Difference between revisions of "Regression coefficient"
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| − | + | A coefficient of an independent variable in a [[Regression|regression]] equation. For example, in the linear regression equation $ {\mathsf E} ( Y \mid X = x ) = \beta _ {0} + \beta _ {1} x $, | |
| + | connecting the random variables $ Y $ | ||
| + | and $ X $, | ||
| + | the regression coefficients $ \beta _ {0} $ | ||
| + | and $ \beta _ {1} $ | ||
| + | are given by | ||
| + | |||
| + | $$ | ||
| + | \beta _ {0} = m _ {2} - \rho | ||
| + | \frac{\sigma _ {2} }{\sigma _ {1} } | ||
| + | m _ {1} ,\ \ | ||
| + | \beta _ {1} = \rho | ||
| + | \frac{\sigma _ {2} }{\sigma _ {1} } | ||
| + | , | ||
| + | $$ | ||
| + | |||
| + | where $ \rho $ | ||
| + | is the [[Correlation coefficient|correlation coefficient]] of $ X $ | ||
| + | and $ Y $, | ||
| + | $ m _ {1} = {\mathsf E} X $, | ||
| + | $ m _ {2} = {\mathsf E} Y $, | ||
| + | $ \sigma _ {1} ^ {2} = {\mathsf D} X $, | ||
| + | and $ \sigma _ {2} ^ {2} = {\mathsf D} Y $. | ||
| + | The calculation of estimates for regression coefficients (sample regression coefficients) is a fundamental problem of [[Regression analysis|regression analysis]]. | ||
Latest revision as of 08:10, 6 June 2020
A coefficient of an independent variable in a regression equation. For example, in the linear regression equation $ {\mathsf E} ( Y \mid X = x ) = \beta _ {0} + \beta _ {1} x $,
connecting the random variables $ Y $
and $ X $,
the regression coefficients $ \beta _ {0} $
and $ \beta _ {1} $
are given by
$$ \beta _ {0} = m _ {2} - \rho \frac{\sigma _ {2} }{\sigma _ {1} } m _ {1} ,\ \ \beta _ {1} = \rho \frac{\sigma _ {2} }{\sigma _ {1} } , $$
where $ \rho $ is the correlation coefficient of $ X $ and $ Y $, $ m _ {1} = {\mathsf E} X $, $ m _ {2} = {\mathsf E} Y $, $ \sigma _ {1} ^ {2} = {\mathsf D} X $, and $ \sigma _ {2} ^ {2} = {\mathsf D} Y $. The calculation of estimates for regression coefficients (sample regression coefficients) is a fundamental problem of regression analysis.
How to Cite This Entry:
Regression coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regression_coefficient&oldid=48474
Regression coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regression_coefficient&oldid=48474
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article