FACTORIAL DESIGN. In factorial design, levels of factors are independently varied, each factor at two or more levels. The effects that can e attributed.

FACTORIAL FACTORIAL DESIGNDESIGN

•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 2n 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

Optimization of chromatographic Optimization of chromatographic conditions for both c8 and c18 conditions for both c8 and c18 columns carried out by a factorial columns carried out by a factorial design which evaluates temperature, design which evaluates temperature, ethanol concentration and mobile ethanol concentration and mobile phase flow rate.phase flow rate.

So design matrix would be 2So design matrix would be 233 factorial design for cfactorial design for c8 8 column.column.

NO. FACTORS LOW LEVEL HIGH LEVEL

1 TEMP (X1) 30 50

2 %ETHANOL (X2) 55 60

3FLOW RATE OF M.

PHASE (X3) 0.1 0.2

In chromatographic condition responses In chromatographic condition responses can be can be

1.1. EfficiencyEfficiency

2.2. Retention factorRetention factor

3.3. AssymetryAssymetry

4.4. Retention timeRetention time

5.5. ResolutionResolution

In this example resolution is considered In this example resolution is considered as responseas response

Experiments for a 2Experiments for a 23 3 Factorial DesignFactorial Design

NO. X1 X2 X3

1 -1 -1 -1

2 -1 1 -1

3 1 -1 -1

4 1 1 -1

5 -1 -1 1

6 -1 1 1

7 1 -1 1

8 1 1 1

Data analysis for 23 factorial design

temp %ethanol flow rate resolution/response

30 55 0.1 2.4

50 55 0.1 1.8

30 60 0.1 1.9

50 60 0.1 1.4

30 55 0.2 2.4

50 55 0.2 1.8

30 60 0.2 1.6

50 60 0.2 1.3

Coding / TransformationCoding / Transformation

The formula for transformation is The formula for transformation is X-the average of the two X-the average of the two

levelslevels one half the difference of the one half the difference of the

levelslevels

NO. X1 X2 X3 X1 X2 X1 X3 X2 X3X1 X2

X3

RESPONSE (Y)

1 -1 -1 -1 1 1 1 -1 2.4

2 -1 1 -1 -1 1 -1 1 1.8

3 1 -1 -1 -1 -1 1 1 1.9

4 1 1 -1 1 -1 -1 -1 1.4

5 -1 -1 1 1 -1 -1 1 2.4

6 -1 1 1 -1 -1 1 -1 1.8

7 1 -1 1 -1 1 -1 -1 1.6

8 1 1 1 1 1 1 1 1.3

The coefficients for polynomial The coefficients for polynomial equation are calculated as equation are calculated as

ΣΣ XY/2 XY/2nn

Where X is the value (+1 or -1) in the Where X is the value (+1 or -1) in the column appropriate for the column appropriate for the coefficient being calculated,coefficient being calculated,

Y is the response.Y is the response.

X1Y X2 Y X3Y X1X2Y X1X3Y X2X3Y X1X2X3Y Y

-2.4 -2.4 -2.4 2.4 2.4 2.4 -2.4 2.4

-1.8 1.8 -1.8 -1.8 1.8 -1.8 1.8 1.8

1.9 -1.9 -1.9 -1.9 -1.9 1.9 1.9 1.9

1.4 1.4 -1.4 1.4 -1.4 -1.4 -1.4 1.4

-2.4 -2.4 2.4 2.4 -2.4 -2.4 2.4 2.4

-1.8 1.8 1.8 -1.8 -1.8 1.8 -1.8 1.6

1.6 -1.6 1.6 -1.6 1.6 -1.6 -1.6 1.6

1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3

averageaverage

B1 b2 b3 b12 b13 b23 b123 b0

-0.275 -0.25 -0.05 0.05 -0.05 0.025 0.025 1.825

Summary output

Regression Statistics

Multiple R 1

R Square 1

Adjusted R Square 65535

Standard Error 0

Observations 8

ANOVA

  Df SS MS FSignificance

F

Regression 7 1.175 0.167857 0 #NUM!

Residual 0 6.9E-31 65535

Total 7 1.175      

 Coefficie

ntsStandard

Error t Stat P-valueLower

95%Upper

95%

Lower 95.0%

Upper 95.0%

Intercept 1.825 0 65535 #NUM! 1.825 1.825 1.825 1.825

X1 -0.275 0 65535 #NUM! -0.275 -0.275 -0.275 -0.275

X2 -0.25 0 65535 #NUM! -0.25 -0.25 -0.25 -0.25

X3 -0.05 0 65535 #NUM! -0.05 -0.05 -0.05 -0.05

X1 X2 0.05 0 65535 #NUM! 0.05 0.05 0.05 0.05

X1 X3 -0.05 0 65535 #NUM! -0.05 -0.05 -0.05 -0.05

X2 X3 0.025 0 65535 #NUM! 0.025 0.025 0.025 0.025

X1 X2 X3 0.025 0 65535 #NUM! 0.025 0.025 0.025 0.025

RESIDUAL OUTPUT

PROBABILITY OUTPUT

ObservationPredicted

RESPONSE (Y) ResidualsStandard Residuals Percentile

RESPONSE (Y)

