Yates's correction for continuity

Yates's correction for continuity, also known as Yates's chi-squared test, is a statistical method applied to Chi-squared tests involving categorical variables to correct for the continuity of the chi-squared distribution. This correction is used when applying the chi-squared test to a 2x2 contingency table to determine if there is a significant association between two categorical variables. It was introduced by Frank Yates in 1934 as an adjustment for the approximation error when the chi-squared statistic is used to test the significance of a 2x2 contingency table.

Background

The chi-squared test is a non-parametric statistical test used to determine if there is a significant association between two categorical variables. It is based on the calculation of a chi-squared statistic, which measures the discrepancy between observed and expected frequencies in a contingency table. However, the chi-squared distribution is continuous, while the data from a 2x2 contingency table are discrete. This discrepancy can lead to an overestimation of the significance level (p-value) of the test, particularly in tables with small sample sizes or when expected frequencies are low.

Yates's Correction

Yates's correction for continuity compensates for the continuity of the chi-squared distribution by adjusting the formula used to calculate the chi-squared statistic. The correction involves subtracting 0.5 from the absolute difference between each observed frequency and its corresponding expected frequency before squaring the result in the chi-squared formula. The formula for the corrected chi-squared statistic (\( \chi^2_{Yates} \)) is:

\[ \chi^2_{Yates} = \sum \frac{(|O_i - E_i| - 0.5)^2}{E_i} \]

where \(O_i\) is the observed frequency for each cell in the table, \(E_i\) is the expected frequency for each cell, and the summation (\(\sum\)) is over all cells in the table.

Application

Yates's correction is primarily used for 2x2 contingency tables with small sample sizes (typically, when the total sample size is less than 40) or when the expected frequency in any cell of the table is less than 5. It is intended to reduce the Type I error rate—rejecting a true null hypothesis—by making the chi-squared test more conservative.

Criticism and Alternatives

While Yates's correction reduces the Type I error rate, it has been criticized for being overly conservative, potentially leading to an increased Type II error rate—failing to reject a false null hypothesis. As a result, some statisticians prefer to use the exact Fisher's exact test for small sample sizes or the uncorrected chi-squared test with a larger sample size.

Conclusion

Yates's correction for continuity is a useful adjustment for the chi-squared test when analyzing 2x2 contingency tables, especially with small sample sizes. However, researchers should be aware of its limitations and consider alternative methods based on the specific context of their study.