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Central Composite Design
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Definition of Experimental Design
It is the methodology of how to conduct and plan experiments in
order to extract the maximum amount of information in the
fewest number of runs.
Cost approach
• Change one separate factor at a time (COST)
• PROBLEMS WITH COST:
– Does not lead to real optimum
– Inefficient, unnecessarily many runs
– Provides no information about what happens when
factors are varied simultaneously (ignores interaction)
– Provides less information about the variability of the
responses
Good Experimental Design
Should help us in following:




Show real effect
Reduce noise
Should provide efficient mapping of functional space
Reduce time and cost
Stages in Experimental Design Process
1) Familiarization
Formulate Question(s) stating the objectives and Goals of
the Investigation.
2) Screening
Screening designs provide simple models with information
about dominating variables, and information about ranges.
In addition they provide few experiments/ factors which
means that relevant information is gained in only a few
experiments.
3) Finding optimal region of operability
4) Response surface modeling and optimization
Types of Experimental Design
Choice of experiments depends on level of knowledge before
experiments, resource available and objectives of the experiments
Discovering important process factors
• Placket-Burman
• Fractional Factorial
Estimating the effect and interaction of several factors
•Full Fractional
•Fractional Factorial
• Tiguchi
For optimization
•Central composite
•Simplex lattice
•D-optimal
•Box Behnken
Design Selection Guideline
Number
of Factors
Comparative
Objective
Screening
Objective
Response Surface
Objective
1
1-factor completely
randomized design
_
_
2-4
Randomized block
design
Full or fractional
factorial
Central composite
or Box-Behnken
5 or more
Randomized block
design
Fractional factorial Screen first to
or Plackettreduce number of
Burman
factors
Central Composite Design
A Box-Wilson Central Composite Design, commonly called `a
central composite design,' contains an imbedded factorial or
fractional factorial design with center points that is enlarged
with a group of `star points' that allow estimation of curvature.
Implementation of design
The design consists of three distinct sets of experimental runs:
1. A factorial(perhaps fractional) design in the factors studied,
each having two levels;
2. A set of center points, experimental runs whose values of
each factor are the medians of the values used in the factorial
portion.
3. A set of axial points (star point), experimental runs identical to
the centre points except for one factor, which will take on
values both below and above the median of the two factorial
levels, and typically both outside their range.
Design matrix
The design matrix for a central composite design experiment
involving k factors is derived from a matrix, d, containing the
following three different parts corresponding to the three types
of experimental runs:
1. The matrix F obtained from the factorial experiment. The
factor levels are scaled so that its entries are coded as +1 and
−1.
2. The matrix C from the center points, denoted in coded
variables as (0,0,0,...,0), where there are k zeros.
3. A matrix E from the axial points, with 2k rows. Each factor is
sequentially placed at ±α and all other factors are at zero.
Central
Composite
Design Type
Terminology
Comments
Circumscribed CCC
CCC designs are the original formed CCD. These designs have
circular, spherical, or hyperspherical symmetry and require 5
levels for each factor. Enlarging an existing factorial or
fractional factorial design with star points can produce this
design.
Inscribed
CCI
CCI design uses the factor settings as the star points and
creates a factorial or fractional factorial design within those
limits (in other words, a CCI design is a scaled down CCC
design with each factor level of the CCC design divided by to
generate the CCI design). This design also requires 5 levels of
each factor.
Face Centered CCF