1 2.4 0 0 6.25 1.3

2 1.8 2.22E-16 0.797724 18.75 1.4

3 1.9 -4.4E-16 -1.59545 31.25 1.6

4 1.4 -4.4E-16 -1.59545 43.75 1.8

5 2.4 0 0 56.25 1.8

6 1.8 -2.2E-16 -0.79772 68.75 1.9

7 1.6 0 0 81.25 2.4

8 1.3 -2.2E-16 -0.79772 93.75 2.4

Multiple Linear Regression (MLR) for first reduced model after least Multiple Linear Regression (MLR) for first reduced model after least variable eliminationvariable elimination

SUMMARY OUTPUT

Regression Statistics 

Multiple R 0.99787

R Square 0.995745

Adjusted R Square 0.970213

Standard Error 0.070711

Observations 8

ANOVA

  df SS MS FSignificanc

e F

Regression 6 1.17 0.195 39 0.121965

Residual 1 0.005 0.005

Total 7 1.175      

  CoefficientsStandard

Errort Stat P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 1.825 0.025 73 0.00872 1.507346 2.142654 1.507346 2.142654

X1 -0.275 0.025 -11 0.057716 -0.59265 0.042654 -0.59265 0.042654

X2 -0.25 0.025 -10 0.063451 -0.56765 0.067654 -0.56765 0.067654

X3 -0.05 0.025 -2 0.295167 -0.36765 0.267654 -0.36765 0.267654

X1 X2 0.05 0.025 2 0.295167 -0.26765 0.367654 -0.26765 0.367654

X1 X3 -0.05 0.025 -2 0.295167 -0.36765 0.267654 -0.36765 0.267654

X2 X3 0.025 0.025 1 0.5 -0.29265 0.342654 -0.29265 0.342654

RESIDUAL OUTPUTPROBABILITY

OUTPUT

Observation

Predicted RESPON

SE (Y) ResidualsStandard Residuals Percentile

RESPONSE (Y)

1 2.425 -0.025 -0.93541 6.25 1.3

2 1.775 0.025 0.935414 18.75 1.4

3 1.875 0.025 0.935414 31.25 1.6

4 1.425 -0.025 -0.93541 43.75 1.8

5 2.375 0.025 0.935414 56.25 1.8

6 1.825 -0.025 -0.93541 68.75 1.9

7 1.625 -0.025 -0.93541 81.25 2.4

8 1.275 0.025 0.935414 93.75 2.4

Multiple linear regression for final reduced modelMultiple linear regression for final reduced model

Regression Statistics

Multiple R 0.969755

R Square 0.940426

Adjusted R Square

0.916596

Standard Error

0.118322

Observations 8

ANOVA

  df SS MS FSignificanc

e F

Regression 2 1.105 0.552539.464

29

0.000866

Residual 5 0.07 0.014

Total 7 1.175      

Normal Probability Plot

y = 0.013x + 1.1774

R2 = 0.937

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100

Sample Percentile

RE

SP

ON

SE

(Y)

  CoefficientsStd

Errort Stat P-value

Lower 95%

Upper 95%

Lower 99.0%

Upper 99.0%

Intercept 1.825 0.04183 43.625 1.19E-07 1.717465 1.932535 1.656324 1.993676

X1 -0.275 0.04183 -6.5737 0.00122 -0.38253 -0.16747 -0.44368 -0.10632

X2 -0.25 0.04183 -5.9761 0.00187 -0.35753 -0.14247 -0.41868 -0.08132

RESIDUAL OUTPUTPROBABILITY

OUTPUT

Observation

Predicted RESPON

SE (Y)Residuals

Standard Residuals

PercentileRESPON

SE (Y)

1 2.35 0.05 0.5 6.25 1.3

2 1.85 -0.05 -0.5 18.75 1.4

3 1.8 0.1 1 31.25 1.6

4 1.3 0.1 1 43.75 1.8

5 2.35 0.05 0.5 56.25 1.8

6 1.85 -0.05 -0.5 68.75 1.9

7 1.8 -0.2 -2 81.25 2.4

8 1.3 -2.2E-16 -2.2E-15 93.75 2.4

Residuals

-0.4

-0.2

0

0.2

0 1 2 3 4 5 6 7 8 9

Residuals

X1 LOW X1 HIGH

X2 LOW

1 2.4 2 1.8

5 2.4 6 1.8

2.4 1.8

X2 HIGH

3 1.9 4 1.4

7 1.6 8 1.3

1.75 1.35

prediction of interaction from graph

2.4

1.81.75

1.35

0

0.5

1

1.5

2

2.5

3

low high

value of factor

avera

ge

low

Series2

Interaction plot showing (by the parallel lines) that factors A and B do not influence each other.

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 Final equation for this final reduced model will be y = 1.825-0.275*temp-model will be y = 1.825-0.275*temp-0.25*(%ethanol).0.25*(%ethanol).

Prediction from equationCoefficients 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.