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

## Motivation: How do we Use the Model for Extrapolation?

Graphical representation of extrapolation Extrapolation is performed similarly to interpolation. For example, the predicted value at temperature T = 375 and time t = 28 is indicated by the "X":
and is computed by substituting the values X1 = +2.0 (T=375) and X2 = +0.8 (t=28) into the prediction equation
$$\hat{Y} = 5 + 2 X_{2} + X_{1}$$
yielding a predicted value of 8.6. Thus we have
Pseudo-data The predicted value from the modeling effort may be viewed as pseudo-data, data obtained without the experimental effort. Such "free" data can add tremendously to the insight via the application of graphical techniques (in particular, the contour plots and can add significant insight and understanding as to the nature of the response surface relating Y to the X's.

But, again, a final word of caution: the "pseudo data" that results from the modeling process is exactly that, pseudo-data. It is not real data, and so the model and the model's predicted values must be validated by additional confirmatory (real) data points. A more balanced approach is that:

Models may be trusted as "real" [that is, to generate predicted values and contour curves], but must always be verified [that is, by the addition of confirmatory data points].
The rule of thumb is thus to take advantage of the available and recommended model-building mechanics for these 2-level designs, but do treat the resulting derived model with an equal dose of both optimism and caution.
Summary In summary, the motivation for model building is that it gives us insight into the nature of the response surface along with the ability to do interpolation and extrapolation; further, the motivation for the use of the cumulative residual standard deviation plot is that it serves as an easy-to-interpret tool for determining a good and parsimonious model.