In this design the star points are at the center of each face of
the factorial space, so = ± 1. This variety requires 3 levels of
each factor.
Comparison of 3 Central composite design
a. CCC explores the largest process
space and the CCI explores the
smallest process space.
b. Both the CCC and CCI are
rotatable designs, but the CCF is
not.
c. Both the CCC and CCI are require
5 level for each factor while CCF
is require 3 level for each factor.
Generation of a Central Composite Design for Two Factor
A central composite design always contains twice as many
star points as there are factors in the design. The star
points represent new extreme values (low and high) for
each factor in the design ±α.
Determining α in central Composite Design
To maintain rotatability, the value of α depends on the number
of experimental runs in the factorial portion of the central
composite design
If the factorial is a full factorial, then
If the factorial is a fractional factorial,
then
Number of
Factors
Factorial
Portion
Scaled Value for
Relative to ±1
2
22
22/4 = 1.414
3
23
23/4 = 1.682
4
24
24/4 = 2.000
5
25-1
24/4 = 2.000
5
25
25/4 = 2.378
6
26-1
25/4 = 2.378
6
26
26/4 = 2.828
Design matrix for two factor experiment
BLOCK
X1
X2
1
-1
-1
1
1
-1
1
-1
1
1
1
1
1
0
0
1
0
0
2
-1.414
0
2
1.414
0
2
0
-1.414
2
0
1.414
2
0
0
2
0
0
Total Runs = 12
Design matrix for three factor experiment
Rep
1
1
1
1
1
1
1
1
1
1
1
1
1
1
6
CCC (CCI)
X1
X2
-1
-1
+1
-1
-1
+1
+1
+1
-1
-1
+1
-1
-1
+1
+1
+1
-1.682
0
1.682
0
0
-1.682
0
1.682
0
0
0
0
0
0
Total Runs = 20
X3
-1
-1
-1
-1
+1
+1
+1
+1
0
0
0
0
-1.682
1.682
0
Table below show how to choose value of α and of center point for CCD
Where
K: number of factor
nf: experiments in factorial design
ne: experiments in star design
Box-Behnken designs
The Box-Behnken design is an independent quadratic design in that
it does not contain an surrounded factorial or fractional factorial
design. In this design the treatment combinations are at the
midpoints of edges of the process space and at the center. These
designs are rotatable (or near rotatable) and require 3 levels of
each factor.
i.
Box-Behnken designs are response surface designs, specially
made to require only 3 levels, coded as -1, 0, and +1.
ii. Box-Behnken designs are available for 3 to 10 factors. It is
formed by combining two-level factorial designs with
incomplete block designs.
iii. This procedure creates designs with desirable statistical
properties but, most importantly, with only a fraction of the
experimental trials required for a three-level factorial. Because
there are only three levels, the quadratic model was found to be
appropriate.
iv. In this design three factors were evaluated, each at three levels,
and experiment design were carried out at all seventeen
possible combinations.
Choosing a Response Surface Design
CCC (CCI)
X1
X2
-1
-1
+1
-1
-1
+1
+1
+1
-1
-1
+1
-1
-1
+1
+1
+1
-1.682
0
X3
-1
-1
-1
-1
+1
+1
+1
+1
0
Rep
1
1
1
1
1
1
1
1
1
X1
-1
+1
-1
+1
-1
+1
-1
+1
-1
X2
-1
-1
+1
+1
-1
-1
+1
+1
0
X3
-1
-1
-1
-1
+1
+1
+1
+1
0
Rep
1
1
1
1
1
1
1
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
0
6
Rep
1
1
1
1
1
1
1
1
1
6
0
0
Total Runs = 20
CCF
0
0
Total Runs = 20
Box-Behnken
X1
X2
-1
-1
+1
-1
-1
+1
+1
+1
-1
0
+1
0
-1
0
+1
0
0
-1
0
Total Runs = 15
X3
0
0
0
0
-1
-1
+1
+1
-1
Factor Settings for CCC and CCI Designs for Three Factors
CCC
Sequence
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X1
10
20
10
20
10
20
10
20
6.6
23.4
15
15
15
15
15
15
15
15
15
15
CCI
X2
10
10
20
20
10
10
20
20
15
15
6.6
23.4
15
15
15
15
15
15
15
15
X3
10
10
10
10
20
20
20
20
15
15
15
15
6.6
23.4
15
15
15
15
15
15
*
*
*
*
*
*
Sequence
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X1
12
18
12
18
12
18
12
18
10
20
15
15
15
15
15
15
15
15
15
15
X2
12
12
18
18
12
12
12
18
15
15
10
20
15
15
15
15
15
15
15
15
X3
12
12
12
12
18
18
18
18
15
15
15
15
10
20
15
15
15
15
15
15
Factor Settings for CCF and Box-Behnken Designs for Three Factors
CCC
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X1
10
20
10
20
10
20
10
20
10
20
15
15
15
15
15
15
15
15
15
15
Box-Behnken
X2
10
10
20
20
10
10
20
20
15
15
10
20
15
