Optimal central composite designs
for fitting second order response surface
regression models
Sung Hyun Park, Hyuk Joo Kim and Jae-Il Cho
2006 International Conference on Design of
Experiments and Its Applications
July 9-13, 2006, Tianjin, P.R. China
1
Contents
2
1
Introduction
2
Orthogonality, rotatability and slope rotatability
3
The alphabetic design optimality
4
Optimal CCDs when the true model is of third order
5
Concluding remarks
1. Introduction
 The central composite design (CCD) is a design
widely used for estimating second order response
surfaces. It is perhaps the most popular class of
second order designs.
 Let x1 , x2 ,
, xk denote the explanatory variables
being considered. Much of the motivation of the
CCD evolves from its use in sequential
experimentation. It involves the use of a two-level
factorial or fraction (resolution Ⅴ) combined with
the following 2k axial points:
3
1. Introduction
x1
x2
xk
-α
0
…
0
α
0
…
0
0
-α
…
0
0
α
…
0
…
0
0
…
-α
0
0
…
α
 As a result, the design involves, say, F=2k factorial
points (or F=2k-p fractional factorial points), 2k axial
points, and n0 center points. The CCDs were first
introduced by Box and Wilson(1951).
4
2. Orthogonality, rotatability and slope rotatability
Let us consider the model represented by
k
k
k
yu  0   i xiu    x   ij xiu x ju   u ,
i 1
i 1
2
ii iu
u  1, 2,
,N
1
i j
where xiu is the value of the variable xi at the uth
experimental point, and εu's are uncorrelated
random errors with mean zero and variance σ2. This
is the second order response surface model.
5
2. Orthogonality, rotatability and slope rotatability
2.1. Orthogonality
In this subsection, we consider the model with the
pure quadratic terms corrected for their means, that
is,
k

k

k
yu     i xiu   ii x  x   ij xiu x ju   u ,
'
0
where
i 1
i 1
k
   0    ii x
'
0
i 1
2
i
2
iu
and
2
i
,N
 2
i j
N
x   xi2 / N
2
i
u  1, 2,
i 1
. In regard to
orthogonality, this model is often used for the sake of
simplicity in calculation.
6
2. Orthogonality, rotatability and slope rotatability
Let bo' , bi , bii , bij denote the least squares estimators of
o' , i , ii , ij respectively. In the CCD, all the covariances
between the estimated regression coefficients except
Cov  bii , b jj 
are zero. But if the  X ' X  matrix is a diagonal
1
matrix, then also becomes zero. This property is called
orthogonality.
It is well-known (See Myers (1976, p.134) and Khuri
and Cornell (1996, p.122).) that the condition for a
CCD to be an orthogonal design is that
1/ 2
 F  F  2k  n0   F 
 

2


7
2. Orthogonality, rotatability and slope rotatability
8
2. Orthogonality, rotatability and slope rotatability
2.2. Rotatability
It is important for a second order design to possess
a reasonably stable distribution of
NVar  y  x   /  2 throu-
ghout the experimental design region. Here y  x  is
the estimated response at the point x   x1 , x2 , , xk ' .
A rotatable design is one for which
NVar  y  x   /  2
has
the same value at any two locations that have the
same distance from the design center. In other words,
is constant on spheres. The rotatability property was
first introduced by Box and Hunter(1957).
9
2. Orthogonality, rotatability and slope rotatability
It is well-known that the condition for a CCD to be
rotatable is that
  F 1/ 4
This means that the value of α for a rotatable CCD
does not depend on the number of center points.
10
2. Orthogonality, rotatability and slope rotatability
Table 2.2 gives the values of α for rotatable CCDs
for various k. Note that for k=5 and 6, a CCD is also
suggested in which a fractional factorial is used
instead of a complete factorial. Also tabulated are F
and T, where T=2k+1.
The designs considered in the table contain a single
center point. This by no means implies that one
would always use only one center point.
11
2. Orthogonality, rotatability and slope rotatability
2.3. Slope rotatability
Suppose that estimation of the first derivative of η
is of interest (η is the expected value of the response
variable y). For the second order model,
y  x 
xi
 bi  2bii xi   bij x j
j i
The variance of this derivative is a function of the
point x at which the derivative is estimated and also
a function of the design.
12
2. Orthogonality, rotatability and slope rotatability
Hader and Park (1978) proposed an analog of the
Box-Hunter rotatability criterion, which requires that
the variance of y  x  / xi be constant on circles (k=2),
spheres (k=3), or hyperspheres (k≥4) centered at
the design origin.
Estimates of the derivative over axial directions
would then be equally reliable for all points x
equidistant from the design origin. They referred to
this property as slope rotatability, and showed that
the condition for a CCD to be a slope-rotatable design
is as follows:
13
2. Orthogonality, rotatability and slope rotatability
 2  F  n0   8   4kF   6  F  N  4  k   kF  8  k  1  4
 8  k  1 F 2   2  2 F 2  k  1 N  F   0
14
2. Orthogonality, rotatability and slope rotatability
15
2. Orthogonality, rotatability and slope rotatability
Table 2.3 gives slope-rotatable values of α for 2≤k≤6.
For k=5 and 6, CCDs involving fractional factorials
are also considered.
16
3. The alphabetic design optimality
3.1. D-optimality
The best known and most often used criterion is Doptimality. D-optimality is based on the notion that
the experimental design should be chosen so as to
achieve certain properties in the matrix . Here is the
following matrix:
1 x11

