CORRELATING PREDICTED AND OBSERVED VALUES

One way we can measure how well the line fits is to calculate a correlation coefficient between the observed values of the dependent variable and those predicted from the regression equation. The value will be one if a perfect fit exists and close to zero if the fit is poor.

The correlation coefficient between the observed and predicted values is exactly the same value we obtained for the correlation of the two variables by using the method of the last chapter. So, now we have yet another interpretation of the correlation coefficient. It is a measure of the strength of the linear relationship between the observed values of the dependent variable and those predicted by the regression line. The correlation coefficient tells us how well the least squares line fits the data. This interpretation applies only to the absolute value of the correlation coefficient (its value if you disregard the sign). That is because even if the relationship between two variables is negative, the relationship between the observed and predicted values will be positive.

If you square the value of the correlation coefficient, you obtain yet another useful statistic. The square of the correlation coefficient tells you what proportion of the variability in the dependent variable is "explained" by the regression. What do we mean when we say that the regression "explains" variability? In general, the distance between a point and the regression line is a measure of how much variability we cannot explain with the regression line. If you compare the sum of the squared distances from the data points to the regression line with the total variability in the dependent variable, you can calculate what percent of the total variability is unexplained by the regression. The remainder of the variability is explained. This is what the square of the correlation coefficient tells you. It is the proportion of the total variability in the dependent variable that can be accounted for by the independent variable.