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FACTORIAL DESIGN

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In factorial design, levels of factors are independently varied, each factor at two or more levels. The effects that can e attributed to the factor and their interactions are assed with maximum efficiency in factorial design. So predictions based on results of an undersigned experiment will be less reliable than those which would be obtained in a factorial design. The optimization procedure is facilitated by costruction of an equation that describes the experimental results as a function of the factorial design. Here in case of a factorial, a polynomial equation can be constructed where the coefficients in the equation are related to effects and interations of the factors. Now factorial design with fators at only two level is called as 2 n factorial design where n is the no. of factors. these designs are simplest and often adequate to achieve the experimental objectives. The optimization procedure is facilitated by fitting of an empirical polynomial equation to the experimental results. The equation from for 2n factorial experiment is of the following form: Y= b0 + b1X1 + b2X2 + b3X3 +………+ b12X1 X2 + b13X1 X3 + b23X2 X3+……+ b123X1 X2 X3

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Optimization of chromatographic conditions for both c8 and c18 columns carried out by a factorial design which evaluates temperature, ethanol concentration and mobile phase flow rate. Optimization of chromatographic conditions for both c8 and c18 columns carried out by a factorial design which evaluates temperature, ethanol concentration and mobile phase flow rate. So design matrix would be 2 3 factorial design for c 8 column. So design matrix would be 2 3 factorial design for c 8 column.

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NO.FACTORSLOW LEVELHIGH LEVEL 1TEMP (X1)3050 2%ETHANOL (X2)5560 3 FLOW RATE OF M. PHASE (X3)0.10.2

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In chromatographic condition responses can be In chromatographic condition responses can be 1. Efficiency 2. Retention factor 3. Assymetry 4. Retention time 5. Resolution In this example resolution is considered as response In this example resolution is considered as response

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Experiments for a 2 3 Factorial Design NO.X1X2X3 1 2 1 31 411 5 1 6 11 71 1 8111

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Data analysis for 2 3 factorial design temp%ethanolflow rate resolution/res ponse 30550.12.4 50550.11.8 30600.11.9 50600.11.4 30550.22.4 50550.21.8 30600.21.6 50600.21.3

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Coding / Transformation The formula for transformation is The formula for transformation is X-the average of the two levels X-the average of the two levels one half the difference of the levels one half the difference of the levels

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NO.X1X2X3X1 X2X1 X3X2 X3 X1 X2 X3 RESPO NSE (Y) 1 111 2.4 21 1 11.8 31 111.9 4111 1.4 5 11 12.4 611 1 1.8 711 1 1.6 811111111.3

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The coefficients for polynomial equation are calculated as The coefficients for polynomial equation are calculated as Σ XY/2 n Σ XY/2 n Where X is the value (+1 or -1) in the column appropriate for the coefficient being calculated, Y is the response. Y is the response.

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X1YX2 YX3YX1X2YX1X3YX2X3YX1X2X3YY -2.4 2.4 -2.42.4 -1.81.8-1.8 1.8-1.81.8 1.9-1.9 1.9 1.4 -1.41.4-1.4 1.4 -2.4 2.4 -2.4 2.4 -1.81.8 -1.8 1.8-1.81.6 -1.61.6-1.61.6-1.6 1.6 1.3 average average B1B1 b2b2 b3b3 b 12 b 13 b 23 b 123 b0b0 -0.275-0.25-0.050.05-0.050.025 1.825

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Summary output Regression Statistics Multiple R1 R Square1 Adjusted R Square65535 Standard Error0 Observations8 ANOVA DfSSMSF Significance F Regression71.1750.1678570#NUM! Residual06.9E-3165535 Total71.175

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Coefficie nts Standard Errort StatP-value Lower 95% Upper 95% Lower 95.0 % Upper 95.0 % Intercept1.825065535#NUM!1.825 X1-0.275065535#NUM!-0.275 X2-0.25065535#NUM!-0.25 X3-0.05065535#NUM!-0.05 X1 X20.05065535#NUM!0.05 X1 X3-0.05065535#NUM!-0.05 X2 X30.025065535#NUM!0.025 X1 X2 X30.025065535#NUM!0.025

