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 operability4) Response surface modeling and optimization

Types of Experimental DesignChoice of experiments depends on level of knowledge before experiments, resource available and objectives of the experiments

Discovering important process factors• Placket-Burman• Fractional FactorialEstimating the effect and interaction of several factors•Full Fractional•Fractional Factorial• TiguchiFor optimization•Central composite•Simplex lattice•D-optimal•Box Behnken

Design Selection Guideline

Numberof Factors

ComparativeObjective

ScreeningObjective

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 or Plackett-

Burman

Screen first to reduce number of

factors

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.

Central Composite 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.

Implementation of design

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.

Design matrix

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 CCFIn 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 ofFactors

FactorialPortion

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

CCC (CCI)

Rep X1 X2 X3

1 -1 -1 -1

1 +1 -1 -1

1 -1 +1 -1

1 +1 +1 -1

1 -1 -1 +1

1 +1 -1 +1

1 -1 +1 +1

1 +1 +1 +1

1 -1.682 0 0

1 1.682 0 0

1 0 -1.682 0

1 0 1.682 0

1 0 0 -1.682

1 0 0 1.682

6 0 0 0

Total Runs = 20

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

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.

Box-Behnken designs

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) CCF Box-Behnken Rep X1 X2 X3 Rep X1 X2 X3 Rep X1 X2 X3

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 Designs for Three Factors

CCC   CCI

SequenceNumber X1 X2 X3  

SequenceNumber X1 X2 X3

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

Factor Settings for CCF and Box-Behnken Designs for Three Factors

CCC   Box-Behnken

Number X1 X2 X3   Number X1 X2 X3

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          

Number of Factors Central Composite Box-Behnken 2 13 (5 center points) -

3 20 (6 centerpoint runs) 15

4 30 (6 center point runs) 27

5 33 (fractional factorial) or 52 (full factorial)

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6 54 (fractional factorial) or 91 (full factorial)

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Number of Runs Required by Central Composite and Box-Behnken Designs

Case study of Box Behnken

Experimental Design

BatchCode

Coded value Actual value

X1 X2 X3 X1 X2 X3

B1 -1 1 0 10 15 25

B 2 0 -1 -1 20 5 15

B 3 1 -1 0 30 5 25

B 4 -1 0 -1 10 10 15

B 5 -1 0 1 10 10 35

B 6 0 -1 1 20 5 35

B 7 1 0 1 30 10 35

B 8 -1 -1 0 10 5 25

B 9 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

Coded and actual values of Box-Behnken design

The amount of HPMC K4M (X1),

amount of Carbopol 934P (X2) and

amount of Sodium alginate (X3)

were selected as independent variables.

Batch X1

(%)X2

(%)X3

(%)FLTSD

(sec)TFTSD

(hr)t50SD

(hr)nSD

B1 10 15 25 2 1 2.50.35 13.10.03 0.480.02

B2 20 5 15 9 2 10.00.41 12.50.06 0.570.01

B3 30 5 25 4 2 24.00.29 13.30.04 0.520.03

B4 10 10 15 11 2 4.20.32 12.00.07 0.600.02

B5 10 10 35 5 2 5.30.28 11.90.04 0.650.07

B6 20 5 35 3 2 24.00.34 14.80.08 0.520.01

B7 30 10 35 26 4 5.60.35 14.70.05 0.510.01

B8 10 5 25 4 2 8.00.44 12.00.01 0.390.02

B9 20 10 25 3 2 2.50.22 15.80.02 0.440.01

B10 30 15 25 15 3 4.40.14 12.00.04 0.520.03

B11 20 15 15 33 4 3.60.26 12.80.03 0.620.02

B12 20 15 25 15 4 4.90.16 11.10.02 0.470.04

B13 30 10 15 3 1 24.00.36 11.30.05 0.360.06

B14 20 10 25 24 3 4.80.18 10.50.04 0.500.01

B15 20 10 25 10 2 6.80.45 15.00.07 0.480.04

B16 20 10 25 6 2 7.00.0.36 13.20.06 0.700.03

B17 20 10 25 12 2 4.20.26 13.20.03 0.450.02

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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

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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

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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.

