Correlation ratio

Suppose each observation is y_xi where x indicates the category that observation is in and i is the label of the particular observation. Let n_x be the number of observations in category x and

${\displaystyle {\overline {y}}_{x}={\frac {\sum _{i}y_{xi}}{n_{x}}}}$ and ${\displaystyle {\overline {y}}={\frac {\sum _{x}n_{x}{\overline {y}}_{x}}{\sum _{x}n_{x}}},}$

where ${\displaystyle {\overline {y}}_{x}}$ is the mean of the category x and ${\displaystyle {\overline {y}}}$ is the mean of the whole population. The correlation ratio η (eta) is defined as to satisfy

${\displaystyle \eta ^{2}={\frac {\sum _{x}n_{x}({\overline {y}}_{x}-{\overline {y}})^{2}}{\sum _{x,i}(y_{xi}-{\overline {y}})^{2}}}}$

which can be written as

${\displaystyle \eta ^{2}={\frac {{\sigma _{\overline {y}}}^{2}}{{\sigma _{y}}^{2}}},{\text{ where }}{\sigma _{\overline {y}}}^{2}={\frac {\sum _{x}n_{x}({\overline {y}}_{x}-{\overline {y}})^{2}}{\sum _{x}n_{x}}}{\text{ and }}{\sigma _{y}}^{2}={\frac {\sum _{x,i}(y_{xi}-{\overline {y}})^{2}}{n}},}$

i.e. the weighted variance of the category means divided by the variance of all samples.

If the relationship between values of ${\displaystyle x}$ and values of ${\displaystyle {\overline {y}}_{x}}$ is linear (which is certainly true when there are only two possibilities for x) this will give the same result as the square of Pearson's correlation coefficient; otherwise the correlation ratio will be larger in magnitude. It can therefore be used for judging non-linear relationships.

The correlation ratio ${\displaystyle \eta }$ takes values between 0 and 1. The limit ${\displaystyle \eta =0}$ represents the special case of no dispersion among the means of the different categories, while ${\displaystyle \eta =1}$ refers to no dispersion within the respective categories. ${\displaystyle \eta }$ is undefined when all data points of the complete population take the same value.