View source for Yates's correction for continuity

In [[Statistics|statistics]], '''Yates's correction for continuity''' (or '''Yates's chi-squared test''') is used in certain situations when testing for [[Independence (probability theory)|independence]] in a [[Contingency table|contingency table]]. It aims
at correcting the error introduced by assuming that the discrete probabilities of frequencies in the table can be approximated by a continuous distribution ([[Chi-squared distribution|chi-squared]]). In some cases, Yates's correction may adjust too far, and so its current use is limited.

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

To reduce the error in approximation, [[Biography:Frank Yates|Frank Yates]], an {{wipe|England}} [[Statistician|statistician]], suggested a correction for continuity that adjusts the formula for [[Pearson's chi-squared test]] by subtracting 0.5 from the difference between each observed value and its expected value in a 2&nbsp;&times;&nbsp;2 contingency table.<ref name=Yates>[[Biography:Frank Yates|Yates, F]] (1934). "Contingency table involving small numbers and the χ<sup>2</sup> test". ''Supplement to the [[Organization:Journal of the Royal Statistical Society|Journal of the Royal Statistical Society]]'' '''1'''(2): 217&ndash;235. {{JSTOR|2983604}}</ref> This reduces the chi-squared value obtained and thus increases its [[P-value|p-value]].

The effect of Yates's 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's correction may tend to overcorrect. This can result in an overly conservative result that fails to reject the [[Null hypothesis|null hypothesis]] when it should (a type II error). So it is suggested that Yates's correction is unnecessary even with quite low sample sizes,<ref name=Sokal1981>Sokal RR, Rohlf F.J. (1981). ''Biometry: The Principles and Practice of Statistics in Biological Research.'' Oxford: W.H. Freeman, {{isbn|0-7167-1254-7}}.</ref> such as:

:<math> \sum_{i=1}^N O_i = 20 \, </math>

The following is Yates's corrected version of [[Pearson's chi-squared test|Pearson's chi-squared statistics]]:

:<math> \chi_\text{Yates}^2 = \sum_{i=1}^{N} {(|O_i - E_i| - 0.5)^2 \over E_i}</math>

where:

:''O<sub>i</sub>'' = an observed frequency
:''E<sub>i</sub>'' = an expected (theoretical) frequency, asserted by the null hypothesis
:''N'' = number of distinct events

== 2 &times; 2 table ==
As a short-cut, for a 2&nbsp;×&nbsp;2 table with the following entries:

{| class="wikitable"
! &nbsp; !! S !! F !! &nbsp;
|-
! A
| ''a'' || ''b'' || a+b
|-
! B
| ''c'' || ''d'' || c+d
|-
! &nbsp;
| a+c || b+d || ''N''
|}

: <math>\chi_\text{Yates}^2 = \frac{N(|ad - bc| - N/2)^2}{(a+b) (c+d) (a+c) (b+d)}.</math>

In some cases, this is better.

: <math>\chi_\text{Yates}^2 = \frac{N( \max(0, |ad - bc| - N/2) )^2}{N_S N_F N_A N_B}.</math>
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Some sources say that this correction should be used when the expected frequency is less than 10 {{Citation needed|date=February 2007}}, yet other sources say that Yates's corrections should always be applied {{Citation needed|date=February 2007}}. However, in situations with large sample sizes, using the correction will have little effect on the value of the test statistic, and hence the p-value obtained.
-->

== See also ==
* [[Continuity correction]]
* [[Binomial proportion confidence interval#Wilson score interval with continuity correction|Wilson score interval with continuity correction]]

==References==
{{reflist}}

[[Category:Statistical tests for contingency tables]]

{{Sourceattribution|Yates's correction for continuity}}