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 Interpolation?

Design table in original data units As for the mechanics of interpolation itself, consider a continuation of the prior k = 2 factor experiment. Suppose temperature T ranges from 300 to 350 and time t ranges from 20 to 30, and the analyst can afford n = 4 runs. A 22 full factorial design is run. Forming the coded temperature as X1 and the coded time as X2, we have the usual:
Temperature Time X1 X2 Y
300 20 - - 2
350 20 + - 4
300 30 - + 6
350 30 + + 8
Graphical representation Graphically the design and data are as follows:
Typical interpolation question As before, from the data, the prediction equation is
$$\hat{Y} = 5 + 2 X_{2} + X_{1}$$
We now pose the following typical interpolation question:
From the model, what is the predicted response at, say, temperature = 310 and time = 26?
In short:
$$\hat{Y}(T = 310, t = 26) = \mbox{?}$$
To solve this problem, we first view the k = 2 design and data graphically, and note (via an "X") as to where the desired (T = 310, t = 26) interpolation point is:
Predicting the response for the interpolated point The important next step is to convert the raw (in units of the original factors T and t) interpolation point into a coded (in units of X1 and X2) interpolation point. From the graph or otherwise, we note that a linear translation between T and X1, and between t and X2 yields
T = 300 => X1 = -1
T = 350 => X1 = +1
thus
X1 = 0 is at T = 325
        |-------------|-------------|
-1     ?       0            +1
300   310     325           350

which in turn implies that
T = 310 => X1 = -0.6
Similarly,
t = 20 => X2 = -1
t = 30 => X2 = +1
therefore,
X2 = 0 is at t = 25
        |-------------|-------------|
-1             0   ?        +1
20             25 26        30

thus
t = 26 => X2 = +0.2
Substituting X1 = -0.6 and X2 = +0.2 into the prediction equation
$$\hat{Y} = 5 + 2 X_{2} + X_{1}$$
yields a predicted value of 4.8.
Graphical representation of response value for interpolated data point Thus