This document discusses the coefficient of determination (R2) in statistical regression models. It notes that R2 can be artificially inflated to 1 by adding extra terms to the model, and that a high R2 does not necessarily mean the model is a good fit to the data or will accurately predict future observations. The dispersion of the x variable can also impact R2 values. R2 does not directly measure the slope or appropriateness of the regression model.

1. Note that each pair of observations satisfies the relationship
is called the residual
The least squares estimates of intercept and slope in the simple linear regression model are
and
Where
and

3. is the total corrected sum of squares of y

4. CONFIDENCE INTERVAL ON THE MEAN RESPONSE

5. Oxygen Purity Confidence Interval on the Mean Response We will construct a 95% confidence interval about the
mean response. The fitted model is
Suppose that we are interested in predicting mean oxygen purity when x0 = 1.00 % Then
95% confidence interval for x=1.00 is given below

6. Prediction Interval

7. Prediction Interval

8. Coefficient of determination

9. The statistic R2 should be used with caution because it is always possible to make R2 unity by simply adding enough
terms to the model. For example, we can obtain a “perfect” fit to n data points with a polynomial of degree n − 1. In
general, R2 will increase if we add a variable to the model, but this does not necessarily imply that the new model is
superior to the old one. Unless the error sum of squares in the new model is reduced by an amount equal to the original
error mean square, the new model will have a larger error mean square than the old one because of the loss of 1 error
degree of freedom. Thus, the new model will actually be worse than the old one. The magnitude of R2 is also impacted
by the dispersion of the variable x. The larger the dispersion, the larger the value of R2 will usually be. There are several
misconceptions about R2 . In general, R2 does not measure the magnitude of the slope of the regression line. A large
value of R2 does not imply a steep slope. Furthermore, R2 does not measure the appropriateness of the model because
it can be artificially inflated by adding higher order polynomial terms in x to the model. Even if y and x are related in a
nonlinear fashion, R2 will often be large. For example, R2 for the regression equation in Fig. 11-6(b) will be relatively
large even though the linear approximation is poor. Finally, even though R2 is large, this does not necessarily imply that
the regression model will provide accurate predictions of future observations.

10. The statistic R2 should be used with caution because it is always possible to make R2 unity by simply adding enough
terms to the model. For example, we can obtain a “perfect” fit to n data points with a polynomial of degree n − 1. In
general, R2 will increase if we add a variable to the model, but this does not necessarily imply that the new model is
superior to the old one. Unless the error sum of squares in the new model is reduced by an amount equal to the original
error mean square, the new model will have a larger error mean square than the old one because of the loss of 1 error
degree of freedom. Thus, the new model will actually be worse than the old one. The magnitude of R2 is also impacted
by the dispersion of the variable x. The larger the dispersion, the larger the value of R2 will usually be. There are several
misconceptions about R2 . In general, R2 does not measure the magnitude of the slope of the regression line. A large
value of R2 does not imply a steep slope. Furthermore, R2 does not measure the appropriateness of the model because it
can be artificially inflated by adding higher order polynomial terms in x to the model. Even if y and x are related in a
nonlinear fashion, R2 will often be large. For example, R2 for the regression equation in Fig. 11-6(b) will be relatively
large even though the linear approximation is poor. Finally, even though R2 is large, this does not necessarily imply that
the regression model will provide accurate predictions of future observations.