15
15
15
15
15
15
15
X3
10
10
10
10
20
20
20
20
15
15
15
15
10
20
15
15
15
15
15
15
*
*
*
*
*
*
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
X1
10
20
10
20
10
20
10
20
15
15
15
15
15
15
15
X2
10
10
20
20
15
15
15
15
10
20
10
20
15
15
15
X3
15
15
15
15
10
10
20
20
10
10
20
20
15
15
15
Number of Runs Required by Central Composite and
Box-Behnken Designs
Number of Factors Central Composite
2
13 (5 center points)
Box-Behnken
-
3
20 (6 centerpoint runs)
15
4
30 (6 center point runs)
27
5
33 (fractional factorial) or 52
(full factorial)
46
6
54 (fractional factorial) or 91
(full factorial)
54
Case study of
Box Behnken
Experimental Design
Coded and actual values of Box-Behnken design
Batch
Code
Actual value
Coded value
X1
X2
X3
X1
X2
X3
B1
-1
1
0
10
15
25
B2
0
-1
-1
20
5
15
B3
1
-1
0
30
5
25
B4
-1
0
-1
10
10
15
B5
-1
0
1
10
10
35
B6
0
-1
1
20
5
35
B7
1
0
1
30
10
35
B8
-1
-1
0
10
5
25
B9
0
0
0
20
10
25
B 10
1
1
0
30
15
25
B 11
0
1
-1
20
15
15
B 12
0
1
1
20
15
25
B 13
1
0
-1
30
10
15
B 14
0
0
0
20
10
25
B 15
0
0
0
20
10
25
B 16
0
0
0
20
10
25
B 17
0
0
0
20
10
25
The amount
K4M (X1),
of
HPMC
amount of Carbopol 934P
(X2) and
amount
of
alginate (X3)
Sodium
were
selected
as
independent variables.
21
TFTSD
(hr)
2.50.35
t50SD
(hr)
13.10.03
0.480.02
15
92
10.00.41
12.50.06
0.570.01
5
25
42
24.00.29
13.30.04
0.520.03
10
10
15
11  2
4.20.32
12.00.07
0.600.02
B5
10
10
35
52
5.30.28
11.90.04
0.650.07
B6
20
5
35
32
24.00.34
14.80.08
0.520.01
B7
30
10
35
26  4
5.60.35
14.70.05
0.510.01
B8
10
5
25
42
8.00.44
12.00.01
0.390.02
B9
20
10
25
32
2.50.22
15.80.02
0.440.01
B10
30
15
25
15  3
4.40.14
12.00.04
0.520.03
B11
20
15
15
33  4
3.60.26
12.80.03
0.620.02
B12
20
15
25
15  4
4.90.16
11.10.02
0.470.04
B13
30
10
15
31
24.00.36
11.30.05
0.360.06
B14
20
10
25
24  3
4.80.18
10.50.04
0.500.01
B15
20
10
25
10  2
6.80.45
15.00.07
0.480.04
B16
20
10
25
62
7.00.0.36 13.20.06
0.700.03
B17
20
10
25
12  2
Batch
X1
(%)
X2
(%)
X3
(%)
FLTSD
(sec)
B1
10
15
25
B2
20
5
B3
30
B4
4.20.26
13.20.03
nSD
0.450.02
Multiple Regression
• It is an extension of linear regression in
which we wish to relate a response, Y
dependent variables to more than one
independent variable
• Linear Regression
– Y = A+ BY
• Multiple Regression
– Y = bo + b1X1 + b2X2+….
– X1, X2, …. Represent factors which influence the
response
28
Y = bo + b1X1 + b2X2 + b3X3…
•
•
•
•
Y is response i.e. dissolution time
Xi is independent variable
bo is the intercept
bi is regression coefficient for the ith
independent variable
• X1, X2, X3.. Are the levels of variables
29
The Polynomial equation generated by this experimental design is
described as:
Yi = b0 + b1x1 +b2x2 + b3x3 + b12x1x2 + b13 x1x3 + b23x2x3 +
b11x12 +b22x22 + b33x32
Where Yi is the dependent variable
b0 is the intercept; bi, bij and bijk represents the regression
coefficients
Xi represents the level of independent variables which were
selected from the preliminary experiments.
30
Correlation Coefficient
• When two variables are correlated with each other it is
important to know the amount or extent of correlation
between them,
r=1
Present direct or positive correlation
r = -1
Present inverse or negative correlation
r=0
No linear correlation/ absence
r = + 0.9 / + 0.8 High degree of relationship
r = + 0.2 or 0.1
Low high degree of relationship
31
FLT  8.41  0.63X1  2.13X2  3.25X3  3.51X1X2  2.75X1X3  1.25X2 X3  5.95X1X1  7.55X2X2  3.32X3X3
R-Square = 0.5996
TFT  5.98  4.6875X1  3.2625X2  0.21X3  4.47X1X2  6.92X1X3  2.35X2 X3  3.40X1X1  0.65X2X2 1.72X3X3
R- Square = 0.898329
t50  14.84  0.14X1  0.28X2  1.21X3  0.45X1X2  1.08X1X3  0.05X2 X3  1.7425X1X1 1.87X2X2  0.66X3X3
R –Square = 0.928214
n  0.38  0.02X1  0.01X2  0.02X3  0.01X1X2  0.03X1X3  0.01X2X3  0.07X1X1  0.02X2X2  0.04X3X3
R-Square = 0.845881
32
ANOVA or Analysis of Variance
• Analysis of variance technique developed by R
A Fisher, to compare two or more groups
means.
• Analysis of variance (ANOVA) is used to find
out the main and interaction effects of
categorical independent variables (called
"factors") on an interval dependent variable.
Steps in Computation of ANOVA
1.Find SST:
(Total sum of squares)
SST 
X

