2. Chapter 13 Multiple Regression
Section 13.4
Checking a Regression Model Using Residual Plots

3. Assumptions for Inference with a Multiple Regression Model
The regression equation approximates well the true relationship between the predictors and the mean of y.
The data were gathered randomly.
y has a normal distribution with the same standard deviation at each combination of predictors.

4. Checking Shape and Detecting Unusual Observations
To test Assumption 3 (the conditional distribution of y is normal at any fixed values of the explanatory variables):
Construction a histogram of the standardized residuals.
The histogram should be approximately bell-shaped.
Nearly all the standardized residuals should fall between -3 and +3. Any residual outside these limits is a potential outlier.

5. Example: House Selling Price
For the house selling price data, a MINITAB histogram of the standardized residuals for the multiple regression model predicting selling price by the house size and the lot size was created and is displayed on the following slide.

6. Example: House Selling Price
Figure 13.4 Histogram of Standardized Residuals for Multiple Regression Model
Predicting Selling Price. Question: Give an example of a shape for this histogram that
would indicate that a few observations are highly unusual.

7. Example: House Selling Price
The residuals are roughly bell shaped about 0.
They fall mostly between about -3 and +3.
No severe nonnormality is indicated.

8. Plotting Residuals against Each Explanatory Variable
Plots of residuals against each explanatory variable help us check for potential problems with the regression model.
Ideally, the residuals should fluctuate randomly about the horizontal line at 0.
There should be no obvious change in trend or change in variation as the values of the explanatory variable increases.

9. Plotting Residuals against Each Explanatory Variable
Figure 13.5 Possible Patterns for Residuals, Plotted Against an Explanatory Variable. Question: Why does the pattern in (b) suggest that the effect of is not linear?