Editing Coefficient of determination (section)

=== As squared correlation coefficient ===
In linear least squares [[multiple regression]] (with fitted intercept and slope), ''R''<sup>2</sup> equals <math>\rho^2(y,f)</math> the square of the [[Pearson correlation coefficient]] between the observed <math>y</math> and modeled (predicted) <math>f</math> data values of the dependent variable.

In a [[simple regression|linear least squares regression with a single explanator]] (with fitted intercept and slope), this is also equal to  <math>\rho^2(y,x)</math> the squared Pearson correlation coefficient between the dependent variable <math>y</math> and explanatory variable <math>x</math>.

It should not be confused with the correlation coefficient between two [[explanatory variable]]s, defined as
: <math>\rho_{\widehat\alpha,\widehat\beta} = {\operatorname{cov}\left(\widehat\alpha,\widehat\beta\right) \over \sigma_{\widehat\alpha} \sigma_{\widehat\beta}},</math>
where the covariance between two coefficient estimates, as well as their [[standard deviation]]s, are obtained from the [[Ordinary least squares#Covariance matrix|covariance matrix]] of the coefficient estimates, <math>(X^T X)^{-1}</math>.

Under more general modeling conditions, where the predicted values might be generated from a model different from linear least squares regression, an ''R''<sup>2</sup> value can be calculated as the square of the [[Pearson product-moment correlation coefficient|correlation coefficient]] between the original <math>y</math> and modeled <math>f</math> data values. In this case, the value is not directly a measure of how good the modeled values are, but rather a measure of how good a predictor might be constructed from the modeled values (by creating a revised predictor of the form {{nowrap|''α'' + ''βƒ''<sub>''i''</sub>}}).{{Citation needed|reason=The citation for the next sentence does not discuss the information in this sentence.|date=March 2017}} According to Everitt,<ref>{{cite book |last=Everitt |first=B. S. |page=78 |year=2002 |title=Cambridge Dictionary of Statistics |edition=2nd |publisher=CUP |isbn=978-0-521-81099-9}}</ref> this usage is specifically the definition of the term "coefficient of determination": the square of the correlation between two (general) variables.