Carefully looking at residuals can tell us whether our assumptions are reasonable and our choice of model is appropriate

Residuals can be thought of as elements of variation unexplained by the fitted model. Since this is a form of error, the same general assumptions apply to the group of residuals that we typically use for errors in general: one expects them to be normally and independently distributed with a mean of 0 and some constant variance. 

These are the assumptions behind ANOVA and classical regression analysis. This means that an analyst should expect a regression model to err in predicting a response in a random fashion; the model should predict values higher than actual and lower than actual with equal probability. In addition, the level of the error should be independent of when the observation occurred in the study, or the size of the observation being predicted, or even the factor settings involved in making the prediction. The overall pattern of the residuals should be similar to the bell-shaped pattern observed when plotting a histogram of normally distributed data. 

We emphasize the use of graphical methods to examine residuals. 

Departures from these assumptions usually mean that the residuals contain structure that is not accounted for in the model. Identifying that structure and adding a term representing it to the original model leads to a better model. 

Tests for Residual Normality 

Any graph suitable for displaying the distribution of a set of data is suitable for judging the normality of the distribution of a group of residuals.  The three most common types are: histograms, normal probability plots, and dot plots. 
 

The histogram is a frequency plot obtained by determining the population of entries in regularly spaced cells and plotting that frequency versus the center of the cell. Figure 2.2 illustrates a normal distribution of residuals produced by a model for a semiconductor process. The software superimposes a normal density function on a histogram if requested.

Sample sizes of residuals are generally small (<50) because experiments have limited treatment combinations, so the histogram may not be the best choice for judging the distribution of residuals. A more sensitive graph is the normal probability plot. To create this graph manually, sort the residuals into ascending order and calculate the cumulative probability of each residual using the formula: 

                       P (i-th residual) = i/(N+1)

where P is the cumulative probability of a point, i is the order of the value in the list and N  is the number of entries in the list. 

Plotting the calculated P values versus the residual value on normal probability paper produces an approximately straight line if the points come from a normal distribution. 

Figure 2.3 below illustrates the normal probability graph created from the same group of residuals used for Figure 2.2 This graph reflects the addition of the dot plot. 
 

In addition it helps explain the unusual dispersion of the markings on the left y-axis. The values on the left y-axis correspond to the area under a normal distribution curve that lies to the left of a particular 'z' value shown on the right y-axis. 

Small departures from the straight line in the normal probability plot are common, but a clearly "S" shaped curve on this graph suggests a bimodal distribution of residuals. Breaks near the middle of this graph are also indications of abnormalities in the residual distribution. 

NOTE: Studentized residuals are residuals converted to a scale approximately representing the standard deviation of an individual residual from the center of the residual distribution. The technique used to convert residuals to this form produces a Student's t distribution of values. 

Independence of Residuals Over Time

If the order of the observations in a data table represents the order of execution of each treatment combination, then a plot of the residuals of those observations versus the case order or time order of the observations will test for any time dependency. 

The residuals in Figure 2.4 suggest a time trend, while those in Figure 2.5 do not. Figure 2.4 suggests that the system was drifting slowly to lower values as the investigation continued. In extreme cases a drift of the equipment will produce models with very poor ability to account for the variability in the data (low R-Square). 

If the investigation includes counterpoints, then plotting them in time order may produce a more clear indication of a time trend if one exists. Plotting the raw responses in time sequence can also sometimes detect trend changes in a process that residual plots might not detect. 

Plot of Residuals Versus Corresponding Predicted Values

Plotting residuals versus the value of a fitted response should produce a distribution of points scattered randomly about 0, regardless of the size of the fitted value. Quite commonly, however, residual values may increase as the size of the fitted value increases. When this happens, the residual cloud becomes "funnel shaped" with the larger end toward larger fitted values; that is, the residuals have larger and larger scatter as the value of the response increases. Plotting the absolute values of the residuals instead of the signed values will produce a "wedge-shaped" distribution; a smoothing function is  added to each graph which helps to show the trend. 

A residual distribution such as that in Figure 2.6 showing a trend to higher absolute residuals as the value of the response increases suggests that one should transform the response, perhaps by  modeling its logarithm or square root, etc., (contractive transformations). Transforming a response in this fashion often simplifies its relationship with a predictor variable and leads to simpler models. Later sections discuss transformation in more detail. Figure 2.7 is the same response after a transformation to reduce the scatter. Notice the difference in scales on the vertical axes. 
 

Independence of Residuals from Factor Settings
 
 

Figure 2.8 shows that the size of the residuals changed as a function of a predictor's settings. A graph like this suggests that the model needs a higher order term in that predictor or that one should transform the predictor using a logarithm or square root, for example. In this case, Figure 2.9 shows the residuals for the same response after adding a quadratic term. Notice the single point widely separated from the other residuals in Figure 2.9. This point is an "outlier." That is, its setting is well within the range of values used for this predictor in the investigation, but its result was somewhat lower than the model predicted. A signal that curvature is present is a trace resembling a "frown" or a "smile" in these graphs. 

The example given in Figures 2.8 and 2.9 obviously involves five levels of the predictor. The experiment is a form of response surface design. For the simple factorial design that includes center points, if the response model being considered lacked one or more higher order terms, the plot of residuals versus factor settings might appear as in Figure 2.10 

While the graph gives a definite signal that curvature is present, identifying the source of that curvature is not possible due to the structure of the design matrix. Graphs generated using the other predictors in that situation would have very similar appearances.

Note: Residuals are an important subject discussed repeatedly in this Handbook. For example, graphical residual plots using Dataplot are discussed in Chapter 1 and the general examination of residuals as a part of model building is discussed in Chapter 4.