Correlation in Linear Regression

For the simple linear regression model, Y=Po+PiX+e, the sum of squares due to regression is  

Referring to the data of Example 1.42 and using Eq. (1.218) we calculate the simple linear correlation coefficient as  

The positive correlation means if X increases or decreases the corresponding values of Y increase or decrease too. The correlation coefficient may also be expressed by covariance of sample SXy as  

Coefficient of determination-regression statistical significance-lack of fit of regression 

It has already been mentioned that the coefficient of determination is that proportion of the total variability in the dependent variable that is accounted for by the regression equation in the independent variable(s). A value for r2 of 1 indicates that the fitted regression equation accounts for all the variability of the values of the dependent variable in the sample data. At the other extreme, a value of 0 for r2 indicates that the regression equation accounts for none of the variability. 

---