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RESIDUAL OUTPUT PROBABILITY OUTPUT Observation Predicted RESPONSE (Y)Residuals Standard ResidualsPercentile RESPO NSE (Y) 12.4006.251.3 21.82.22E-160.79772418.751.4 31.9-4.4E-16-1.5954531.251.6 41.4-4.4E-16-1.5954543.751.8 52.40056.251.8 6 -2.2E-16-0.7977268.751.9 71.60081.252.4 81.3-2.2E-16-0.7977293.752.4

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Multiple Linear Regression (MLR) for first reduced model after least variable elimination SUMMARY OUTPUT Regression Statistics Multiple R0.99787 R Square0.995745 Adjusted R Square0.970213 Standard Error0.070711 Observations8 ANOVA dfSSMSF Significanc e F Regression61.170.195390.121965 Residual10.005 Total71.175

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Coefficients Standard Error t StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept1.8250.025730.008721.5073462.1426541.5073462.142654 X1-0.2750.025-110.057716-0.592650.042654-0.592650.042654 X2-0.250.025-100.063451-0.567650.067654-0.567650.067654 X3-0.050.025-20.295167-0.367650.267654-0.367650.267654 X1 X20.050.02520.295167-0.267650.367654-0.267650.367654 X1 X3-0.050.025-20.295167-0.367650.267654-0.367650.267654 X2 X30.025 10.5-0.292650.342654-0.292650.342654

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RESIDUAL OUTPUT PROBABILITY OUTPUT Observati on Predicted RESPON SE (Y)Residuals Standard ResidualsPercentile RESPON SE (Y) 12.425-0.025-0.935416.251.3 21.7750.0250.93541418.751.4 31.8750.0250.93541431.251.6 41.425-0.025-0.9354143.751.8 52.3750.0250.93541456.251.8 61.825-0.025-0.9354168.751.9 71.625-0.025-0.9354181.252.4 81.2750.0250.93541493.752.4

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Multiple linear regression for final reduced model Regression Statistics Multiple R0.969755 R Square0.940426 Adjusted R Square 0.916596 Standard Error 0.118322 Observations8 ANOVA dfSSMSF Significanc e F Regression21.1050.5525 39.464 2 9 0.000866 Residual50.070.014 Total71.175

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Coefficients Std Error t StatP-value Lower 95% Upper 95% Lower 99.0 % Upper 99.0 % Intercept 1.8250.0418343.6251.19E-071.7174651.9325351.6563241.993676 X1-0.2750.04183-6.57370.00122-0.38253-0.16747-0.44368-0.10632 X2-0.250.04183-5.97610.00187-0.35753-0.14247-0.41868-0.08132

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RESIDUAL OUTPUT PROBABILITY OUTPUT Observatio n Predicted RESPON SE (Y) Residuals Standard Residuals Percentile RESPON SE (Y) 12.350.050.56.251.3 21.85-0.05-0.518.751.4 31.80.1131.251.6 41.30.1143.751.8 52.350.050.556.251.8 61.85-0.05-0.568.751.9 71.8-0.2-281.252.4 81.3-2.2E-16-2.2E-1593.752.4

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X1 LOWX1 HIGH X2 LOW 12.421.8 52.461.8 2.41.8 X2 HIGH 31.941.4 71.681.3 1.751.35 Interaction plot showing (by the parallel lines) that factors A and B do not influence each other.

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Diagnostic Checking: Adjusted 2 R Rule of Thumb: Values > 0.8 typically indicate that the regression model is a good fit. Otherwise, a second order model is required because the linear regression is not fit for our experiment. Final equation for this final reduced model will be y = 1.825-0.275*temp-0.25*(%ethanol). Final equation for this final reduced model will be y = 1.825-0.275*temp-0.25*(%ethanol).

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Prediction from equation Coefficients of both temperature and %ethanol are having (-) negative value. So if we put lesser the value for both we will get good/ highest response / resolution. Now, batch 5 is good, so we can say that batch 5 is best which give good resolution.

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