Additional Response Surface Topics
RSM Model Terms
The degree of the model indicates the presence of certain terms.
Degree Description
1 Linear - A, B, C…
2 Quadratic - AB, AC, BC, A², B², C²…
3 Cubic - ABC, A²B, A²C, AB², A3, B3, C3…
The designs differ in their ability to estimate higher order terms. Most designs are adequate for the quadratic model only. One factor designs can be built for up to a cubic model. Optimal designs can be extended up to sixth order polynomial models.
Factor Coding in Response Surface Designs
Design descriptions and analyses are best done with coded factors. Coding reduces the range of each factor to a common scale, -1 to +1, regardless of its relative magnitude. Scaling establishes factor levels that can be orthogonal (or nearly so). Also, it is easier to think in terms of changes from low to high for the factors than to think about their actual values - especially when thinking about squared terms and interactions. For example, one factor may vary from 100 to 200 while another varies from 0.1 to 0.5. Typical coding has -1 as the lower level of a factor, +1 as the upper level, and 0 as the middle level. Box-Behnken designs use only these three levels.
Central composite designs typically use a more extended range and five factor levels.
Candidate based designs have default factor ranges of –1 to +1.
After the design is built, you can switch between coded and actual display of factors at any time via the Display Options menu.
RSM Design Points Needed
You may want to determine for yourself how many design points are needed for a good response surface design. The response surface method produces a mathematical model that you can use to predict a response. You should provide the following design features to build a good model:
Enough unique design points to estimate all the terms in the postulated model: linear, 2FI, quadratic, cubic, etc. The number of model terms increases in proportion to the number of factors studied.
Extra unique design points in addition to the above to test how the model fits the data. These points must be at locations in the design space that are different from the model points. They are used in a “Lack of Fit” test for the model. At least three of these extra points to give an adequate statistical test.
Replicates of some design points to estimate the experimental (pure) error. This is the error to be expected in the response if the experiment is repeated starting from scratch. Typically, the center point of the design is repeated, often four or more times. This gives an adequate estimate of the variation of the response and provides the number of degrees of freedom needed for an adequate statistical test of the model. You may choose to duplicate other points in the design if you desire better estimates of the response at those areas in the experimental space.
As a guide, here is a table of the number of coefficients in linear, quadratic, and cubic equations for the given number of factors.
# FACTORS |
LINEAR |
QUADRATIC |
CUBIC |
|---|---|---|---|
2 |
3 |
6 |
10 |
3 |
4 |
10 |
20 |
4 |
5 |
15 |
35 |
5 |
6 |
21 |
56 |
6 |
7 |
28 |
84 |
7 |
8 |
36 |
120 |
In summary, to ensure success in RSM modeling, you should allow for sufficient model points, plus at least 4 replicate points, plus at least 3 other points to determine the fit of the model.
One Factor RSM Design
Custom designs are used to build designs with a single numeric factor. The user-defined design is the closest parallel to the “one factor design” found in previous versions.