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

How to Interpret: Contour Curves

Non-linear 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 non-linear (fan-shaped, in this case) appearance. Such non-linearity implies a strong X1*X3 interaction effect. If the X1*X3 interaction were small, the contour plot would consist of a series of near-parallel 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 model-fitting results that were achieved in the cumulative residual standard deviation plot. In that step, we derived the following good-fitting prediction equation:
$$\hat{Y} = 71.25 + 11.5 X_{1} + 5 X_{1}X_{3} - 2.5 X_{2}$$
The contour plot has axes of X1 and X3. X2 is not included and so a fixed value of X2 must be assigned. The response variable is the percentage of acceptable springs, so we are attempting to maximize the response. From the ordered data plot, the main effects plot, and the interaction effects matrix plot of the general analysis sequence, we saw that the best setting for factor X2 was "-". The best observed response data value (Y = 90) was achieved with the run (X1, X2, X3) = (+, -, +), which has X2 = "-". Also, the average response for X2 = "-" was 73 while the average response for X2 = "+" was 68. We thus set X2 = -1 in the prediction equation to obtain
$$\hat{Y} = 71.25 + 11.5 X_{1} + 5 X_{1}X_{3} - 2.5 (-1)$$
$$\hat{Y} = 73.75 + 11.5 X_{1} + 5 X_{1}X_{3}$$
This equation involves only X1 and X3 and is immediately usable for the X1 and X3 contour plot. The raw response values in the data ranged from 52 to 90. The response implies that the theoretical worst is Y = 0 and the theoretical best is Y = 100.

To generate the contour curve for, say, Y = 70, we solve

70 = 73.75 + 11.5*X1 + 5*X1*X3
by rearranging the equation in X3 (the vertical axis) as a function of X1 (the horizontal axis). By substituting various values of X1 into the rearranged equation, the above equation generates the desired response curve for Y = 70. We do so similarly for contour curves for any desired response value Y.
Values for X1 For these X3 = g(X1) equations, what values should be used for X1? Since X1 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 X1 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 X1 and X3 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 X1*X3 interaction.