Central Composite Design WWW.PARASSHAH.WEEBLY.COM

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 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 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 Fractional factorial or Plackett-Burman Screen first to reduce number of 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: A factorial(perhaps fractional) design in the factors studied, each having two levels; A set of center points, experimental runs whose values of each factor are the medians of the values used in the factorial portion. 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: The matrix F obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1 and −1. The matrix C from the center points, denoted in coded variables as (0,0,0,...,0), where there are k zeros. 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 CCC explores the largest process space and the CCI explores the smallest process space. Both the CCC and CCI are rotatable designs, but the CCF is not. 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

Scaled Value for Relative to ±1 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 25 25/4 = 2.378 6 26-1 26 26/4 = 2.828

Design matrix for two factor experiment BLOCK X1 X2 1 -1 2 -1.414 1.414 0  Total Runs = 12

Design matrix for three factor experiment CCC (CCI) Rep X1 X2 X3 1 -1 +1 -1.682 1.682 6 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

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.

Box-Behnken designs are response surface designs, specially made to require only 3 levels, coded as -1, 0, and +1. Box-Behnken designs are available for 3 to 10 factors. It is formed by combining two-level factorial designs with incomplete block designs. 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. 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 1 -1 +1 -1.682 1.682 3   6 Total Runs = 20 Total Runs = 15

Factor Settings for CCC and CCI Designs for Three Factors   CCI Sequence Number X1 X2 X3 1 10 12 2 20 18 3 4 5 6 7 8 9 6.6 15 * 23.4 11 13 14 16 17 19

Factor Settings for CCF and Box-Behnken Designs for Three Factors CCC   Box-Behnken Number X1 X2 X3 1 10 15 2 20 3 4 5 6 7 8 9 * 11 12 13 14 16 17 18 19

Number of Runs Required by Central Composite and Box-Behnken Designs 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) 46 6 54 (fractional factorial) or 91 54

Box Behnken Experimental Design Case study of Box Behnken Experimental Design

Coded and actual values of Box-Behnken design Batch Code Coded value Actual value X1 X2 X3 B1 -1 1 10 15 25 B 2 20 5 B 3 30 B 4 B 5 35 B 6 B 7 B 8 B 9 B 10 B 11 B 12 B 13 B 14 B 15 B 16 B 17 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 FLTSD (sec) TFTSD (hr) t50SD nSD B1 10 15 25 2  1 2.50.35 13.10.03 0.480.02 B2 20 5 9  2 10.00.41 12.50.06 0.570.01 B3 30 4  2 24.00.29 13.30.04 0.520.03 B4 11  2 4.20.32 12.00.07 0.600.02 B5 35 5  2 5.30.28 11.90.04 0.650.07 B6 3  2 24.00.34 14.80.08 0.520.01 B7 26  4 5.60.35 14.70.05 0.510.01 B8 8.00.44 12.00.01 0.390.02 B9 2.50.22 15.80.02 0.440.01 B10 15  3 4.40.14 12.00.04 B11 33  4 3.60.26 12.80.03 0.620.02 B12 15  4 4.90.16 11.10.02 0.470.04 B13 3  1 24.00.36 11.30.05 0.360.06 B14 24  3 4.80.18 10.50.04 0.500.01 B15 10  2 6.80.45 15.00.07 0.480.04 B16 6  2 7.00.0.36 13.20.06 0.700.03 B17 12  2 4.20.26 13.20.03 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

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

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.

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

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) Correction factor

4. Calculate the degrees of freedom: dfb = k-1 and dfw = N – k 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

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.

Conclusion: 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.

One Way analysis of variance – Example 1 Method A Method B Method C 102 99 103 101 100 104 98  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 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 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

One Way analysis of variance – Example 2 Source Df SS MS F Between analysis 2 0.0593 0.297 F = 19.41 Within analysis 9 0.0138 0.00153 Total 11 0.0733 A B C 22.53 22.48 22.57 22.60 22.40 22.62 22.54 22.61 22.43 22.65 Ftabulated = 8.02 Fcalculated > Ftabulated (19.41 > 8.02) Shows significant difference in results.

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.

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.

Two Way ANOVA - Example Carry out two way ANOVA for Subject A B C 1 12 16 30 2 5 10 18 3 7 28 35 4 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

Two Way ANOVA - Example 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. 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 -

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.

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

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

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.

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

Types of RSP and its contour plot contiue…

RSP and CP illustration surface with maximum

RSP and CP illustration surface with maximum

RSP and CP illustration surface with saddle point or minimax

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