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SCREENING- FULL & FRACTIONAL DESIGN

Tuan Amran Tuan Abdullah, PhD

Institute of Hydrogen Economy

Universiti Teknologi Malaysia

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Slide 2

Screening

• Aim – identify significant factors (variables)

• A factor is ‘significant’ if its influence is greater than the ‘noise’ level (experimental error)

• Usually carry out screening using reduced designs such as factorial or Plackett-Burman designs

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Slide 3

The trials in a factorial design can be represented as points on an n-dimensional cube (n=3 in this case)

1,1,1

-1,1,1

-1,1,-1

1,-1,1

-1,-1,1

-1,-1,-1

1,-1,-1 11,-1

Slide 4

Case Study – HPLC method

• Aim: to optimise the separation of peaks in a HPLC analysis

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Slide 5

Define the Response

• The CRF (chromatographic response function) is used to quantify separation of peaks. This function thus gives a single number to the ‘quality’ of a chromatogram. The aim is thus to maximise the CRF

Slide 6

Define the Factors

• The factors studied in this study were levels in the eluent of:-

• Acetic Acid

• Methanol

• Citric Acid

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Slide 7

Experimental Domain

Low High

Acetic Acid

(mol/L)

0.004 0.01

% Methanol 70 80

Citric Acid (g/L) 2 6

Slide 8

Factorial design (Coded form)

Run

Number

Acetic

Acid

Methanol Citric Acid CRF

1 - - -

2 + - -

3 - + -

4 + + -

5 - - +

6 + - +

7 - + +

8 + + +

This design gives all combinations of the factors at 2 levels

‘+’, high ‘-’, low

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Slide 9

Factorial design (Uncoded)

Run

Number

Acetic

Acid

Methanol Citric Acid CRF

1 0.004 70 2

2 0.01 70 2

3 0.004 80 2

4 0.01 80 2

5 0.004 70 4

6 0.01 70 4

7 0.004 80 4

8 0.01 80 4

This table shows the actual levels of the variables used in the experiments. Normally the order of experiments is randomised but we will keep it in this structured forms so you can see the patterns

Results are inserted here when the experiments are performed

Slide 10

Factorial design (Uncoded)

Run

Number

Acetic

Acid

Methanol Citric Acid CRF

1 0.004 70 2 10

2 0.01 70 2 9.5

3 0.004 80 2 11

4 0.01 80 2 10.7

5 0.004 70 4 9.3

6 0.01 70 4 8.8

7 0.004 80 4 11.9

8 0.01 80 4 11.7

The CRF values are now inserted after the experiments (chromatographic runs) are carried out

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Slide 11

Analysis of the results - Excel

• Calculate Main Effects – this calculates the effect on the response solely due to one factor

• Main effects are the difference between average response at high level of the factor – average response at low level

Slide 12

Acetic

Acid

Methanol Citric

Acid

CRF

-1 -1 -1 10

+1 -1 -1 9.5

-1 +1 -1 11

+1 +1 -1 10.7

-1 -1 +1 9.3

+1 -1 +1 8.8

-1 +1 +1 11.9

+1 +1 +1 11.7

10.18 11.33 10.43

10.55 9.40 10.30

-0.37 1.93 0.13

Average of the ‘high’ values of CRF for each variable e.g for AA = (9.5+10.7+8.8+11.7)/4

Average of ‘low’ values of CRF for each variable e.g for AA = (10+11+9.3+11.9)/4

The ’Main Effect’ is the difference between the ‘high’ and ‘low’ average e.g for AA = (10.19-10.55)

Calculation of Main Effects

The main effects can also be calculated by multiplying the variable column by the CRF column pairwise , adding up the column and then dividing by 4

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Slide 13

Calculation of Interactions

Acetic

Acid

Methanol Citric

Acid

AA*M AA*CA M*CA CRF AA*M*CRF

-1 -1 -1 +1 +1 +1 10 +10

+1 -1 -1 -1 -1 +1 9.5 -9.5

-1 +1 -1 -1 +1 -1 11 -11

+1 +1 -1 +1 -1 -1 10.7 +10.7

-1 -1 +1 +1 -1 -1 9.3 +9.3

+1 -1 +1 -1 +1 -1 8.8 -8.8

-1 +1 +1 -1 -1 +1 11.9 -11.9

+1 +1 +1 +1 +1 +1 11.7 +11.7

Sum = 0.5

0.125 0.025 0.825 0.5/4 =

0.125

Interactions coefficients –found by multiplying the appropriate variable columns

Interactions calculated by multiplying the CRF column and the appropriate variable interactions column. To get the interaction effect add up the column and divide by 4

Slide 14

Factorial Calculations using Minitab

• The program ‘Minitab’ can be used to carry out calculations as follows:-

• To set up the design:

Stat > DOE > Factorial > Create Factorial Design

Type of Design: 2 level factorial design 9default generators)

Number of factors: 3

Designs: Full Factorial

Factors see screen dump on next slide

Options: do not randomize (normally should randomize but for

the tutorial not randomizing makes it easier to see patterns in the layout)

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Slide 15

Slide 16

The generated FFD design as it should appear in Minitab

Type the CRF responses here after performing the experiments

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Slide 17

Slide 18

Factorial Calculations using Minitab

• To analyse the design:

• Stat > DOE > Factorial > Analyse factorial Design

• Click on ‘C8 CRF’ as the response . Accept the default values

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Slide 19

Factorial Calculations - Minitab

• 31/07/2007 11:26:30 ————————————————————

• Welcome to Minitab, press F1 for help.

