Download - 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
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• 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?
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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 )
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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
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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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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
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Case Study – HPLC method• Aim: to optimise the separation of peaks
in a HPLC analysis
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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
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Define the Factors• The factors studied in this study were
levels in the eluent of:-• Acetic Acid • Methanol• Citric Acid
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Experimental DomainLow High
Acetic Acid (mol/L)
0.004 0.01
% Methanol 70 80
Citric Acid (g/L) 2 6
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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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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
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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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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
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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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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
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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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The generated FFD design as it should appear in Minitab
Type the CRF responses hereafter performing the experiments
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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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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
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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
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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)
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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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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’)
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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
Note that Methanol has the steepestslope, indicating the strongest effect
CRF average at ‘high’ methanol
CRF average at ‘low’ methanol
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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
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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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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
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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
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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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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
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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
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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
Acetic Acid Methanol
Citric Acid
Point TypeCornerCenter
Main Effects Plot (data means) for CRF
The centre point responses are allon the linear response line. Thus no curvature is indicated.
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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
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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
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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
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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
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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
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Author: Dr Jeff Hughes
School of Applied Science, RMIT University
Melbourne , Australia
http://www.rmit.edu.au/staff/hughes j