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Box Behnken RSM |
This example validates the calculation of the Box-Behnken RSM method in DOE++.
Reference Case
Data is from Example 7.5 on page 347 in the book Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 2nd Edition, by Raymond H. Myers and Douglas C. Montgomery, John Wiley & Sons, Inc. 2002.
Data
| Level (coded value) | Temperature (A) | Agitation (B) | Rate (C) |
|---|---|---|---|
| High (+1) | 200 | 10 | 25 |
| Center (0) | 175 | 7.5 | 20 |
| Low (-1) | 150 | 5 | 15 |
The viscosity of the resin was recorded as an indirect measure of molecular weight.
| Standard order | A | B | C | Y |
|---|---|---|---|---|
| 1 | -1 | -1 | 0 | 53 |
| 2 | +1 | -1 | 0 | 58 |
| 3 | -1 | +1 | 0 | 59 |
| 4 | +1 | +1 | 0 | 56 |
| 5 | -1 | 0 | -1 | 64 |
| 6 | +1 | 0 | -1 | 45 |
| 7 | -1 | 0 | +1 | 35 |
| 8 | +1 | 0 | +1 | 60 |
| 9 | 0 | -1 | -1 | 59 |
| 10 | 0 | +1 | -1 | 64 |
| 11 | 0 | -1 | +1 | 53 |
| 12 | 0 | +1 | +1 | 65 |
| 13 | 0 | 0 | 0 | 65 |
| 14 | 0 | 0 | 0 | 59 |
| 15 | 0 | 0 | 0 | 62 |
Result
From the book, the ANOVA table (partial sum of squares) is:
| Source | Partial SS | DF | Mean Square | F value | Prob>F |
|---|---|---|---|---|---|
| Model | 882.48 | 9 | 98.05 | 9.57 | 0.0114 |
| A | 8 | 1 | 8.00 | 0.78 | 0.4174 |
| B | 55.12 | 1 | 55.12 | 5.38 | 0.0681 |
| C | 45.13 | 1 | 45.13 | 4.40 | 0.09 |
| [math]\displaystyle{ A^2 }[/math] | 200.83 | 1 | 200.83 | 19.59 | 0.0069 |
| [math]\displaystyle{ B^2 }[/math] | 12.98 | 1 | 12.98 | 1.27 | 0.3115 |
| [math]\displaystyle{ C^2 }[/math] | 48.52 | 1 | 48.52 | 4.73 | 0.0816 |
| AB | 16.00 | 1 | 16.00 | 1.56 | 0.2668 |
| AC | 484.00 | 1 | 484.00 | 47.22 | 0.001 |
| BC | 12.25 | 1 | 12.25 | 1.22 | 0.3241 |
| Residual | 51.25 | 5 | 10.25 | ||
| Lack of Fit | 33.25 | 3 | 11.08 | 1.23 | 0.4774 |
| Pure error | 18 | 2 | 9.00 | ||
| Cor total | 933.73 | 14 |
The final equation in terms of actual factors:
Results in DOE++
The software results match the book results. The ANOVA table is:
The final equation in terms of actual factors is:
The maximum viscosity is shown in the following optimal solution plot.
