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Linear Regression Analysis 5E Montgomery, Peck and Vining
1
Chapter 6
Diagnostics for Leverage
and Influence
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.1 Importance of Detecting Influential Observations
• Leverage Point:– unusual x-value; – very little effect
on regression coefficients.
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.1 Importance of Detecting Influential Observations
• Influence Point: unusual in y and x;
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.2 Leverage
• The hat matrix is:
H = X(XX)- 1 X• The diagonal elements of the hat matrix are
given by
hii = xi(XX)-1xi
• hii – standardized measure of the distance of
the ith observation from the center of the x-space.
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.2 Leverage
• The average size of the hat diagonal is p/n.
• Traditionally, any hii > 2p/n indicates a
leverage point.
• An observation with large hii and a large
residual is likely to be influential
Linear Regression Analysis 5E Montgomery, Peck and Vining
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Linear Regression Analysis 5E Montgomery, Peck and Vining
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Example 6.1 The Delivery Time Data
• Examine Table 6.1; if some possibly influential points are removed here is what happens to the coefficient estimates and model statistics:
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.3 Measures of Influence
• The influence measures discussed here are those that measure the effect of deleting the ith observation.
1. Cook’s Di, which measures the effect on
2. DFBETASj(i), which measures the effect on
3. DFFITSi, which measures the effect on
4. COVRATIOi, which measures the effect on the variance-covariance matrix of the parameter estimates.
j
iY
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.3 Measures of Influence: Cook’s D
What contributes to Di:1. How well the model fits the ith observation, yi
2. How far that point is from the remaining dataset.
Large values of Di indicate an influential point, usually if Di > 1.
)1()(
)ˆ(
)ˆˆ()'ˆˆ(),'(
22
Re
)()(Re
ii
iii
i
ii
s
iiisi
h
h
p
r
eVar
yVar
p
r
pMSDpMSXXD
XX'
Linear Regression Analysis 5E Montgomery, Peck and Vining
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Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.4 Measures of Influence: DFFITS and DFBETASDFBETAS – measures how much the regression
coefficient changes in standard deviationunits if the ith observation is removed.
where is an estimate of the jth coefficient when the ith observation is removed.
• Large DFBETAS indicates ith observation has considerable influence. In general, |DFBETASj,i| > 2/
jji
ijjij
CSDFBETAS
2)(
)(,
ˆˆ
n
)(ˆ
ij
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.4 Measures of Influence: DFFITS and DFBETAS
DFFITS – measures the influence of the ith observation on the fitted value, again in standard deviation units.
• Cutoff: If |DFFITSi| > 2 , the point is
most likely influential.
iii
iii
hS
yyDFFITS
2)(
)(ˆˆ
np /
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.4 Measures of Influence: DFFITS and DFBETAS
Equivalencies
• See the computational equivalents of both DFBETAS and DFFITS (page 217). You will see that they are both functions of R-student and hii.
Linear Regression Analysis 5E Montgomery, Peck and Vining
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Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.5 A Measure of Model Performance
• Information about the overall precision of estimation can be obtained through another statistic, COVRATIOi
iips
pi
s
iiii
hMS
S
MS
SCOVRATIO
1
1)(
|)(|
|)(|
Re
2)(
Re1
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1)()(
XX'
XX
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.5 A Measure of Model Performance
Cutoffs and Interpretation• If COVRATIOi > 1, the ith observation
improves the precision. • If COVRATIOi < 1, ith observation can
degrade the precision. Or,• Cutoffs: COVRATIOi > 1 + 3p/n or
COVRATIOi < 1 - 3p/n; (the lower limit is really only good if n > 3p).
Linear Regression Analysis 5E Montgomery, Peck and Vining
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Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.6 Detecting Groups of Influential Observations
• Previous diagnostics were “single-observation”
• It is possible that a group of points have high-leverage or exert undue influence on the regression model.
• Multiple-observation deletion diagnostic can be implemented.
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.6 Detecting Groups of Influential Observations
• Cook’s D can be extended to incorporate multiple observations:
where i denotes the m 1 vector of indices specifying the points to be deleted.
• Large values of Di indicate that the set of m points are influential.
ss pMS
DpMSDRe
)()(Re
)ˆˆ()'ˆˆ(),'(
ii
ii
XX'XX
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.7 Treatment of Influential Observations
• Should an influential point be discarded?
Yes, if– there is an error in recording a measured value;– the sample point is invalid; or,– the observation is not part of the population that
was intended to be sampled
No, if– the influential point is a valid observation.
Linear Regression Analysis 5E Montgomery, Peck and Vining
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6.7 Treatment of Influential Observations
• Robust estimation techniques– These techniques offer an alternative to deleting
an influential observation.– Observations are retained but downweighted in
proportion to residual magnitude or influence.