1 x12
X 


1 x1N
17
xk1
x112
xk21
x11 x12
xk 2
x122
xk22
x12 x22
xkN
x12N
2
xkN
x1N x2 N
xk 1,1 xk1 

xk 1,2 xk 2 


xk 1, N xkN 
3. The alphabetic design optimality
Suppose the maximum, arithmetic mean, and
geometric mean of the eigenvalues 1 , 2 , ,  p of

X 'X

are indicated by max ,  and  . It turns out that an
important norm on the moment matrix is the
determinant; that is,
p
D  X X   i1    p
'
i 1
where p is the number of parameters in the model.
18
1
3. The alphabetic design optimality
Under the assumption of independent normal
errors with constant variance, the determinant of X ' X
is inversely proportional to the square of the volume
of the confidence region for the regression
coefficients. The volume of the confidence region is
relevant because it reflects how well the set of
coefficients are estimated. A D-optimal design is one
in which X ' X is maximized; that is,
max  X ' X  
where maxξ implies that the maximum is taken over
all design ξ’s.
19
3. The alphabetic design optimality
3.3. E-optimality
The criterion E, evaluation of the smallest
eigenvalue, also gains in understanding by a passage
to variances. It is the same as minimizing the largest
eigenvalue of the dispersion matrix; that is,
E  max i i
where i=1,2,…,p
In terms of variance, it is a minimax approach.
Thus the E-optimal design is defined as
min maxi  i 
20
3. The alphabetic design optimality
3.4. Application to the CCD
For fitting the two factor second order model, we
can consider the following CCD. It consists of (i) a 22
factorial, at levels ±1, (ii) a one-factor-at-a-time
array and (iii) n0 center points. That is, the matrix X
is given by
21
3. The alphabetic design optimality
Then the matrix X ' X is given by
N
0

0

a
a

0
0
0
a
a
a
0
0
0
0
a
0
0
0
0
b
F
0
0
F
b
0
0
0
0
0
0 
0

0
0

F
where N is the number of experimental points, F is
the number of factorial points, a=F+2α2 and b=F+2α4
For the two factor CCD, for example, the value of D
is


D  n0 Fa 2  N b2  F 2  2a 2  b  F 
where n0 is the number of center points.
22
3. The alphabetic design optimality
Figure 3.1 shows a plot of D versus for α the
indicated values of n0 for a CCD in k=2 factors.
23
3. The alphabetic design optimality
In CCDs, the determinant of moment matrix has a
tendency of increase as α increases. That is, a larger
value of α is recommendable for D-optimal sense. But
in a practical experiment, the region of interest is
usually restricted and the conditions of experiment
cannot be set for a large α. So it is necessary for the
experimenter to choose as large as possible within
the controllable region of interest.
On the other hand, for the two factor CCD, the value of A
is
2 1
1
N b F
A   i   

