what shape is your method in?

What Shape is your Method In?A Tutorial on the Application of Experimental Designs to

Development of Chromatographic MethodsBy Dr Jeff Hughes

School of Applied Science, RMIT UniversityMelbourne , Australia

Mean o

f CRF

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

RMIT University Slide 2

• The aim of this tutorial is to demonstrate how the methods of experimental design can be used to investigate chromatographic procedures

• In this tutorial we will look at using Factorial Designs to answer the following questions:

• Which factors significantly influence the of separation of peaks in our chromatogram?

• Which factor has the greatest influence on separation?


Data for this tutorial is taken from “Chemometrics: Experimental Design” by Ed Morgan

Calculations are demonstrated using Excel, but can be carried out using the commercial program Minitab ( a demo version can be downloaded from http://minitab.com )

http://minitab.com/


Factorial Designs• The most suitable type of design for

screening• Each variable (factor) has a set number of

possible levels or values• If there are k variables, each set at 2

possible levels (‘high’ and ‘low’) then there are 2k possible combinations

• These designs are called two-level factorial designs. If all combinations are used they are called full factorial designs


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


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,-111,-1


Case Study – HPLC method• Aim: to optimise the separation of peaks

in a HPLC analysis


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


Define the Factors• The factors studied in this study were

levels in the eluent of:-• Acetic Acid • Methanol• Citric Acid


Experimental DomainLow High

Acetic Acid (mol/L)

0.004 0.01

% Methanol 70 80

Citric Acid (g/L) 2 6


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


Factorial design (Uncoded)

Run Number

Acetic Acid


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


Factorial design (Uncoded)

Run Number

Acetic Acid


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


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


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


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 effectadd up the column and divide by 4


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)


The generated FFD design as it should appear in Minitab

Type the CRF responses hereafter performing the experiments


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


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


Factorial Fit: CRF versus Acetic Acid, Methanol, Citric Acid

Estimated Effects and Coefficients for CRF (coded units)

Term Effect CoefConstant 10.3625Acetic Acid -0.3750 -0.1875Methanol 1.9250 0.9625Citric Acid 0.1250 0.0625Acetic Acid*Methanol 0.1250 0.0625Acetic Acid*Citric Acid 0.0250 0.0125Methanol*Citric Acid 0.8250 0.4125Acetic Acid*Methanol*Citric Acid 0.0250 0.0125

Main Effects

Interactions

Coefficients – from fitting to a second order equationY = 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 simplytwice the effects sono new information


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)


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.


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’)


Mean o

f CRF

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


Citric Acid


Note that Methanol has the steepestslope, indicating the strongest effect

CRF average at ‘high’ methanol

CRF average at ‘low’ methanol


Acetic Acid

8070 62

11

10

9

Methanol

11

10

9

Citric Acid

AceticAcid0.0040.010

Methanol7080

Interaction Plot (data means) for CRF

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


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


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)


Effect

Perc

ent

2.01.51.00.50.0-0.5

99

95

90

80

70605040

30

20

10

5

1

Factor NameA Acetic AcidB MethanolC Citric Acid

Effect TypeNot SignificantSignificant

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 indicatesMethanol and the Methanol/Citric Acidinteraction are significant effects


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


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


Acetic Acid Methanol Citric Acid CRF-1 -1 -1 10.01 -1 -1 9.5-1 1 -1 111 1 -1 10.7-1 -1 1 9.31 -1 1 8.8-1 1 1 11.91 1 1 11.70 0 0 10.20 0 0 10.4 Results added

for the centre points


Estimated Effects and Coefficients for CRF (coded units)

Term Effect Coef SE Coef T PConstant 10.362 0.05000 207.25 0.003Acetic Acid -0.3750 -0.1875 0.05000 -3.75 0.166Methanol 1.9250 0.9625 0.05000 19.25 0.033Citric Acid 0.1250 0.0625 0.05000 1.25 0.430Acetic Acid*Methanol 0.1250 0.0625 0.05000 1.25 0.430Acetic Acid*Citric Acid 0.0250 0.0125 0.05000 0.25 0.844Methanol*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% levelalthough the methanol-citric acid effect is just above the 5% level


Mean o

f CRF

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


Citric Acid

Point TypeCornerCenter


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


Standardized Effect

Perc

ent

20151050-5

99

95

90

80

70

605040

30

20

10

5

1

Factor NameA Acetic AcidB MethanolC Citric Acid

Effect TypeNot SignificantSignificant

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


Next Phase?• Factorial designs give indication of

significant effects and interactions• Designs such as the Central Composite

Design (CCD) can be used to find the best (optimal) settings of the variables and plot Response Surfaces

• CCD designs involve adding extra points (trials) to the Factorial Designs


CCD design for 3 factors. The 8 factorial points are corners of the cube.In this study the cube points correspond to the FFD .6 axial points are added to form a CCD

Cube (factorial) points

Axial points


This is an example of a response surface plot for the optimization of capillary electrophoresis separation of a mixture of ranitidine-related compounds. Ranitidine is a drug used in the treatment of gastric and duodenal ulcers. The two variables being optimized are pH and applied voltage. The response variable used in the optimization is the logarithm of the CEF function, a parameter devised to assess the quality of a chromatogram (this is an alternative response function to the

CRF). This function takes into account peak separation and total elution time (J. Chrom. A 766 245-54. Optimization of the capillary electrophoresis separation of ranitidine and related compounds. V.M. Morris, C. Hargraeves, K. Overall, P.J.Marriott, J.G.Hughes)

Example ofa CCD design and the generated response surface


Extra Information• Notes on factorial and central composite

designs can be found at the author’s website:

• Chemometrics in Australia

This site also has an Excel spreadsheet which sets out the calculations for the FFD design used in this tutorial

http://www.rmit.edu.au/appliedsciences/chemometrics-in-aust


Author: Dr Jeff Hughes

School of Applied Science, RMIT University

Melbourne , Australia

http://www.rmit.edu.au/staff/hughes j

http://www.rmit.edu.au/staff/hughes%20j

what shape is your method in?

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