Chapter 8: Regression Wisdom
• Patterns on residual
plots p277
• Example: Population (in
millions) in a country
for 2000-2005
(recorded as 0, 1, 2, 3,
4, 5):
1
Example
• The regression equation
is
• population = 5.19 + 0.686 year
• R-Sq = 93.5%
2
Example - continued
• Nonlinearity is more
prominent
3
Sifting Residuals for Groups
• No regression analysis is complete without a
display of the residuals to check that the linear
model is reasonable.
• Residuals often reveal subtleties that were not
clear from a plot of the original data
• Sometimes they reveal violations of the
regression conditions that require our attention
4
Example: Regression Analysis: Self-
Esteem versus Age
• It is a good idea to look
at both a histogram of
the residuals and a
scatterplot of the
residuals vs. predicted
values:
5
• Looks like two groups
6
Real Data
7
Example
• An examination of residuals
often leads us to discover
groups of observations that
are different from the rest.
• Histograms might show
multiple modes.
• When we discover that there
is more than one group in a
regression, we may decide to
analyze the groups separately,
using a different model for
each group.
8
Outliers, Leverage, and Influence
• Any point that stands away from the others
can be called an outlier and deserves your
special attention.
• Outlying points can strongly influence a
regression. Even a single point far from the
body of the data can dominate the analysis.
9
High Leverage point
Examples
• A data point that has an x-
value far from the mean of
the x-values is called a high
leverage point.
10
High Leverage point
Examples
• A data point that has an x-
value far from the mean of
the x-values is called a high
leverage point.
11
Influential observations
Example
• A data point is influential if
omitting it from the analysis
gives a very different model.
12
Influential observations
Example
• A data point is influential if
omitting it from the analysis
gives a very different model.
P 284
13
Example (A high leverage point that is
not influential)
16
Restricted-range problem
• When one of the variables is restricted (you
only look at some of the values), the
correlation can be surprisingly low.
17
Restricted Range
Restricted Range
Restricted Range
Restricted Range
Working with summary statistics
• There is a strong,
positive, linear
association between
weight (in pounds) and
height (in inches) for
men
22
Working with summary statistics
• If instead of data on
individuals we only had
the mean weight for
each height value, we
would see an even
stronger association
23
