) percent confidence interval on the
regression coefficient, 
, is obtained as follows:
(27)Calculation of confidence intervals for multiple linear regression models are similar to those for simple linear regression models explained in Chapter 4, Simple Linear Regression Analysis.
A 100() percent confidence interval on the
regression coefficient, , is obtained as follows:(27)
The confidence interval on the regression coefficients are displayed in the Regression Information table under the Low CI and High CI columns as shown in Figure 5.13.
A 100(
) percent confidence interval on any
fitted value, 
, is given by:
(28)
where: 
In Example
5.1 (Chapter 5, Estimating
Regression Models Using Least Squares), the fitted value corresponding
to the fifth observation was calculated as 
. The 90% confidence interval on this
value can be obtained as shown in Figure 5.17.
The values of 47.3 and 29.9 used in the figure are the values of the predictor
variables corresponding to the fifth observation in Table 5.1.
|
Figure 5.17: Confidence interval for the fitted value corresponding to the fifth observation in Table 5.1. |
As explained in Chapter 4, Simple
Linear Regression Analysis, the confidence interval on a new observation
is also referred to as the prediction interval. The prediction
interval takes into account both the error from the fitted model and the
error associated with future observations. A 100(
) percent confidence interval on a
new observation, 
, is obtained as follows:
where: 
,..., 
are the levels of the predictor variables
at which the new observation, 
, needs to be obtained.
In multiple linear regression, prediction intervals should only be obtained
at the levels of the predictor variables where the regression model applies.
In the case of multiple linear regression it is easy to miss this. Having
values lying within the range of the predictor variables does not necessarily
mean that the new observation lies in the region to which the model is
applicable. For example, consider Figure 5.18 where
the shaded area shows the region to which a two variable regression model
is applicable. The point corresponding to 
th level of first predictor variable,
, and 
th level of the second predictor variable,
, does not lie in the shaded area,
although both of these levels are within the range of the first and second
predictor variables respectively. In this case, the regression model is
not applicable at this point.
|
Figure 5.18: Predicted values and region of model application in multiple linear regression. |
