5.
Process Improvement
5.5. Advanced topics 5.5.9. An EDA approach to experimental design 5.5.9.10. DOE contour plot


Nonlinear appearance of contour curves implies strong interaction 
Based on the fitted model (cumulative
residual standard deviation plot) and the best data
settings for all of the remaining factors, we draw contour curves
involving the two dominant factors. This yields a graphical
representation of the response surface.
Before delving into the details as to how the contour lines were generated, let us first note as to what insight can be gained regarding the general nature of the response surface. For the defective springs data, the dominant characteristic of the contour plot is the nonlinear (fanshaped, in this case) appearance. Such nonlinearity implies a strong X_{1}*X_{3} interaction effect. If the X_{1}*X_{3} interaction were small, the contour plot would consist of a series of nearparallel lines. Such is decidedly not the case here. 

Constructing the contour curves 
As for the details of the construction of the contour plot, we
draw on the modelfitting results that were achieved in the
cumulative residual standard deviation
plot. In that step, we derived the following goodfitting
prediction equation:
\( \hat{Y} = 73.75 + 11.5 X_{1} + 5 X_{1}X_{3} \) To generate the contour curve for, say, Y = 70, we solve


Values for X1  For these X_{3} = g(X_{1}) equations, what values should be used for X_{1}? Since X_{1} is coded in the range 1 to +1, we recommend expanding the horizontal axis to 2 to +2 to allow extrapolation. In practice, for the DOE contour plot generated previously, we chose to generate X_{1} values from 2, at increments of 0.05, up to +2. For most data sets, this gives a smooth enough curve for proper interpretation.  
Values for Y  What values should be used for Y? Since the total theoretical range for the response Y (= percent acceptable springs) is 0 % to 100 %, we chose to generate contour curves starting with 0, at increments of 5, and ending with 100. We thus generated 21 contour curves. Many of these curves did not appear since they were beyond the 2 to +2 plot range for the X_{1} and X_{3} factors.  
Summary  In summary, the contour plot curves are generated by making use of the (rearranged) previously derived prediction equation. For the defective springs data, the appearance of the contour plot implied a strong X_{1}*X_{3} interaction. 