# Analyzing Residuals

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There are a variety of residuals and versions of the dependent variable that can be examined and plotted in a regression analysis. SPSS produces the following residuals (as described in the SPSS Guide to Data Analysis by Marija J. Norusis, New Jersey: Prentice Hall, 2008, p. 465) by clicking Plots to display the Linear Regression Plots dialogue in a linear regression analysis:

DEPENDENT The dependent variable

ZPRED            The standardized predicted values of the dependent variable.

ZRESID           The standardized residuals
DRESID          Deleted residuals, the residuals when the case is excluded from the regression.
ADJPRED       Adjusted predicted values, the predicted value for a case when it is excluded from the regression

SRESID           Studentized residuals

SDRESID        Studentized deleted residuals

The standardized residual (ZPRED) is the residual divided by its standard error. Standardizing is a method for transforming data so that its mean is zero and standard deviation is one. If the distribution of the residuals is approximately normal, then 95% of the standardized residuals should fall between -2 and +2. If many of the residuals fall outside of + or – 2, then they could be considered unusual. However, about 5% of the residuals could fall outside of this region due to chance.

The studentized residuals (SRESID) take into account that the variance of the predicted value used in calculating residuals is not constant. As Norusis discusses in Chapter 22, the variability of cases close to the sample mean for an independent variable have smaller variance compared to cases further away from the mean. The studentized residual takes this change in variability into account by dividing the observed residual by an estimate of the standard deviation of the residual at that point. Norusis argues that this adjustment makes violation of regression assumptions more visible, so it is preferred to standardized residuals.

Studentized deleted residuals (SDRESID), as discussed by Norusis, p. 515, take advantage of the fact that you can see the impact of a case on the slope by calculating a regression with and without the slope. As the name implies the studentized deleted residual is the Studentized residual when the case is excluded from the regression. Norusis argues that you can see departures from regression assumptions more easily with studentized deleted residuals than other residuals. Fox refers to these residuals as studentized residuals (rstudent in R).  Fox considers these to be the appropriate residuals to examine. See Fox (An R and S-Plus Companion to Applied Regression, Thousand Oaks, CA: Sage, 2002, page 192) for a discussion of how these residuals are calculated.

These residuals can be plotted on the y-axis with either the dependent variable (DEPENDENT) or the adjusted predicted variable (ADJPRED) on the x-axis. In all of the cases of standardized residuals, a rough rule of thumb is that residuals greater than two are worth examining. Remember, that if one observation is deleted, all of the residuals will have to be recalculated and reconsidered.

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