Appendix 8B: Small Composite Designs (SCDs) to Minimize Runs

In the jargon of response surface methods, the adjective "composite" describes an experimental design made up of discrete blocks geared to fit a quadratic model. Since Box and Wilson introduced these RSM designs in the 1951, much work has been devoted to reduce the number of required runs. The minimum number of runs to fit a quadratic model with k factors equals the parameters (p), which can be broken down by degree:

• Zero: 1 constant ( ? ₀) often described as the intercept or mean.

• First: k main-effect coefficients ( ? _i) the slopes for each individual (i) factor.

• Second:

  1. k(k-1)/2 coefficients for two-factor interactions (2FI).

  2. k pure quadratic (squared terms) for curvature coefficients.

This simplifies to:

p = (k+1)(k+2)/2

For example, a minimum-run RSM on 10 factors requires 66 (= (10+1)(10+2)/2) unique combinations of the factors. Designs like this are considered to be saturated with the maximum number of factors in the given number of runs.

An experimenter may want a minimal-run composite design when runs are extremely expensive, difficult or time-consuming.

Early on, statisticians seeking minimal-run composite designs realized that it would be acceptable if the factorial core aliased main effects with 2fi's, provided it did not alias these second-order terms with each other. (See sidebar titled "An Asterisk On Resolution" for details.)