# 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).