a F b  F N  b  F   2a 2
24
3. The alphabetic design optimality
Figure 3.2 shows plots of A versus for the indicated
values of n0 for CCDs in k=2 factors. Table 3.1 shows
the results of optimal α values for two factor CCDs.
25
3. The alphabetic design optimality
26
4. Optimal CCDs when the true model is of third order
Suppose that we fit the second order response
surface model, but the true model is of third order.
For this case, what value of α should be used in the
CCD?
We can generally formulate the problem by
supposing that the experimenter fits a model yˆ  x1 , x2 , , xk 
of order d1 in a region R of the explanatory variables.
However, the true model is a polynomial g  x1 , x2 , , xk 
of order d2 , where d2>d1 . Then, a reasonable design
criterion is the minimization of
M 
27
N

2

R
E  y  x   g  x   dx/  dx
2
R
 3
4. Optimal CCDs when the true model is of third order
The multiple integral in Eq. (3) actually represents
the average of the expected squared deviations of
the true response from the estimated response over
the region R.
Writing the integral

R
dx  1/ k ,
2
NK


E
y
x

g
x
     dx
2 R


2
NK
 2  E  y  x   E  y   E  y   g  x  dx
 R
2
2
NK
 2  E  y  x   E  y   dx    E  y   g  x   dx .
R
R

M

28

 4
4. Optimal CCDs when the true model is of third order
The first quantity in Eq. (4) is the variance of y,
integrated or, rather averaged over the region R,
whereas the second quantity is the square of the bias,
similarly averaged. Thus M is naturally divided as
follows:
M=V+B
where V is the average variance of y, and B is the
average squared bias of y.
In this section, as a reasonable choice of design we
will consider the design which minimizes B. Such a
design is called the all-bias design.
29
4. Optimal CCDs when the true model is of third order
It is assumed here that the experimenter desires to
fit a quadratic response surface in a cuboidal region
R but that the true function is best described by a
cubic polynomial. The actual measured variables
have been transformed to x1 , x2 ,
, xk which are scaled
so that the region of interest R is a unit cube. Also
the assumption on the design is made that its center
of gravity is at the origin (0,0,…,0) of the cube.
30
4. Optimal CCDs when the true model is of third order
The equation of the fitted model is
y  x1' 1 ,
where
x1'  1, x1 ,
xk ; x12 ,
1  b0 , b1 ,
bk ; b11 ,
, xk2 ; x1 x2 ,
, xk 1 xk 
, bkk ; b12 ,
, bk 1,k  .
The true relationship is written as
E  y   x1' 1  x '2 2
where x '2  [ x13 , x1 x22 ,
x1 xk2 ; x23 , x2 x12 ,
x1 x2 x3 , x1 x2 x4 ,
, x2 xk2 ;
; xk3 , xk x12 ,
, xk xk21;
, xk 2 xk 1 xk ]
contains the cubic contribution to the actual model.
The vector  2' contains the coefficients corresponding
to terms in x '2 ; terms such as 111 , 122 ,
31
are included.
4. Optimal CCDs when the true model is of third order
The matrix X1 is given by
1 x11

1 x12
X1  


1 x1N
xk1
x112
xk21
x11 x21
xk 2
x122
xk22
x12 x22
xkN
x12N
2
xkN
x1N x2 N
xk 1,1 xk ,1 

xk 1,2 xk ,2 


xk 1, N xk , N 
In this case the matrix X2 is
3
 x11
 3
x
X 2   12

 3
 x1 N
32
2
x11 x21
x11 xk21
3
x21
2
x21 x11
x21 xk21
2
x12 x22
x12 xk22
3
x22
2
x22 x12
x22 xk22
x1 N x22N
2
x1N xkN
x23 N
x2 N x12N
2
x2 N xkN
xk31
2
xk 1 x11
xk 1 xk21,1
x11 x21 x31
xk32
2
xk 2 x12
xk 2 xk21,2
x12 x22 x32
3
xkN
xkN x12N
xkN xk21, N
x1N x2 N x3 N
xk  2,1 xk 1,1 xk1 

xk  2,2 xk 1,2 xk 2 


xk  2, N xk 1, N xkN 
4. Optimal CCDs when the true model is of third order
Let us now write
M11  N 1 X1' X1 ,
M12  N 1 X 1' X 2 ,
11  K  x1x1' dx,
12  K  x1x '2 dx,
R
R
where K 1   dx. One can write the bias term as
R

