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Residuals

Remember that the predicted values arey i = 0 + 1 x 1 i + + m x mi , i = 1 , . . . , n .

The residuals are e1 , . . . , e n , where

e i = yi y i , i = 1 , . . . , n .

Plots to consider:

1) Construct a histogram, boxplot or normalprobability plot of residuals to check on

normality assumption.

2) Plot residuals against the predicted values.This is a good plot for checking the equalvariances assumption.

3) If the independent variables are not highlyrelated, plot residuals against each inde-pendent variable.

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4) If the data are collected over time, plot theresiduals against time. If time does not af-

fect the response, this plot should show nopattern. Durbin-Watson test can be usedto test for time effect. The Durbin-Watsonstatistic can be gotten in SPSS via Re-gression Linear Statistics Durbin-Watson . Values of the statistic larger than

2.5 or less than 1.5 are indicative of a timeeffect.

Outliers

As in simple regression, outliers that occur near

the boundary of the x-region may not show upin a residual plot. So, methods besides residu-als are needed to spot outliers.

Dene

DFFITS ( i ) = y i y ( i )

scale factor,

where y i is as usual and y ( i ) is the ith pre-dicted value obtained after removing the ithobservation from the data set.

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A large value of DF F ITS ( i ) indicates that the

i th observation may be an outlier. Values big-ger than 2 in absolute value indicate potentialoutliers.

The DFFITS statistics are obtained in SPSSas follows: Regression Linear Save Standardized DfFit .

Plot DFFITS ( i ) against i or one of the inde-pendent variables to check for outliers.

Always plot both residuals and DFFITS .

Residuals may miss outliers near boundaryof x-region.

DFFITS 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 anerror was made in recording the data. If anerror was made, correct it and re-estimate

the model using all the data.

If no errors were made, there are at leasttwo courses of action:

Throw out the outlier(s) and estimatethe model with the remaining data. Con-sult a statistician if you want to predictthe response at values of x near the onesthrown out.

Use an alternative to least squares anal-ysis, such as robust regression .

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