5.
Process Improvement
5.3.
Choosing an experimental design
5.3.3.
How do you select an experimental design?
5.3.3.6.
Response surface designs
5.3.3.6.3.

Comparisons of response surface designs



Choosing a Response Surface Design

Various CCD designs and BoxBehnken designs are compared and their
properties discussed

Table 3.24 contrasts the structures of four common quadratic designs
one might use when investigating three factors. The table combines
CCC and CCI designs because they are structurally identical.
For three factors, the BoxBehnken design offers some advantage in
requiring a fewer number of runs. For 4 or more factors, this
advantage disappears.

Structural comparisons of CCC (CCI), CCF, and BoxBehnken designs
for three factors

TABLE 3.24: Structural Comparisons of CCC (CCI), CCF, and
BoxBehnken Designs for Three Factors
CCC (CCI)

CCF

BoxBehnken

Rep

X_{1
} 
X_{2}

X_{3}

Rep

X_{1}

X_{2}

X_{3}

Rep

X_{1}

X_{2}

X_{3}

1

1

1

1

1

1

1

1

1

1

1

0

1

+1

1

1

1

+1

1

1

1

+1

1

0

1

1

+1

1

1

1

+1

1

1

1

+1

0

1

+1

+1

1

1

+1

+1

1

1

+1

+1

0

1

1

1

+1

1

1

1

+1

1

1

0

1

1

+1

1

+1

1

+1

1

+1

1

+1

0

1

1

1

+1

+1

1

1

+1

+1

1

1

0

+1

1

+1

+1

+1

1

+1

+1

+1

1

+1

0

+1

1

1.682

0

0

1

1

0

0

1

0

1

1

1

1.682

0

0

1

+1

0

0

1

0

+1

1

1

0

1.682

0

1

0

1

0

1

0

1

+1

1

0

1.682

0

1

0

+1

0

1

0

+1

+1

1

0

0

1.682

1

0

0

1

3

0

0

0

1

0

0

1.682

1

0

0

+1





6

0

0

0

6

0

0

0





Total Runs = 20

Total Runs = 20

Total Runs = 15


Factor settings for CCC and CCI three factor designs

Table 3.25 illustrates the factor settings required for a central
composite circumscribed (CCC) design and for a central composite
inscribed (CCI) design (standard order), assuming three factors, each
with low and high settings of 10 and 20, respectively. Because the
CCC design generates new extremes for all factors, the investigator
must inspect any worksheet generated for such a design to make certain
that the factor settings called for are reasonable.
In Table 3.25, treatments 1 to 8 in each case are the factorial points
in the design; treatments 9 to 14 are the star points; and 15 to 20 are
the systemrecommended center points. Notice in the CCC design how
the low and high values of each factor have been extended to create the
star points. In the CCI design, the specified low and high values
become the star points, and the system computes appropriate settings for
the factorial part of the design inside those boundaries.
TABLE 3.25: Factor Settings for CCC and CCI Designs for Three
Factors
Central Composite
Circumscribed CCC


Central Composite
Inscribed CCI

Sequence
Number

X_{1}

X_{2}

X_{3}


Sequence
Number

X_{1}

X_{2}

X_{3}

1

10

10

10


1

12

12

12

2

20

10

10


2

18

12

12

3

10

20

10


3

12

18

12

4

20

20

10


4

18

18

12

5

10

10

20


5

12

12

18

6

20

10

20


6

18

12

18

7

10

20

20


7

12

12

18

8

20

20

20


8

18

18

18

9

6.6

15

15

*

9

10

15

15

10

23.4

15

15

*

10

20

15

15

11

15

6.6

15

*

11

15

10

15

12

15

23.4

15

*

12

15

20

15

13

15

15

6.6

*

13

15

15

10

14

15

15

23.4

*

14

15

15

20

15

15

15

15


15

15

15

15

16

15

15

15


16

15

15

15

17

15

15

15


17

15

15

15

18

15

15

15


18

15

15

15

19

15

15

15


19

15

15

15

20

15

15

15


20

15

15

15

* are star points

Factor settings for CCF and BoxBehnken three factor designs

Table 3.26 illustrates the factor settings for the corresponding central
composite facecentered (CCF) and BoxBehnken designs. Note that
each of these designs provides three levels for each factor and that the
BoxBehnken design requires fewer runs in the threefactor case.
TABLE 3.26: Factor Settings for CCF and BoxBehnken Designs
for Three Factors
Central Composite
FaceCentered CCC