 



'
B  a '2  22  12' 11112  M111M12  11112 11 M111M12  11112  a 2 ,


where the a2 vector is merely 2 n / .
(See Myers (1976, p.213))
33
 5
4. Optimal CCDs when the true model is of third order
The first term in the square brackets in Eq. (5)
contains only the region moment matrices and thus is
independent of the design. The bias term can be no
smaller than the positive semidefinite quadratic form


a '2 22  12' 11112 a 2 . So the experimenter has to use
designs which minimize the positive semidefinite
quadratic form


'


a '2 M 111M 12  11112 11 M 111M 12  11112 a 2
34
4. Optimal CCDs when the true model is of third order
Now we will find out the value of which makes the
optimal design in the CCDs. But, let's assume that a2
is a vector of ones. That is, a2 is (1,1,1,1) for the two
factor CCDs when d1=2 and d2=3. And, if we assume
that the region of interest -1≤xi≤1 is where i=1,2,…,k
then we can obtain region moment matrices(μ11 and
μ12).
35
4. Optimal CCDs when the true model is of third order
For example, let's consider the second order CCD
which minimizes the squared bias from the third
order terms for k=2. The design consists of four
factorial points, four axial points at a distance α from
the origin, and two center points. Then we obtain the
following design moment matrices and region
moment matrices.
 10

 0
1  0
M 11  
10  4  2 2
 4  2 2

 0
36
0
0
4  2 2
4  2 2
4  2 2
0
0
0
0
4  2 2
0
0
0
0
4  2 2
4
0
0
4
4  2 2
0
0
0
0
0

0
0

0
0

4 
4. Optimal CCDs when the true model is of third order
 0

4
 4  2
 4  2 2

1 
M 12 
0
10 
 0

 0
 0


4

0


0
1
11  
4 4

3
4

3

 0
37
0
0
4
3
4
3
4
3
0
0
0
0
4
3
0
0
0
0
0
0
0
0
4
5
4
9
4
9
4
5
0
0

0

0


0


0


0

4
9 
0
0
0
2
4  2 2
2
4  2 2
0
0
0
0
0
0
0
0
12

5
1
12   0
4
0

0
 0


0 


4  2 4 
4  2 2 
0 

0 
0 
0
0
0
4
3
0
0
0
0

4
0
3

12 
0

5
0 0

0 0
0 0 
0
4. Optimal CCDs when the true model is of third order
So


'

 is obtained as
2  32  14  15 
f   
.
675  2   
a '2 M 111M 12  11112 11 M 111M 12  11112 a 2
2
4
2
2
2
The value of which minimizes is found to be
1/ 2
 2 2  1


 0.91018
A very interesting fact is that f(α) has nothing to do
with the number of center points. Table 4.1 gives the
appropriate values of α for second order CCD which
minimize the squared bias from the third order terms
for k factors.
38
4. Optimal CCDs when the true model is of third order
39
5. Concluding remarks
In this paper, we found out values of α which
optimize CCDs for fitting second order response
surface models under several criteria. Table 5.1 gives
the value of α in Tables 2.1, 2.2 and 4.1.
40
5. Concluding remarks
From Table 5.1, we can find that the values of tend
to increase in the following order :
Minimum bias<Orthogonality<Rotatability
<Slope rotatability<Alphabetic optimality
41
5. Concluding remarks
Note that the optimal value of α under the minimum
bias and rotatability criteria does not depend on the
number of center points. Also, an interesting fact is
that the optimal value of α under the minimum bias
criterion is very similar to that under the
orthogonality criterion with one center point.
In conclusion, we will consider reasonable choice of
CCD for fitting the second order model according to
the following cases:
1. when the true model is of second order (d2=2)
2. when the true model is of third order (d2=3)
42
5. Concluding remarks
Table 5.2 shows values of α recommended for the
CCD considering the order d2.
43
Thank you
44