Similarly the overall suitability of the sample regression function can be expressed using a measure of goodness of fit known as the coefficient of determination. It is denoted with notation r2 or R2 depending on the number of variables involved in the regression modeling.
The coefficient of determination simply tells the fraction of variability in the dependent variable that is explained using the independent variables. Fraction of variability in the dependent variable is nothing but the variance of the dependent variable.
Variability explained using the independent variable (in case of two variable regression model discussed in the previous section) is nothing but;
The fraction of variability explained using the regression model is;
In other words the same quantity can be expressed as given below;
The square root of the coefficient of determination is termed as the sample correlation coefficient and it gives a measure of association between two variables.
Example
For the example problem discussed in the previous lecture the coefficient of determination and the sample correlation coefficient between the gap and the fraction of TWs can be found as shown below;
The coefficient of determination is,
The sample correlation coefficient between the gap and the fraction of TWs is .52.