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A characteristic of dependence between random variables. The correlation ratio of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265801.png" /> relative to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265802.png" /> is the expression
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265803.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265804.png" /> is the variance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265806.png" /> is the conditional variance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265807.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265808.png" />, which characterizes the spread of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c0265809.png" /> about its conditional mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658010.png" /> for a given value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658011.png" />. Invariably, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658012.png" />. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658013.png" /> corresponds to non-correlated random variables; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658014.png" /> if and only if there is an exact functional relationship between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658016.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658017.png" /> is linearly dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658018.png" />, the correlation ratio coincides with the squared correlation coefficient. The correlation ratio is non-symmetric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658020.png" />, and so, together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658021.png" />, one considers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658022.png" /> (the correlation ratio of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658023.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658024.png" />, defined analogously). There is no simple relationship between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026580/c02658026.png" />. See also [[Correlation (in statistics)|Correlation (in statistics)]].
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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}  = \
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1 - {\mathsf E} \left [
 +
 
 +
\frac{ {\mathsf D} ( Y \mid  X) }{ {\mathsf D} Y }
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 +
\right ] ,
 +
$$
 +
 
 +
where $  {\mathsf D} Y $
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is the variance of $  Y $,  
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$  {\mathsf D} ( Y \mid  X) $
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is the conditional variance of $  Y $
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given $  X $,  
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which characterizes the spread of $  Y $
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about its conditional mathematical expectation $  {\mathsf E} ( Y \mid  X) $
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for a given value of $  X $.  
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Invariably, 0 \leq  \eta _ {Y \mid  X }  ^ {2} \leq  1 $.  
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The equality $  \eta _ {Y \mid  X }  ^ {2} = 0 $
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corresponds to non-correlated random variables; $  \eta _ {Y \mid  X }  ^ {2} = 1 $
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if and only if there is an exact functional relationship between $  Y $
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and $  X $;  
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if $  Y $
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is linearly dependent on $  X $,  
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the correlation ratio coincides with the squared correlation coefficient. The correlation ratio is non-symmetric in $  X $
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and $  Y $,  
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and so, together with $  \eta _ {Y \mid  X }  ^ {2} $,  
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one considers $  \eta _ {X \mid  Y }  ^ {2} $(
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the correlation ratio of $  X $
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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)|Correlation (in statistics)]].

Latest revision as of 17:31, 5 June 2020


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).

How to Cite This Entry:
Correlation ratio. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correlation_ratio&oldid=13666
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article