X


2
2
N
Correction factor

 X
2
2. Find SSB:
(between sum of squares)
SSB 
3. Find SSW by subtraction:
(within sum of squares)
SSW  SST  SSB
Ti

Mi
N
4. Calculate the degrees of freedom: dfb = k-1 and dfw = N – k.
(N is total number of observations, k – number of methods
to be compared)
5. Construct the mean square (MS) estimates by dividing SSB
and SSW by their degrees of freedom:
MSw = SSW / dfw
MSb = SSB / dfb
6. Find F ratio by Formula:
F = MSb / MSw
35
One-Way ANOVA
• It is also known as Completely Randomized
Design (CRD).
• We can take two independent groups ‘t’
test to analyze in ANOVA.
Ex: Two treatment are randomly assigned to
different patients. The results in two groups,
each group representing one of the two
treatments.
36
Analysis of Variance table
Source
SS
DF
MS
Treatme
nts
244.14
02
122.07
Errors
168.80
12
Total
412.94
14
F calculated
F tabulated
8.68
14.07
Conclusion:
If, F calculated > F Tabulated, then the Null hypothesis is rejected and if
F calculated < F tabulated, then we accept the Null hypothesis.
37
One Way analysis of variance – Example 1
SST 



Method A
Method B
Method C
102
99
103
101
100
100
101
99
99
100
101
104
102
98
102
 Xa = 506
 Xb = 497
 Xc = 508
Xa mean =
101.2
Xb mean = 99.4
Xc mean = 101.6
s.d. = 0.84
s.d. = 1.14
s.d. = 2.07

 X
X
  X
2
2
N
SST  152 ,247  152 ,208 .07  38 .93
T  X 
SSB  

N
N
2
i
i
 (506 )2 (497 )2 (508 )2 
  152 ,208 .07  13 .73
SSB  


 5

5
5


SSW  SST  SSB
SSW  38.93  13.73  25.20
2
 152 ,208 .07
N  3x5  15
Ti – sum of observations in
treatment groups
Ni – number of observations
in treatment group
Degree of Freedom (df) = N - 1 = 15 – 1 = 14
Between treatment df = k - 1 = 3-1 = 2
Within treatment df = N - k = 15 – 3 = 12
38
One Way analysis of variance – Example
1 (cont.)
Source
Df
SS
MS
F
Between methods
2
13.73
6.87
F = 3.27
Within methods
12
25.20
2.10
Total
14
38.93
Ftabulated = 3.89
Fcalculated < Ftabulated
All means are equal.
Therefore,
Method A = Method B =
Method C
One Way analysis of variance – Example 2
Source
A
B
C
22.53
22.48
22.57
22.60
22.40
22.62
22.54
22.48
22.61
22.62
22.43
22.65
Df
SS
MS
F
Between analysis
2
0.0593
0.297
F = 19.41
Within analysis
9
0.0138
0.001
53
Total
11
0.0733
Ftabulated = 8.02
Fcalculated > Ftabulated (19.41 > 8.02)
Shows significant difference in results.
40
Two Way ANOVA
• In two way ANOVA, one can test sets of
hypothesis with the same data at the same
time.
• SST = SSR + SSC + SSE
SST – Total sum of square
SSR – Sum of square due to rows
SSC – Sum of square due to column
SSE – sum of square due to error.
41
Two Way ANOVA - Example
• The determination of maximum plasma
concentration of drug in mcg/ml for 3
different formulation A, B & C, was the subject
of a recent experiment. Four different subjects
chosen at random for a group were used for
this purpose.
42
Two Way ANOVA - Example
Subject
A
B
C
1
12
16
30
2
5
10
18
3
7
28
35
4
10
26
51
Grandmean 
SSR 
SSC 
SST 
248
12
Correlationfactor  0.33
Ri
 CF  340 .34
C

