Special Topics Example 3 - Residuals Analysis

In this example, we will first look at the untransformed dependent variable values to illustrate a residual analysis indicating a faulty model and/or dependent variable. To begin the analysis, open the example data file Textile2.sta and start Experimental Design (DOE).

Ribbon bar. Select the Home tab. In the File group, click the Open arrow and from the menu, select Open Examples to display the Open a Statistica Data File dialog box. Double-click the Datasets folder, and open the Textile2.sta data set. Then, select the Statistics tab, and in the Industrial Statistics group, click DOE to display the Design & Analysis of Experiments Startup Panel.

In the Startup Panel, double-click 3**(k-p) and Box-Behnken designs to display the Design and Analysis of Experiments with Three-Level factors dialog box.

Select the Analyze design tab. Click the Variables button, and select Cycles and Log_Cycl as the Dependent variables; Length, Amplitud, and Load as the Indep (factors); and click OK.

Click OK in the Design and Analysis of Experiments with Three-Level factors dialog box to display the Analysis of an Experiment with Three-Level Factors dialog box.

On the Model tab, select the No interactions option button, select the Ignore some effects check box, and in the Customized (Pooled) Error Term dialog box, highlight  the quadratic effects to pool them into the error term.

Click OK in the Customized (Pooled) Error Term dialog box.

On either the Quick tab or the ANOVA/Effects tab, click the Summary: Effect estimates button to produce the Effect Estimates spreadsheet.

Shown above are the last four columns of the spreadsheet, with the coefficients for the recoded (-1,0,1) factor values. The column of t-values (not shown in the illustration above) shows that all three linear effects are highly significant.  

Now, click the Histogram of predicted values button. Here we see a hint of serious trouble; four of the predicted values are negative, which does not make physical sense on the number-of-cycles-to-failure dependent variable, although there is no clear evidence that the predicted values are negatively skewed.  

Next, click the Normal plot button in the Probability plots of residuals group box. You can see evidence of non-linearity in the plot of Expected Normal Values and corresponding probabilities against the Residuals, with very low and very high residuals falling below and moderate residuals falling above the straight line that is expected under the ANOVA model assumptions.  

The Observed vs. predicted values plot shows the same pattern, and the Detrended normal plot shows the pattern even more clearly. The Predicted vs. residual values plot shows that the relation between the Residuals and Predicted Values clearly is not uniform, and the Observed vs residual values plot shows severe non-uniformity in the relation between the Observed values and Residuals. Fortunately, the uniformity of the Residuals vs. deleted residuals plot and the Residuals vs. case numbers plot provide no evidence of outliers or serial correlation in the observations, respectively, but nevertheless it is clear the residuals do not behave as would be expected if the usual ANOVA assumptions are met.

Perhaps most troubling of all, there is clear evidence that the cell variances on Cycles are strongly related to the cell means. It does take a little bit of ingenuity to make this determination, because there is only one case in each of the 27 cells of the 3 x 3 x 3 design, so the variance in each cell is, strictly speaking, undefined. However, the relations between marginal cell means and corresponding marginal cell standard deviations for pairs of factors collapsing across the third factor can be assessed.

For example, on the Means tab (or the Quick tab), in the under Observed marginal means group box, click the Display button to produce a spreadsheet with the marginal cell means and standard deviations for Length and Amplitude, collapsing across Load (highlight Length and Amplitude in the Compute marginal means for dialog box). If you were to perform a correlation between these marginal means and standard deviations it would be .96, or nearly perfect.

One issue that remains, however, is whether it is better to transform a dependent variable with poorly-behaved residuals or to add higher-order terms to the prediction model to account for the non-linearity in the diagnostic plots of the residuals from the first degree polynomial (linear) model. For this example, Box and Draper (1987) present evidence that the former alternative is preferable.  

Shown below are the last four columns of the Effect estimates spreadsheet for the Log_Cycl  dependent variable (produced by clicking the Summary: Effect estimates button on either the Quick tab or the Anova/Effects tab), with the coefficients for the recoded (-1,0,1) factor values.

Of greatest interest is the R-square value of .96585, showing that virtually all of the variability in the transformed dependent variable is accounted for by the simple, first degree polynomial model. If you were to redo the analysis using the second degree polynomial model to fit the untransformed dependent variable Cycles (by selecting 2-way interactions (linear x linear) on the Model tab), the R-square value of .93788 is actually smaller. Thus, the greater complexity of a model with added higher order terms does not pay off in producing greater predictability of the untransformed dependent variable. Parsimony favors the adoption of the simpler model with the transformed dependent variable over the more complex model with the untransformed dependent variable, because in the latter case responses are explained no better than in the former.