Correlation ratio

A characteristic of dependence between random variables. The correlation ratio of a random variable $ Y $ relative to a random variable $ X $ is the expression

$$ \eta _ {Y \mid X } ^ {2} = \ 1 - {\mathsf E} \left [ \frac{ {\mathsf D} ( Y \mid X) }{ {\mathsf D} Y } \right ] , $$

where $ {\mathsf D} Y $ is the variance of $ Y $, $ {\mathsf D} ( Y \mid X) $ is the conditional variance of $ Y $ given $ X $, which characterizes the spread of $ Y $ about its conditional mathematical expectation $ {\mathsf E} ( Y \mid X) $ for a given value of $ X $. Invariably, $ 0 \leq \eta _ {Y \mid X } ^ {2} \leq 1 $. The equality $ \eta _ {Y \mid X } ^ {2} = 0 $ corresponds to non-correlated random variables; $ \eta _ {Y \mid X } ^ {2} = 1 $ if and only if there is an exact functional relationship between $ Y $ and $ X $; if $ Y $ is linearly dependent on $ X $, the correlation ratio coincides with the squared correlation coefficient. The correlation ratio is non-symmetric in $ X $ and $ Y $, and so, together with $ \eta _ {Y \mid X } ^ {2} $, one considers $ \eta _ {X \mid Y } ^ {2} $( the correlation ratio of $ X $ relative to $ Y $, defined analogously). There is no simple relationship between $ \eta _ {Y \mid X } ^ {2} $ and $ \eta _ {X \mid Y } ^ {2} $. See also Correlation (in statistics).