Convert Coded response surface model to Actual

The Design-Expert ® software by default performs regression computations using the coded scale where the low setting for each factor is set to -1 and the high set to +1 per the formula below.

\[Coded = \frac{2 \cdot (Actual\ setting\ -\ Average\ actual\ setting)}{(Range\ between\ low\ and\ high\ actual\ settings)}\]

For example, if a factor has a low setting of 30 and a high setting of 50, the average is 40, and the range is 20. If the value 35 is used in a run, the coded value is, \((35 - 40)/(20/2) = -5/10 = -0.5\).

Design-Expert provides both the coded and actual scale models for your convenience. The conversion from a coded model to an actual model requires knowing all of the actual factor ranges and the coded model. Full precision (all decimal points) must be available as rounding errors are the typical reason manual conversion to the actual model does not match Design-Expert output.

The conversion is done by substituting the above formula for each factor in every term of the model.

For this case, the model in coded units is:

\[Y = 70.0625\, +\, 10.8125\, \cdot\, A\, +\, 4.9375\, \cdot\, C\, +\, 7.3125\, \cdot\, D\, -\, 9.0625\, \cdot\, AC\, +\, 8.3125\, \cdot\, AD\]

Where the factor settings are laid out in the table below:

Low

High

Range

Average

A

24

35

11

29.5

B

10

15

5

12.5

C

2

4

2

3.0

D

15

30

15

22.5

Wherever there is a factor in the coded model, substitute the formula and simplify the algebra. Follow the example below, starting with the higher-order terms.

Convert the AD term as follows:

\(+8.3125 \cdot AD\)

\(= 8.3125 \cdot \begin{bmatrix} \frac{2\, \cdot\, (A\, - 29.5)}{11}\, \cdot\, \frac{2\, \cdot\, (D\, -\, 22.5)}{15} \end{bmatrix}\)

\(= \frac{8.3125}{11 \cdot 15} \cdot 4 \cdot \begin{bmatrix} (A - 29.5) \cdot (D - 22.5) \end{bmatrix}\)

\(= 0.20\bar{15} \cdot \begin{bmatrix} AD - 22.5 \cdot A - 29.5 \cdot D +29.5 \cdot 22.5 \end{bmatrix}\)

Convert the AC term as follows:

\(-9.0625 \cdot AC\)

\(= -9.0625 \cdot \begin{bmatrix} \frac{2\, \cdot\, (A\, -\, 29.5)}{11} \cdot \frac{2\, \cdot\, (C\, -\, 3)}{2} \end{bmatrix}\)

\(= \frac{-9.0625}{11 \cdot 2} \cdot 4 \cdot \begin{bmatrix} (A - 29.5) \cdot (C - 3) \end{bmatrix}\)

\(= -1.647\bar{72} \cdot \begin{bmatrix}AC\, -\, 3\, \cdot\, A\, -\, 22.5\, \cdot\, C\, +\, 29.5\, \cdot\, 3 \end{bmatrix}\)

Convert the D term as follows:

\(+7.3125 \cdot D\)

\(= 7.3125 \cdot \begin{bmatrix}\frac{2\, \cdot\, (D\, -\, 22.5)}{15} \end{bmatrix}\)

\(= \frac{7.3125}{15}\, \cdot\, 2\, \cdot\, \begin{bmatrix}D\, -\, 22.5\end{bmatrix}\)

\(= 0.975 \cdot \begin{bmatrix}D\, -\, 22.5\end{bmatrix}\)

Convert the C term as follows:

\(+4.9375 \cdot C\)

\(= 4.9375 \cdot \begin{bmatrix}\frac{2\, \cdot\, (C\, -\, 3)}{2} \end{bmatrix}\)

\(= \frac{4.9375}{2}\, \cdot\, 2\, \cdot\, \begin{bmatrix}C\, -\, 3 \end{bmatrix}\)

\(= 4.9375 \cdot \begin{bmatrix}C\, -\, 3 \end{bmatrix}\)

Convert the A term as follows:

\(+10.8125 \cdot A\)

\(= 10.8125 \cdot \begin{bmatrix}\frac{2\, \cdot\, (A\, -\, 29.5)}{11} \end{bmatrix}\)

\(= \frac{10.8125}{11}\, \cdot\, 2\, \cdot\, \begin{bmatrix}A\, -\, 29.5 \end{bmatrix}\)

\(= 1.9659\bar{09}\, \cdot\, \begin{bmatrix}A\, -\, 29.5 \end{bmatrix}\)

When all conversions are expanded and like terms are added together the result is:

\[Y = -36.75\, +\, 2.375\, \cdot\, A\, +\, 53.\bar{54}\, \cdot\, C\, -\, 4.\bar{96}\, \cdot\, D\, -\, 1.647\bar{72}\, \cdot\, AC\, +\, 0.02\bar{15}\, \cdot\, AD\]

As you can see in the conversions, the two-factor interaction (2FI) terms expand into corrections to the interaction, both main effects, and the intercept. Main effect terms expand into corrections to the main effect and the intercept.