Residuals and outliers

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First found May 22, 2018

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Residuals
Remember that the predicted values are
ŷi = β̂0 + β̂1x1i + · · · + β̂mxmi,
i = 1, . . . , n.
The residuals are e1, . . . , en, where
ei = yi − ŷi,
i = 1, . . . , n.
Plots to consider:
1) Construct a histogram, boxplot or normal
probability plot of residuals to check on
normality assumption.
2) Plot residuals against the predicted values.
This is a good plot for checking the equal
variances assumption.
3) If the independent variables are not highly
related, plot residuals against each independent variable.
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4) If the data are collected over time, plot the
residuals against time. If time does not affect the response, this plot should show no
pattern. Durbin-Watson test can be used
to test for time effect. The Durbin-Watson
statistic can be gotten in SPSS via Regression → Linear → Statistics → DurbinWatson. Values of the statistic larger than
2.5 or less than 1.5 are indicative of a time
effect.
Outliers
As in simple regression, outliers that occur near
the boundary of the x-region may not show up
in a residual plot. So, methods besides residuals are needed to spot outliers.
Define
DF F IT S(i) =
ŷi − ŷ(i)
,
scale factor
where ŷi is as usual and ŷ(i) is the ith predicted value obtained after removing the ith
observation from the data set.
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A large value of DF F IT S(i) indicates that the
ith observation may be an outlier. Values bigger than 2 in absolute value indicate potential
outliers.
The DF F IT S statistics are obtained in SPSS
as follows: Regression → Linear → Save →
Standardized DfFit.
Plot DF F IT S(i) against i or one of the independent variables to check for outliers.
Always plot both residuals and DF F IT S.
• Residuals may miss outliers near boundary
of x-region.
• DF F IT S may miss outliers in ”middle” of
x-region.
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What should one do with outliers?
• After spotting an outlier, check to see if an
error was made in recording the data. If an
error was made, correct it and re-estimate
the model using all the data.
• If no errors were made, there are at least
two courses of action:
– Throw out the outlier(s) and estimate
the model with the remaining data. Consult a statistician if you want to predict
the response at values of x near the ones
thrown out.
– Use an alternative to least squares analysis, such as robust regression.
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