BoxBehnken

Sequence
Number

X_{1}

X_{2}

X_{3}


Sequence
Number

X_{1}

X_{2}

X_{3}

1

10

10

10


1

10

10

15

2

20

10

10


2

20

10

15

3

10

20

10


3

10

20

15

4

20

20

10


4

20

20

15

5

10

10

20


5

10

15

10

6

20

10

20


6

20

15

10

7

10

20

20


7

10

15

20

8

20

20

20


8

20

15

20

9

10

15

15

*

9

15

10

10

10

20

15

15

*

10

15

20

10

11

15

10

15

*

11

15

10

20

12

15

20

15

*

12

15

20

20

13

15

15

10

*

13

15

15

15

14

15

15

20

*

14

15

15

15

15

15

15

15


15

15

15

15

16

15

15

15






17

15

15

15






18

15

15

15






19

15

15

15






20

15

15

15






* are star points for the CCC

Properties of classical response surface designs

Table 3.27 summarizes properties of the classical quadratic designs.
Use this table for broad guidelines when attempting to choose from
among available designs.
TABLE 3.27: Summary of Properties of Classical Response
Surface Designs
Design Type

Comment

CCC

CCC designs provide high quality predictions over the entire
design space, but require factor settings outside the range
of the factors in the factorial part. Note: When the
possibility of running a CCC design is recognized before starting
a factorial experiment, factor spacings can be reduced to ensure
that ±
for each
coded factor corresponds to feasible (reasonable) levels.
Requires 5 levels for each factor.

CCI

CCI designs use only points within the factor ranges originally
specified, but do not provide the same high quality prediction
over the entire space compared to the CCC.
Requires 5 levels of each factor.

CCF

CCF designs provide relatively high quality predictions over the
entire design space and do not require using points outside the
original factor range. However, they give poor precision for
estimating pure quadratic coefficients.
Requires 3 levels for each factor.

BoxBehnken

These designs require fewer treatment combinations than a central
composite design in cases involving 3 or 4 factors.
The BoxBehnken design is rotatable (or nearly so) but it contains
regions of poor prediction quality like the CCI. Its "missing
corners" may be useful when the experimenter should avoid
combined factor extremes. This property prevents a potential
loss of data in those cases.
Requires 3 levels for each factor.


Number of runs required by central composite and BoxBehnken designs

Table 3.28 compares the number of runs required for a given number of
factors for various Central Composite and BoxBehnken designs.
TABLE 3.28 Number of Runs Required
by Central Composite and BoxBehnken Designs
Number of Factors

Central Composite

BoxBehnken

2

13 (5 center points)



3

20 (6 centerpoint runs)

15

4

30 (6 centerpoint runs)

27

5

33 (fractional factorial) or 52 (full factorial)

46

6

54 (fractional factorial) or 91 (full factorial)

54



Desirable Features for Response Surface
Designs

A summary of desirable properties for response surface designs

G. E. P. Box and N. R. Draper in "Empirical Model Building and Response
Surfaces," John Wiley and Sons, New York, 1987, page 477, identify
desirable properties for a response surface design:
 Satisfactory distribution of information across the experimental
region.
 Fitted values are as close as possible to observed values.
 minimize residuals or error of prediction
 Good lack of fit detection.
 Internal estimate of error.
 Constant variance check.
 Transformations can be estimated.
 Suitability for blocking.
 Sequential construction of higher order designs from simpler
designs
 Minimum number of treatment combinations.
 Good graphical analysis through simple data patterns.
 Good behavior when errors in settings of input variables occur.
