5. Process Improvement
5.5.9. An EDA approach to experimental design
5.5.9.9. Cumulative residual standard deviation plot

## Motivation: What is the Best Confirmation Point for Interpolation?

Augment via center point For the usual continuous factor case, the best (most efficient and highest leverage) additional model-validation point that may be added to a 2k or 2k-p design is at the center point. This center point augmentation "costs" the experimentalist only one additional run.
Example For example, for the k = 2 factor (Temperature (300 to 350), and time (20 to 30)) experiment discussed in the previous sections, the usual 4-run 22 full factorial design may be replaced by the following 5-run 22 full factorial design with a center point.
X1 X2 Y
- - 2
+ - 4
- + 6
+ + 8
0 0
Predicted value for the center point Since "-" stands for -1 and "+" stands for +1, it is natural to code the center point as (0,0). Using the recommended model
$$\hat{Y} = 5 + 2 X_{2} + X_{1}$$
we can substitute 0 for X1 and X2 to generate the predicted value of 5 for the confirmatory run.
Importance of the confirmatory run The importance of the confirmatory run cannot be overstated. If the confirmatory run at the center point yields a data value of, say, Y = 5.1, since the predicted value at the center is 5 and we know the model is perfect at the corner points, that would give the analyst a greater confidence that the quality of the fitted model may extend over the entire interior (interpolation) domain. On the other hand, if the confirmatory run yielded a center point data value quite different (e.g., Y = 7.5) from the center point predicted value of 5, then that would prompt the analyst to not trust the fitted model even for interpolation purposes. Hence when our factors are continuous, a single confirmatory run at the center point helps immensely in assessing the range of trust for our model.
Replicated center points In practice, this center point value frequently has two, or even three or more, replications. This not only provides a reference point for assessing the interpolative power of the model at the center, but it also allows us to compute model-free estimates of the natural error in the data. This in turn allows us a more rigorous method for computing the uncertainty for individual coefficients in the model and for rigorously carrying out a lack-of-fit test for assessing general model adequacy.