5. Process Improvement

## What are small composite designs?

Small composite designs save runs, compared to Resolution V response surface designs, by adding star points to a Resolution III design Response surface designs (RSD) were described earlier. A typical RSD requires about 13 runs for 2 factors, 20 runs for 3 factors, 31 runs for 4 factors, and 32 runs for 5 factors. It is obvious that, once you have four or more factors you wish to include in a RSD, you will need more than one lot (i.e., batch) of experimental units for your basic design. This is what most statistical software today will give you. However, there is a way to cut down on the number of runs, as suggested by H.O. Hartley in his paper 'Smallest Composite Designs for Quadratic Response Surfaces', published in Biometrics, December 1959.

This method addresses the theory that using a Resolution V design as the smallest fractional design to create a RSD is unnecessary. The method adds star points to designs of Resolution III and uses the star points to clear the main effects of aliasing with the two-factor interactions. The resulting design allows estimation of the higher-order interactions. It also provides poor interaction coefficient estimates and should not be used unless the error variability is negligible compared to the systematic effects of the factors.

Useful for 4 or 5 factors This could be particularly useful when you have a design containing four or five factors and you wish to only use the experimental units from one lot (i.e., batch).
Table containing design matrix for four factors The following is a design for four factors. You would want to randomize these runs before implementing them; -1 and +1 represent the low and high settings, respectively, of each factor.
TABLE 5.11 Four factors: Factorial design section is based on a generator of I = X1*X2*X3, Resolution III; -α and +α are the star points, calculated beyond the factorial range; 0 represents the midpoint of the factor range.
Row X1 X2 X3 X4

1 +1 -1 -1 -1
2 -1 +1 -1 -1
3 -1 -1 +1 -1
4 +1 +1 +1 -1
5 +1 -1 -1 +1
6 -1 +1 -1 +1
7 -1 -1 +1 +1
8 +1 +1 +1 +1
9 -α 0 0 0
10 α 0 0 0
11 0 -α 0 0
12 0 α 0 0
13 0 0 -α 0
14 0 0 α 0
15 0 0 0 -α
16 0 0 0 α
17 0 0 0 0
18 0 0 0 0
19 0 0 0 0
20 0 0 0 0
Determining α in Small Composite Designs
α based on number of treatment combinations in the factorial portion To maintain rotatability for usual CCD's, the value of α is determined by the number of treatment combinations in the factorial portion of the central composite design:
$$\alpha = \left[ \mbox{number of factorial runs} \right] ^{1/4}$$
Small composite designs not rotatable However, small composite designs are not rotatable, regardless of the choice of α. For small composite designs, α should not be smaller than [number of factorial runs]1/4 nor larger than k1/2.