Page 343 - The Handbook for Quality Management a Complete Guide to Operational Excellence

330 C o n t i n u o u s I m p r o v e m e n t A n a l y z e S t a g e 331

Analysis of Residuals
The experimenter should carefully examine the residuals. The residuals
rep resent the variation “left over” after subtracting the variation explained
by the model. The examination of residuals is done to answer the ques-
tion “What might explain the rest of the variation?” Potential clues to
the answer might arise if a pattern can be detected. For example, the
experimenter might notice that the residuals tend to be associated with
certain experimental conditions, or they might increase or decrease over
time. Other clues might be obtained if certain residuals are outliers, that is,
errors much larger than would be expect ed from chance alone. Residu-
als that exhibit patterns or that contain out liers are evidence that the
linear model is incorrect. There are many reasons why this might be so.
The response might be non-linear. The model may leave out important
variables. Or, our assumptions may not be valid.
There are four common ways of plotting the residuals:

1. Overall
2. In time sequence (if the order is known)
3. Against the predicted values
4. Against the independent variables

Overall Plot of Residuals. When the assumptions are correct, we expect
to see residuals that follow an approximately normal distribution with
zero mean. An overall plot of the residuals, such as a histogram, can be
used to evaluate this. It is often useful to plot standardized residuals
rather than actual residuals. Standardized resid uals are obtained by
dividing each residual by the standard error; the result is the residual
expressed in standard deviations. The standardized residuals should
then be plotted on a normal probability plot to verify normality.
When performing the other three types of analysis on the list, the exper-
imenter should look for any non-randomness in the patterns. Figure 15.10
illustrates some common patterns of residuals behavior.
Pattern #1 is the overall impression that will be conveyed when the
model fits satisfactorily. Pattern #2 indicates that the size of the residual
increases with respect to time or the value of the independent variable.
It suggests the need for performing a transformation of the y values
prior to performing the regression analysis. Pattern #3 indicates that the
linear effect of the indepen dent variable was not removed, perhaps due
to an error in calculations. Pattern #4 appears when a linear model is fit-
ted to curvilinear data. The solution is to perform a linearizing transfor-
mation of the y’s, or to fit the appropriate non-linear model.
In addition to the above, the analyst should always bring his or her
knowl edge of the process to bear on the problem. Patterns may become

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