 Coefficient of Determination Section 4.3 Alan Craig

 2 Objectives Compute and interpret the coefficient of determination.

 3 Coefficient of Determination The coefficient of determination, R 2, measures the percentage of the total variation in the response variable that is explained by the least-squares regression. R 2 is calculated by squaring the linear correlation coefficient, r.

 4 On the Calculator r and R 2 are part of the calculator output for a linear regression with DiagnosticsOn.

 5 Least-Squares Regression Recall that the least-squares regression line minimizes the sum of the squared errors (residuals) =  residuals 2 Error or residual = actual - predicted

 6 Estimating y If I have no information about values of the predictor variable x, then my best guess for y is the mean of y: and the deviation is the actual value minus the mean.

 7 Actual Deviation Mean Total Deviation

 8 Estimating y However, if I have additional data on the values of x and corresponding values of y, I can often do better by calculating the regression of y on x. Part of the total deviation is now explained by the regression equation although some of the deviation is still unexplained (unless there is a perfect linear correlation).

 9 Actual Deviation Mean Predicted Unexplained Deviation Deviation explained by the regression

 10 Deviation Note that That is, y = mean of y + explained deviation + unexplained deviation

 11 Deviation Or That is, Total deviation = explained deviation + unexplained deviation

 12 Variation The total variation of y is The explained variation is The unexplained variation is

 13 Variation R2R2

 14 Interpreting R 2 Thus, R 2 is the percentage of variation in the response variable, y, that is explained by the predictor variable x.

 15 Interpreting R 2 Using our example from Sections 4.1 and 4.2 (problem 10, p. 172), R 2 =0.9835=98.35%, so the predictor variable, Carats, explains 98.35% of the variation in the response variable, Price.

 16 Questions ???????????????