• Results for: Worksheet 2 • • Full Factorial Design

• Factors: 3 Base Design: 3, 8 • Runs: 8 Replicates: 1 • Blocks: 1 Center pts (total): 0

• All terms are free from aliasing.

• Design Table

• Run A B C • 1 - - - • 2 + - - • 3 - + - • 4 + + - • 5 - - + • 6 + - + • 7 - + + • 8 + + +

•

Minitab Output

Slide 20

Factorial Fit: CRF versus Acetic Acid, Methanol, Citric Acid Estimated Effects and Coefficients for CRF (coded units) Term Effect Coef Constant 10.3625 Acetic Acid -0.3750 -0.1875 Methanol 1.9250 0.9625 Citric Acid 0.1250 0.0625 Acetic Acid*Methanol 0.1250 0.0625 Acetic Acid*Citric Acid 0.0250 0.0125 Methanol*Citric Acid 0.8250 0.4125 Acetic Acid*Methanol*Citric Acid 0.0250 0.0125

Main Effects

Interactions

Coefficients – from fitting to a second order equation Y = bo+b1*x1+b2*x2+b3*x3+b12*x1*x2+b13*x1*x3+b23*x2*x3+b123*x1*x2*x3

Where x1 is acetic acid , x2 is methanol and x3 is citric acid

Note , however, coefficients are simply twice the effects so no new information

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Slide 21

What do the results tell us?

• The main effects tell us which variable has the strongest effect on the response (CRF) – in this case methanol has the strongest effect on CRF

• A negative effect means the response is reduced as the variable increases. The negative effect for acetic acid means that as we increase the concentration of acetic acid, the CRF gets smaller (and hence our separation is worse)

Slide 22

What about interactions?

• An interaction effect is where the effect on the response of one variable depends on the level of another variable.

• In this study methanol and citric acid seem to have the largest interaction.

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Slide 23

Main Effects and Interactions Plots

• Main effects plots help to visually display the variable effect. They graph the average response at the high and low levels. The steeper the graph, the stronger the effect

• The plots can be drawn in Miniab as follows:-

• Stat > DOE >Factorial > factorial Plots

• Tick ‘Main Effects Plots’

• Setup > Select CRF as the response and choose all 3 variables (>>)

• The Interactions plots can be produced similarly (just select ‘Interactions’ instead of ‘Main Effects Plots’)

Slide 24

Me

an

of

CR

F

0.0100.004

11.5

11.0

10.5

10.0

9.5

8070

62

11.5

11.0

10.5

10.0

9.5

Acetic Acid Methanol

Citric Acid

Main Effects Plot (data means) for CRF

Note that Methanol has the steepest slope, indicating the strongest effect

CRF average at ‘high’ methanol

CRF average at ‘low’ methanol

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Slide 25

A cetic A cid

8070 62

11

10

9

Methanol

11

10

9

Citr ic A cid

Acetic

Acid

0.004

0.010

Methanol

70

80

Interaction Plot (data means) for CRF

The plots show there is an interaction effect with methanol and citric acid at high methanol CA has a Positive effect but at low methanol it has a negative effect on CRS

Slide 26

Conclusions

• Methanol has the largest effect on CRF

• The Methanol effect strongly depends on the Citric Acid level. Citric acid has a positive effect at high Methanol but a negative effect at low Methanol

• All 3 variables do seem to affect the result. Citric acid has the smallest main effect but large interaction effect

• Hence probably can’t ‘screen out’ any of these variables from further study

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Significance – Normal Probability Plots

• Normal Probability Plots are used to test whether data is normally distributed.

• In our case, we can use such a plot to test for significance of the effects/coefficients

• If the effects are not significant we expect variations just to be due to random error and this can be tested with the plots. It is only a guide, however, as we have no real estimate of the experimental error

• In Minitab the plot can be generated:

• Stat > DOE >factorial > Analyse Factorial Design > Graphs and Select Effects Plots (Normal)

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Slide 28

Effect

Pe

rce

nt

2.01.51.00.50.0-0.5

99

95

90

80

70

60

50

40

30

20

10

5

1

Factor Name

A A cetic A cid

B Methanol

C C itric A cid

Effect Type

Not Significant

Significant

BC

B

Normal Probability Plot of the Effects(response is CRF, Alpha = .05)

Lenth's PSE = 0.1875

Effects due to random errors should be on a straight line. This plot indicates Methanol and the Methanol/Citric Acid interaction are significant effects

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Slide 29

Problems

• We have not replicated any experiments so no determination of error. We cannot tell if the coefficients (effects) overall are significant (although normal probability plots help). We can only compare them to see which is the most significant

• We also cannot test for curvature – i.e are the effects of the variables linear. A non-linear effect can be when the response at the high and low levels is similar but at intermediate values is much higher or lower. pH effects are often non-linear

Slide 30

Solution?

