CHAPTER 13 Simple Linear Regression

530 CHAPTER THIRTEEN Simple Linear Regression
The third assumption, normality, requires that the errors (ε i
) are normally distributed at
each value of X. Like the t test and the ANOVA F test, regression analysis is fairly robust
against departures from the normality assumption. As long as the distribution of the errors at
each level of X is not extremely different from a normal distribution, inferences about β 0
and β 1
are not seriously affected.
The fourth assumption, equal variance or homoscedasticity, requires that the variance of
the errors (ε i
) are constant for all values of X. In other words, the variability of Y values is the
same when X is a low value as when X is a high value. The equal variance assumption is
important when making inferences about β 0
and β 1
. If there are serious departures from this
assumption, you can use either data transformations or weighted least-squares methods (see
reference 4).
13.5 RESIDUAL ANALYSIS
In Section 13.1, regression analysis was introduced. In Sections 13.2 and 13.3, a regression
model was developed using the least-squares approach for the Sunflowers Apparel data. Is this
the correct model for these data? Are the assumptions introduced in Section 13.4 valid? In this
section, a graphical approach called residual analysis is used to evaluate the assumptions and
determine whether the regression model selected is an appropriate model.
The residual or estimated error value, e i
, is the difference between the observed (Y i
) and
predicted ( Yˆ i ) values of the dependent variable for a given value of X i
. Graphically, a residual
appears on a scatter plot as the vertical distance between an observed value of Y and the prediction
line. Equation (13.14) defines the residual.
RESIDUAL
The residual is equal to the difference between the observed value of Y and the predicted
value of Y.
ei = Yi −Yˆ
i
(13.14)
Evaluating the Assumptions
Recall from Section 13.4 that the four assumptions of regression (known by the acronym
LINE) are linearity, independence, normality, and equal variance.
Linearity To evaluate linearity, you plot the residuals on the vertical axis against the corresponding
X i
values of the independent variable on the horizontal axis. If the linear model is
appropriate for the data, there is no apparent pattern in this plot. However, if the linear model is
not appropriate, there is a relationship between the X i
values and the residuals, e i
. You can see
such a pattern in Figure 13.9. Panel A shows a situation in which, although there is an increasing
trend in Y as X increases, the relationship seems curvilinear because the upward trend
decreases for increasing values of X. This quadratic effect is highlighted in Panel B, where
there is a clear relationship between X i
and e i
. By plotting the residuals, the linear trend of X
with Y has been removed, thereby exposing the lack of fit in the simple linear model. Thus, a
quadratic model is a better fit and should be used in place of the simple linear model. (See
Section 15.1 for further discussion of fitting quadratic models.)
To determine whether the simple linear regression model is appropriate, return to the evaluation
of the Sunflowers Apparel data. Figure 13.10 provides the predicted and residual values
of the response variable (annual sales) computed by Microsoft Excel.