Contingency Coefficient

A contingency table, as in the chi-squared test of independence, reveals if two sets of data or groups are independent or not. It does not reveal the strength of the dependence. The contingency coefficient is a non-parametric measure of the association for cross-classification data.

After calculating the chi-squared value from the contingency table exercise, we can use that value to determine the contingency coefficient with the following formula.

$$ \large\displaystyle C=\sqrt{\frac{{{\chi }^{2}}}{n+{{\chi }^{2}}}}$$

where

$$ \large\displaystyle {{\chi }^{2}}=\sum{\frac{{{\left( O-E \right)}^{2}}}{E}}$$

and, n = total sample size.

See the article on the  $- \chi^2 -$ test of independence for information on the contingency table and calculations.

If C is zero of very near zero there is no association between the two groups. If the C value is closer to 1 there is a strong negative or positive association.

One of the disadvantages to the coefficient is it generally does not achieve 1 even if there is completely dependent on one another. You can determine the theoretical max C value with r (the number of rows and columns – which must be equal)

$$ \large\displaystyle {{C}_{\max }}=\sqrt{\frac{r-1}{r}}$$

For a two by two table, C has a possible maximum of 0.707.

Example

Using the value of chi-squared from the test of independence example of 7.5, which has a sample size of 40, we find

$$ \large\displaystyle C=\sqrt{\frac{7.5}{40+7.5}}=0.397$$

Which suggest a modest association between a person’s position and their opinion.

Related:

Kendall Coefficient of Concordance (article)

Chi-Square Test of Independence (article)

Spearman Rank Correlation Coefficient (article)