5.
Process Improvement
5.5.
Advanced topics
5.5.8.

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 twofactor
interactions. The resulting design allows estimation of the
higherorder 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 = X_{1}*X_{2}*X_{3}, Resolution
III; α and +α are the star points,
calculated beyond the factorial range; 0 represents the midpoint of
the factor range.
Row

X_{1}

X_{2}

X_{3}

X_{4}


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 k^{1/2}.
