Residual Analysis in Regression
Because a linear regression model is not always appropriate for the data,
you should assess the appropriateness of the model by defining
residuals and examining residual plots.
Residuals
The difference between the observed value of the dependent variable
(y) and the predicted value (ŷ) is called the
residual (e). Each data point has one
residual.
Residual = Observed value - Predicted value
e = y - ŷ
Both the sum and the mean of the residuals are equal to zero. That is,
Σ e = 0 and e = 0.
Residual Plots
A residual plot is a graph that shows the
residuals on the vertical axis and the independent variable
on the horizontal axis. If the points in a residual plot
are randomly dispersed
around the horizontal axis, a linear regression model is
appropriate for the data; otherwise, a non-linear model is more
appropriate.
The table below shows inputs and outputs from a simple linear regression
analysis.
x |
y |
ŷ |
e |
60 |
70 |
65.411 |
4.589 |
70 |
65 |
71.849 |
-6.849 |
80 |
70 |
78.288 |
-8.288 |
85 |
95 |
81.507 |
13.493 |
95 |
85 |
87.945 |
-2.945 |
And the chart below displays the residual (e) and independent
variable (X) as a residual plot.
The residual plot shows a fairly random pattern - the first residual is positive,
the next two are negative, the fourth is positive, and the last residual is negative.
This random pattern indicates that a linear model provides a decent fit to
the data.
Below, the residual plots show three typical patterns. The
first plot shows a random pattern, indicating a good
fit for a linear model.
Random pattern
Non-random: U-shaped
Non-random: Inverted U
The other plot patterns are
non-random (U-shaped and inverted U), suggesting a better fit
for a non-linear model.
In the
next lesson, we will work on a problem, where the residual plot shows a
non-random pattern. And we will show how to "transform"
the data to use a linear model with nonlinear data.
Test Your Understanding
In the context of
regression
analysis,
which of the following statements are true?
I. When the sum of the residuals is greater than zero, the data set is
nonlinear.
II. A random pattern of residuals supports a linear model.
III. A random pattern of residuals supports a non-linear model.
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III
Solution
The correct answer is (B).
A random pattern of residuals supports a linear model; a non-random
pattern supports a non-linear model.
The sum of the residuals is always zero, whether the data set is
linear or nonlinear.