Correlation (Multiple)

Let Y be one variable, and (X₁, X₂ , ..., X_n ) a set of other variables. Let X be a linear combination of the X_i's :

X = S_i a_iX_i 

and consider the correlation coefficient r(X, Y).

When the coefficients a_i are made to vary in every possible way, the value of r changes. It can be shown that, in general, there is a single set of values of the coefficients that maximizes r. This largest possible value of r(X, Y) is usually denoted R, and is called the Multiple Correlation Coefficient between Y and the set of variables (X₁, X₂ , ..., X_n ).

The Multiple Correlation Coefficient plays a central role in Multiple Linear Regression, as R² is then equal to the ratio of the explained variance to the total variance, and is therefore a measure of the quality of the regression. So, in this respect, there is a complete similarity between Simple and Multiple Linear Regression.

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