STA291
Statistical Methods
Lecture 28
Extrapolation and Prediction
Extrapolating – predicting a y value by extending
the regression model to regions outside the range of
the x-values of the data.
Extrapolation and Prediction
Why is extrapolation dangerous?
 It introduces the questionable and untested
assumption that the relationship between x and y
does not change.
Extrapolation and Prediction
Cautionary Example: Oil Prices in Constant Dollars
Price = – 0.85 + 7.39 Time
Model Prediction
(Extrapolation):
On average, a barrel
of oil will increase
$7.39 per year from
1983 to 1998.
Extrapolation and Prediction
Cautionary Example: Oil Prices in Constant Dollars
Price = – 0.85 + 7.39 Time
Actual Price Behavior
Extrapolating the 1971-1982 model to the ’80s and
’90s lead to grossly erroneous forecasts.
Extrapolation and Prediction
Remember: Linear models ought not be trusted
beyond the span of the x-values of the data.
If you extrapolate far into the future, be prepared
for the actual values to be (possibly quite)
different from your predictions.
Unusual and Extraordinary Observations
Outliers, Leverage, and Influence
In regression, an outlier can stand out in two
ways. It can have…
1) a large residual:
Unusual and Extraordinary Observations
Outliers, Leverage, and Influence
In regression, an outlier can stand out in two
ways. It can have…
2) a large distance from
x:
“High-leverage
point”
A high leverage point is influential if omitting it
gives a regression model with a very different
slope.
Unusual and Extraordinary Observations
Outliers, Leverage, and Influence
Tell whether the point is a high-leverage point, if
it has a large residual, and if it is influential.
Not high-leverage
Large residual
Not very influential
Unusual and Extraordinary Observations
Outliers, Leverage, and Influence
Tell whether the point is a high-leverage point, if
it has a large residual, and if it is influential.
 High-leverage
 Small residual
 Not very influential
Unusual and Extraordinary Observations
Outliers, Leverage, and Influence
Tell whether the point is a high-leverage point, if
it has a large residual, and if it is influential.
 High-leverage
 Medium (large?)
residual
 Very influential
(omitting the red
point will change the
slope dramatically!)
Unusual and Extraordinary Observations
Outliers, Leverage, and Influence
What should you do with a high-leverage point?
 Sometimes, these points are important.
They can indicate that the underlying
relationship is in fact nonlinear.
 Other times, they simply do not belong
with the rest of the data and ought to be
omitted.
When in doubt, create and report two models:
one with the outlier and one without.
Unusual and Extraordinary Observations
Example: Hard Drive Prices
Prices for external hard drives are linearly associated
with the Capacity (in GB). The least squares regression
line without a 200 GB drive that sold for $299.00 was
•  18.64  0.104Capacity.
found to be Price
The regression equation with the original data is
•  66.57  0.088Capacity
Price
How are the two equations different?
The intercepts are different, but the slopes are similar.
Does the new point have a large residual? Explain.
Yes. The hard drive’s price doesn’t fit the pattern since it
pulled the line up but didn’t decrease the slope very
much.
Working with Summary Values
Scatterplots of summarized (averaged) data tend to
show less variability than the un-summarized data.
Example:
Wind speeds at two locations, collected at 6AM,
noon, 6PM, and midnight.
Raw data:
R2 = 0.736
Daily-averaged
data:
R2 = 0.844
Monthly-averaged
data:
R2 = 0.942
Working with Summary Values
WARNING:
Be suspicious of conclusions based on regressions
of summary data.
Regressions based on summary data may look
better than they really are!
In particular, the strength of the correlation will be
misleading.
Autocorrelation
Time-series data are sometimes autocorrelated, meaning
points near each other in time will be related.
First-order autocorrelation:
Adjacent measurements are related
Second-order autocorrelation:
Every other measurement is related
etc…
Autocorrelation violates the independence condition.
Regression analysis of autocorrelated data can produce
misleading results.
Transforming (Re-expressing) Data
An aside
On using technology:
Transforming (Re-expressing) Data
Linearity
Some data show departures from linearity.
Example: Auto Weight vs. Fuel Efficiency
Linearity condition is not satisfied.
Transforming (Re-expressing) Data
Linearity
In cases involving upward bends of negativelycorrelated data, try analyzing –1/y (negative reciprocal
of y) vs. x instead.
Linearity condition now appears satisfied.
Transforming (Re-expressing) Data
The auto weight vs. fuel economy example illustrates
the principle of transforming data.
There is nothing sacred about the way x-values or yvalues are measured. From the standpoint of
measurement, all of the following may be equallyreasonable:
x vs. y
x vs. –1/y
x2 vs. y
x vs. log(y)
One or more of these
transformations may be
useful for making data
more linear, more
normal, etc.
Transforming (Re-expressing) Data
Goals of Re-expression
Goal 1 Make the distribution of a variable more symmetric.
y vs. x
y vs. log  x 
Transforming (Re-expressing) Data
Goals of Re-expression
Goal 2 Make the spread of several groups more alike.
y vs. x
log  y  vs. x
We’ll see methods later in the book that can be applied only to groups
with a common standard deviation.
Looking back
Make sure the relationship is straight
enough to fit a regression model.
Beware of extrapolating.
Treat unusual points honestly.You must
not eliminate points simply to “get a good
fit”.
Watch out when dealing with data that
are summaries.
Re-express your data when necessary.