Coefficient of multiple correlation: Difference between revisions

Revision as of 13:03, 5 April 2013

Revision as of 14:05, 5 April 2013

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==Properties==

The coefficient of multiple correlation is not computationally [[commutative]]: a regression of ''y'' on ''x'' and ''z'' will in general have a different R² than will a regression of ''z'' on ''x'' and ''y''. For example, suppose that in a particular sample the variable ''z'' is [[Correlation and dependence|uncorrelated]] with both ''x'' and ''y'', while ''x'' and ''y'' are linearly related to each other. Then a regression of ''z'' on ''y'' and ''x'' will yield an R² of zero, while a regression of ''y'' on ''x'' and ''z'' will yield a strictly positive R². This follows since the correlation of ''y'' with the best predictor based on ''x'' and ''z'' mustis bein all cases at last as large as the correlation of ''y'' with the best predictor based on ''x'' alone, and in this case with ''z'' providing no explanatory power it will be exactly as large.

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