Stat 101 – Lecture 15
Getting the “Bends”
• A fundamental assumption is that
the relationship is a straight line.
• What looks straight on a scatter
plot may show a curve when one
looks at the plot of residuals versus
the explanatory variable.
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Example
• y, Stopping distance (feet)
• x, Speed (miles per hour)
R 2 = 0 . 984
yˆ = − 62 . 8 + 3 . 48 x
2
200
Distance
150
100
50
0
0
10
20
30
40
50
Speed
60
70
80
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Stat 101 – Lecture 15
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Residual
5
0
-5
-10
0
10
20
30
40
50
Speed
60
70
80
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Interpretation
• There is a curved pattern in the
residuals.
– under predicts at 20 mph
– over predicts at 30, 40 and 50 mph
– under predicts at 60 mph
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Interpretation
• Although the straight line does a
very good job explaining the
variation in stopping distance, a
curved relationship model would
do even better.
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Stat 101 – Lecture 15
Regression Wisdom
• Sifting Residuals for Groups
– Display residuals versus the
explanatory variable.
– Look at the distribution of residuals.
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Example
• y, Life Expectancy (years)
• x, Wealth Index
r = 0 . 874
yˆ = 2 . 41 + 7 . 71 x
8
10
6
Count
8
4
2
-15
-10
-5
0
5
10
Residual Life Expectancy
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Stat 101 – Lecture 15
Interpretation
• There appear to be several groups
of residuals.
– two large negative residuals.
– a large group between –5 and 0.
– another group between 5 and 10.
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90
Life Expectancy
80
70
60
50
40
30
5
6
7
8
9
Wealth Index
10
11
11
15
Residual
10
5
0
-5
-10
-15
5
6
7
8
9
Wealth Index
10
11
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Stat 101 – Lecture 15
Dangers of Extrapolation
• Suppose we use the least squares
equation relating speed to stopping
distance for a vehicle traveling at 5
mph?
• The predicted stopping distance is
– 45.4 feet.
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Special Points
• Outlier – In regression, this is a
point with a large residual.
• Leverage – In regression a point
has high leverage if it is an extreme
value for the explanatory variable.
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Influence
• Outliers and high leverage points
can greatly influence what the
intercept and the slope of the least
squares line will be.
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