• Add centre points!!

• Centre points are experiments with all variables set at 0 (coded) i.e. mid values

• Replication of the centre point allows determination of error

Coded Uncoded

Acetic

Acid

0 0.007

Methanol 0 75

Citric Acid 0 4

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Slide 31

Acetic Acid Methanol Citric Acid CRF

-1 -1 -1 10.0

1 -1 -1 9.5

-1 1 -1 11

1 1 -1 10.7

-1 -1 1 9.3

1 -1 1 8.8

-1 1 1 11.9

1 1 1 11.7

0 0 0 10.2

0 0 0 10.4

Results added for the centre points

Slide 32

Estimated Effects and Coefficients for CRF (coded units) Term Effect Coef SE Coef T P Constant 10.362 0.05000 207.25 0.003 Acetic Acid -0.3750 -0.1875 0.05000 -3.75 0.166 Methanol 1.9250 0.9625 0.05000 19.25 0.033 Citric Acid 0.1250 0.0625 0.05000 1.25 0.430 Acetic Acid*Methanol 0.1250 0.0625 0.05000 1.25 0.430 Acetic Acid*Citric Acid 0.0250 0.0125 0.05000 0.25 0.844 Methanol*Citric Acid 0.8250 0.4125 0.05000 8.25 0.077

P is the probability a coefficient is not significantly different from zero i.e no effect on CRF. A low probability (< 0.05 at the 5% level) indicates high significance. The methanol effect is the only significant one at the 5% level although the methanol-citric acid effect is just above the 5% level

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Slide 33

Me

an

of

CR

F

0.0100.0070.004

11.5

11.0

10.5

10.0

9.5

807570

642

11.5

11.0

10.5

10.0

9.5

Acetic Acid Methanol

Citric Acid

Point Type

Corner

Center

Main Effects Plot (data means) for CRF

The centre point responses are all on the linear response line. Thus no curvature is indicated.

Slide 34

Standardized Effect

Pe

rce

nt

20151050-5

99

95

90

80

70

60

50

40

30

20

10

5

1

Factor Name

A A cetic A cid

B Methanol

C C itric A cid

Effect Type

Not Significant

Significant

B

Normal Probability Plot of the Standardized Effects(response is CRF, Alpha = .05)

The Normal Plot shows also that Methanol is the only significant effect but Methanol/CA interaction is

probably also significant

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Fractional Factorial Design in Minitab

• The method for creating fractional factorial very similar to the method for creating full factorial design

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Example Fractional design

StdOrder RunOrder CenterPt Blocks A B C D Rate 1 1 1 1 -1 -1 -1 -1 45 2 2 1 1 1 -1 -1 -1 71 3 3 1 1 -1 1 -1 -1 48 4 4 1 1 1 1 -1 -1 65 5 5 1 1 -1 -1 1 -1 68 6 6 1 1 1 -1 1 -1 60 7 7 1 1 -1 1 1 -1 80 8 8 1 1 1 1 1 -1 65 9 9 1 1 -1 -1 -1 1 43

10 10 1 1 1 -1 -1 1 100 11 11 1 1 -1 1 -1 1 45 12 12 1 1 1 1 -1 1 104 13 13 1 1 -1 -1 1 1 75 14 14 1 1 1 -1 1 1 86 15 15 1 1 -1 1 1 1 70 16 16 1 1 1 1 1 1 96

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1050-5

99

95

90

80

70

60

50

40

30

20

10

5

1

Standardized Effect

Pe

rce

nt

A A

B B

C C

D D

Factor Name

Not Significant

Significant

Effect Type

AD

AC

D

C

A

Normal Plot of the Standardized Effects(response is Rate, Alpha = 0.05)

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9876543210

98

95

90

85

80

70

60

50

40

30

20

10

0

Absolute Standardized Effect

Pe

rce

nt

A A

B B

C C

D D

Factor Name

Not Significant

Significant

Effect Type

AD

AC

D

C

A

Half Normal Plot of the Standardized Effects(response is Rate, Alpha = 0.05)

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Pareto Chart

AB

BD

CD

BC

B

C

D

AD

AC

A

9876543210

Te

rm

Standardized Effect

2.571

A A

B B

C C

D D

Factor Name

Pareto Chart of the Standardized Effects(response is Rate, Alpha = 0.05)

The significant rank order A>D>C>B

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