factors:
(5)This section is divided into the following subsections:
A second order model is generally used to approximate the response once
it is realized that the experiment is close to the optimum response region
where a first order model is no longer adequate. The second order model
is usually sufficient for the optimum region, as third order and higher
effects are seldom important. The second order regression model takes
the following form for factors:(5)
The model contains 
regression parameters that include
coefficients for main effects (
), coefficients for quadratic main
effects (
) and coefficients for two factor interaction
effects (
.
). A full factorial design with all
factors at three levels would provide estimation of all the required regression
parameters. However, full factorial three level designs are expensive
to use as the number of runs increases rapidly with the number of factors.
For example, a three factor full factorial design with each factor at
three levels would require 
runs while a design with four factors
would require 
runs. Additionally, these designs
will estimate a number of higher order effects which are usually not of
much importance to the experimenter. Therefore, for the purpose of analysis
of response surfaces, special designs are used that help the experimenter
fit the second order model to the response with the use of a minimum number
of runs. Examples of these designs are the central composite
and Box-Behnken designs.
Central composite designs are two level full factorial (2
) or fractional factorial (2
) designs augmented by a number of
center points and other chosen runs. These designs are such that they
allow the estimation of all the regression parameters required to fit
a second order model to a given response.
The simplest of the central composite designs can be used to fit a second
order model to a response with two factors. The design consists of a 2
full factorial design augmented by
a few runs at the center point (such a design is shown in Figure 9.10 (a)). A central composite design is obtained when
runs at four other points - (
), (
), (
) and (
) are added to this design. These points
are referred to as axial points or star points and represent
runs where all but one of the factors are set at their mid-levels. The
number of axial points in a central composite designs having 
factors is 2
. The distance of the axial points
from the center point is denoted by 
and is always specified in terms of
coded values. For example, the central composite design in Figure 9.10 (b) has 
, while for the design of Figure 9.10 (c) 
. It can be noted that when 
, each factor is run at five levels
(
, 
, 
, 
and 
) instead of the three levels of 
, 
and 
. The reason for running central composite
designs with 
is to have a rotatable design, which
is explained next.
|
Figure 9.10: Central composite designs - (a) shows the
2 |
A central composite design is said to be rotatable if the variance of
any predicted value of the response, 
, for any level of the factors depends
only on the distance of the point from the center of the design, regardless
of the direction. In other words, a rotatable central composite design
provides constant variance of the estimated response corresponding to
all new observation points that are at the same distance from the center
point of the design (in terms of the coded variables). The variance of
the predicted response at any point, 
, is given as follows:
(6)
The contours of ![$V[/hat{y}_{p}]$](graphics/chapter9__108.png){kind=small}
for the central composite design in
Figure 9.10 (c) are shown in Figure 9.11.
The contours are concentric circles indicating that the central composite
design of Figure 9.10 (c) is rotatable. Rotatability
is a desirable property because the experimenter does not have any prior
information about the location of the optimum. Therefore, a design that
provides equal precision of estimation in all directions would be preferred.
Such a design will assure the experimenter that no matter what direction
is taken to search for the optimum, he/she will be able to estimate the
response value with equal precision. A central composite design is rotatable
if the value of 
for the design satisfies the following
equation:
(7)
where 
is the number of replicates
of the runs in the original factorial design and 
is the number of replicates of the
runs at the axial points. For example, a central composite design with
two factors, having a single replicate of the original factorial design,
and a single replicate of all the axial points, would be rotatable for
the following 
value:
Thus, a central composite design in two factors, having a single replicate
of the original 2
design and axial points, and with
, is a rotatable design. This design
is shown in Figure 9.10 (c).
|
Figure 9.11: The contours of |
A central composite design is said to be spherical if all factorial
and axial points are at same distance from the center of the design. Spherical
central composite designs are obtained by setting 
. For example, the rotatable design
in Figure 9.10 (c) is also a spherical design because
for this design 
.
Central composite designs in which the axial points represent the mid
levels for all but one of the factors are also referred to as face-centered
central composite designs. For these designs, 
and all factors are run at three levels,
which are 
, 
and 
in terms of the coded values (see
Figure 9.12).
|
Figure 9.12: Face-centered central composite design for three factors. |
In Chapter 8, highly fractional
designs introduced by Plackett and Burman were discussed. Plackett-Burman
designs are used to estimate main effects in the case of two level fractional
factorial experiments using very few runs. G. E. P. Box and D. W. Behnken
(1960) introduced similar designs for three level factors that are widely
used in response surface methods to fit second-order models to the response.
The designs are referred to as Box-Behnken designs. The designs were developed
by the combination of two level factorial designs with incomplete block
designs. For example, Figure 9.13 shows the Box-Behnken
design for three factors. The design is obtained by the combination of
2
design with a balanced incomplete
block design having three treatments and three blocks (for details see
[1]).
|
Figure 9.13: Box-Behnken design for three factors - (a) shows the geometric representation and (b) shows the design. |
The advantages of Box-Behnken designs include the fact that they are
all spherical designs and require factors to be run at only three levels.
The designs are also rotatable or nearly rotatable. Some of these designs
also provide orthogonal blocking. Thus, if there is a need to
separate runs into blocks for the Box-Behnken design, then designs are
available that allow blocks to be used in such a way that the estimation
of the regression parameters for the factor effects are not affected by
the blocks. In other words, in these designs the block effects are orthogonal
to the other factor effects. Yet another advantage of these designs is
that there are no runs where all factors are at either the 
or 
levels. For example, in Figure 9.13 the representation of the Box-Behnken design for
three factors clearly shows that there are no runs at the corner points.
This could be advantageous when the corner points represent runs that
are expensive or inconvenient because they lie at the end of the range
of the factor levels. A few of the Box-Behnken designs available in DOE++ are presented
in Appendix E.
Continuing with the example in Chapter 9, Method
of Steepest Ascent, the first order model was found to be inadequate
for the region near the optimum. Once the experimenter realized that the
first order model was not adequate (for the region with a reaction temperature
of 350 
and reaction time of 165 minutes),
it was decided to augment the experiment with axial runs to be able to
complete a central composite design and fit a second order model to the
response. [Note]
Notice the advantage of using a central composite design, as the experimenter
only had to add the axial runs to the 2
design with center point runs, and
did not have to begin a new experiment. The experimenter decided to use
to get a rotatable design. The obtained
response values are shown in Figure 9.14. Such
a design can be set up in DOE++ using the properties shown in Figure 9.15. The resulting design is shown in Figure 9.16
and results from the analysis of the design are shown in Figure
9.17.
|
Figure 9.14: Response values for the two factor central composite design in Example 9.2. |
|
Figure 9.15: Properties for the central composite design in Example 9.2. |
|
Figure 9.16: Central composite design for the experiment in Example 9.2. |
|
Figure 9.17: Results for the central composite design in Example 9.2. |
The results in Figure 9.17 show that the main
effects, 
and 
, the interaction, 
, and the quadratic main effects, 
and 
, (represented as AA and BB in the
figure) are significant. The lack-of-fit test also shows that the second
order model with these terms is adequate and a higher order model is not
needed. Using these results, the model for the experiment, in terms of
the coded values, is:
The response surface and the contour plot for this model, in terms of the actual variables, are shown in Figures 9.18 (a) and (b), respectively.
|
Figure 9.18: Response surface and contour plot for the experiment in Example 9.2. |
design with center point runs, (b) shows
the two-factor central composite design with 
and (c) shows the two-factor central composite
design with 
.![$V[/hat{y}_{p}]$](graphics/chapter9__117.png){kind=small}