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• 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 correlationr = 0 No linear correlation/ absencer = + 0.9 / + 0.8 High degree of relationshipr = + 0.2 or 0.1 Low high degree of relationship

Correlation Coefficient

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1 2 3 1 2 1 3 2 3 1 1 2 2 3 3FLT 8.41 0.63X 2.13X 3.25X 3.51X X 2.75X X 1.25X X 5.95X X 7.55X X 3.32X X

1 2 3 1 2 1 3 2 3 1 1 2 2 3 3TFT 5.98 4.6875X 3.2625X 0.21X 4.47X X 6.92X X 2.35X X 3.40X X 0.65X X 1.72X X

50 1 2 3 1 2 1 3 2 3 1 1 2 2 3 314.84 0.14X 0.28X 1.21X 0.45X X 1.08X X 0.05X X 1.7425X X 1 .87X X 0.66X Xt

1 2 3 1 2 1 3 2 3 1 1 2 2 3 30.38 0.02 0.01 0.02 0.01 0.03 0.01 0.07 0.02 0.04X X X X X X X X X X X X X X Xn

R-Square = 0.5996

R- Square = 0.898329

R –Square = 0.928214

R-Square = 0.845881

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)

2. Find SSB:(between sum of squares)

3. Find SSW by subtraction:(within sum of squares)

N

XXSST

2

2

N

X

M

TSSB

2

i

i

SSBSSTSSW

Correction factor

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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

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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.

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Analysis of Variance table

Source SS DF MS F calculated F tabulated

Treatments

244.14 02 122.07  8.68

 

Errors 168.80 12 14.07

Total 412.94 14  

Conclusion: If, F calculated > F Tabulated, then the Null hypothesis is rejected and if F calculated < F tabulated, then we accept the Null hypothesis.

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One Way analysis of variance – Example 1

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

93.3807.208,152247,152SST

Degree of Freedom (df) = N - 1 = 15 – 1 = 14Between treatment df = k - 1 = 3-1 = 2Within treatment df = N - k = 15 – 3 = 12

N

X

N

TSSB

2

i

i

N

XXSST

2

2

155x3N 07.208,152X2

73.1307.208,1525

)508(

5

)497(

5

)506(SSB

222

SSBSSTSSW

20.2573.1393.38SSW

Ti – sum of observations in treatment groups

Ni – number of observations in treatment group

One Way analysis of variance – Example 1 (cont.)

Ftabulated = 3.89

Fcalculated < Ftabulated

All means are equal. Therefore,

Method A = Method B = Method C

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

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One Way analysis of variance – Example 2

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

Ftabulated = 8.02

Fcalculated > Ftabulated (19.41 > 8.02)

Shows significant difference in results.

Source Df SS MS F

Between analysis 2 0.05930.29

7F = 19.41

Within analysis 9 0.01380.00153

Total 11 0.0733

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Two Way ANOVA

• In two way ANOVA, one can test sets of hypothesis with the same data at the same time.

• SST = SSR + SSC + SSESST – 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.

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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.

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Two Way ANOVA - Example

Subject A B C

1 12 16 30

2 5 10 18

3 7 28 35

4 10 26 51

• 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

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248Grandmean 33.0nfactorCorrelatio

34.340CFC

RSSR i

67.1012CFR

CSSC

2

i

367,154CFXSST 2ij

66.190SSCSSTSSE

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Two Way ANOVA - ExampleF(3,6) tablated = 4.76Fcalculated < Ftabulated

Therefore H0 is accepted at 5% level, no difference between subjects

Fcalculated > Ftabulated

15.37 > 5.14H0 is rejected at 5% level of significance.Hence, there is significance difference between maximum plasma concentration of different formulations.

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 - -

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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).

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.

GBSHAH KBIPER 47

Key Concepts

• ANOVA can be used in situations where the researcher is interested in the differences in sample means across three or more categories.

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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

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Strategies for Experimentation

Contour Plot

A contour plot is a graphical technique for representing a 3-dimensional 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.

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Response surface plot (RSP) and contour plot

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

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Types of RSP and its contour plot contiue…

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RSP and CP illustration surface with maximum

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RSP and CP illustration surface with maximum

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RSP and CP illustration surface with saddle point or minimax

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