Chapter 9 Multiple Regression 363
predictors can be predicted from the others, any single predictor can be
eliminated without losing much.
You might think that you could just drop from the model all the terms
that are not signifi cant. However, it is important to bear in mind that the in-
dividual tests are correlated, so each of them changes when you drop one of
the terms. If you drop the least-signifi cant term, others might then become
signifi cant. A frequently used strategy for reducing the number of predictors
involves the following steps:
- Eliminate the least-signifi cant predictor if it is not signifi cant.
- Refi t the model.
- Repeat Steps 1 and 2 until all predictors are signifi cant.
In the exercises, you’ll get a chance to rerun this model and eliminate all
non signifi cant variables. For now, examine the model and see whether any
assumptions have been violated.
Testing Regression Assumptions
There are a number of useful ways to look at the results produced by mul-
tiple linear regression. This section reviews the four common plots that can
help you assess the success of the regression.
- Plotting dependent variables against the predicted values shows how
well the regression fi ts the data.
- Plotting residuals against the predicted values magnifi es the vertical
spread of the data so you can assess whether the regression assumptions
are justifi ed. A curved pattern to the residuals indicates that the model
does not fi t the data. If the vertical spread is wider on one side of the
plot, it suggests that the variance is not constant.
- Plotting residuals against individual predictor variables can sometimes
reveal problems that are not clear from a plot of the residuals versus the
predicted values.
- Creating a normal plot of the residuals helps you assess whether the
regression assumption of normality is justifi ed.
Observed versus Predicted Values
How successful is the regression? To see how well the regression fi ts the
data, plot the actual Calculus values against the predicted values stored in
B29:B109. (You can scroll down to view the residual output.) To plot the ob-
served calculus scores versus the predicted scores, you must fi rst place the
data on the same worksheet.
