tables and suggests, instead, using an alternative chi-square, The N-1 chi-square, which performs well provided all expected counts are 1 or greater. This chi-square is outputted by SPSS CROSSTABS and is called the linear-by-linear chi-square test and it may also be computed using the on-line calculator on Ian Campbell's website.

References

Campbell I (2007) Chi-squared and Fisher-Irwin tests of two-by-two tables with small sample recommendations. Statistics in Medicine, 26, 3661 - 3675. A pre-print of this paper is available in pdf format from here.

Sokal RR, Rohlf FJ (1981). Biometry: The Principles and Practice of Statistics in Biological Research. Oxford: W.H. Freeman, ISBN 0716712547.

Yates, F (1934). "Contingency table involving small numbers and the χ2 test". Supplement to the Journal of the Royal Statistical Society **1(2)** 217-235.