Coefficient of multiple correlation: Difference between revisions

Revision as of 22:42, 20 December 2016

Revision as of 22:43, 20 December 2016

Line 31:

==Properties==

With more than two variables being related to each other, the value of the coefficient of multiple correlation depends on the choice of dependent variable: a regression of <math>y</math> on <math>x</math> and <math>z</math> will in general have a different <math>R</math> than will a regression of <math>z</math> on <math>x</math> and <math>y</math>. For example, suppose that in a particular sample the variable <math>z</math> is [[Correlation and dependence|uncorrelated]] with both <math>x</math> and <math>y</math>, while <math>x</math> and <math>y</math> are linearly related to each other. Then a regression of <math>z</math> on <math>y</math> and <math>x</math> will yield an <math>R</math> of zero, while a regression of <math>y</math> on <math>x</math> and <math>z</math> will yield a strictly positive <math>R</math>. This follows since the correlation of <math>y</math> with theits best predictor based on <math>x</math> and <math>z</math> is in all cases at least as large as the correlation of <math>y</math> with theits best predictor based on <math>x</math> alone, and in this case with <math>z</math> providing no explanatory power it will be exactly as large.

==References==