Definition
Suppose each observation is yxi where x indicates the category that observation is in and i is the label of the particular observation. Let nx be the number of observations in category x and
- and
where is the mean of the category x and is the mean of the whole population. The correlation ratio η (eta) is defined as to satisfy
which can be written as
i.e. the weighted variance of the category means divided by the variance of all samples.
It is worth noting that if the relationship between values of and values of is linear (which is certainly true when there are only two possibilities for x) this will give the same result as the square of the correlation coefficient, otherwise the correlation ratio will be larger in magnitude. It can therefore be used for judging non-linear relationships.
Read more about this topic: Correlation Ratio
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