2
C
i
 CF  1012 .67
X
 CF  154 ,367
R
2
ij
• Carry out two way ANOVA
for
– There is no significant
difference
between
subjects and
– There is no significance
difference
between
maximum
plasma
concentration of different
formulations
SSE  SST  SSC  190.66
43
Two Way ANOVA - Example
Source
Df
SS
MS
F
SSR
R-1 = 3
340.34
113.45
F(3,6) = 3.569
SSC
C-1 = 2
1012.67
506.34
F(2,6) = 15.93
SSE
(R-1) (C-1) = 6
190.66
31.78
SST Total
1543.67
11
-
-
F(3,6) tablated = 4.76
Fcalculated < Ftabulated
Therefore H0 is accepted at
5% level, no difference
between subjects
Fcalculated > Ftabulated
15.37 > 5.14
H0 is rejected at 5% level of
significance.
Hence, there is significance
difference
between
maximum
plasma
concentration of different
formulations.
44
ANOVA - Overview
• Analysis of variance tests the null hypotheses that group
means do not differ. It is not a test of differences in
variances, but rather assumes relative homogeneity of
variances. Thus some key ANOVA assumptions are that
the groups formed by the independent variables are
relatively equal in size and have similar variances on the
dependent variable ("homogeneity of variances").
• Like regression, ANOVA is a parametric procedure which
assumes multivariate normality (the dependent has a
normal distribution for each value category of the
independents).
45
ANOVA - Overview
• The key statistic in ANOVA is the F-test of
difference of group means, testing if the
means of the groups formed by values of the
independent variable are different enough not
to have occurred.
• If the group means do not differ significantly
then it is inferred that the independent
variables did not have an effect on the
dependent variable.
Key Concepts
• ANOVA can be used in situations where the
researcher is interested in the differences in
sample means across three or more
categories.
GBSHAH KBIPER
47
Key Concepts (cont.)
• Examples:
– Reduction in pain/BP by various drugs
– Percent distribution after 15 min for tablets for a single batch
tested in 5 laboratories
– Comparison of dissolution of various tablet formulations
– Replicate tablet dissolution for number of Laboratories
– Change in BP during pre-clinical study comparing 2 drugs and
control
– Increase in exercise time for 3 treatments of anti-histaminics at
three clinical sites
– HB level of no. of groups of children fed by 3 different diets
– Performance of 3 salesman
48
Strategies for Experimentation
49
Contour Plot
A contour plot is a graphical technique for representing a 3dimensional surface by plotting constant z slices, called contours,
on a 2-dimensional format. That is, given a value for z, lines are
drawn for connecting the (x, y) coordinates where that z value
occurs.
Response surface plot (RSP) and contour plot
51
The contour plot is formed by:
• Vertical axis: Independent variable 2
• Horizontal axis: Independent variable 1
• Lines: iso-response values
The independent variables are usually restricted to a regular grid.
The actual techniques for determining the correct iso-response
values are rather complex and are almost always computer
generated.
An additional variable may be required to specify the Z
values for drawing the iso-lines. Some software packages
require explicit values. Other software packages will
determine them automatically.
Types of RSP and its contour plot
Types of RSP and its contour plot
contiue…
55
RSP and CP illustration surface with maximum
56
RSP and CP illustration surface with maximum
57
RSP and CP illustration surface with saddle point or minimax
58
59