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CHEM7111

EXPERIMENTAL DESIGN

Aug 2011

Diako Ebrahimi

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Definition

Experimental design is a process to organise

the experiments properly to ensure that the right

type of data, and enough of it will be available to

answer the questions of interest as clearly and

efficiently as possible.

Set up the

questions

(purposes)

Design the

experiments

Interpret the

result

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Purposes

1. Optimisation:

maximizing or minimizing the output of a process by

systematically changing input variables.

examples:

Maximizing the yield of a chemical synthesis by changing

temperature, pH, solvent, etc.

Maximizing the sensitivity of a GC/MS instrument by changing

the setup, i.e. temperature program of GC, voltage of analyser,

angle of the grids, etc.; or by twiddling the knobs

Minimizing the sum of squares of residuals in regression by

changing the function parameters

“Simplex, Mixture design and Central Composite design”

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Purposes

2. Screening:

Examine the importance of factors for the process and

then decide which one to be eliminated and which one to

be studied in detail

example: Studying the effect of time, temperature, pH, solvent, … (many

factors) on a chemical synthesis

“Factorial, Taguchi and Plackett-Burman designs”

3. Quantitative modeling:To build a mathematical model of the system, such as

simple linear calibration

“Central Composite and calibration designs”

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3

5

Model of factor effects

Temperature (x1)%

Yie

ld

40 60Temperature (x1)

%Y

ield

40 60

b0

Temperature (x1)

%Y

ield

40 60

b0

Y = b0^

Y = b1x1

^Y = b0 + b1x1

^

When there are more than

two factors, interactions are

also need to be considered.

pH (x2)%

Yie

ld3 8

Y = b0 + b2x2

b2<1

^

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%Y

ield

Model of factor effects: Interaction

22110ˆ xbxbby

Response

Constant Factor 1 Factor 2

Effect of factor 1 Effect of factor 2

%Y

ield

211222110ˆ xxbxbxbby

Effect of interaction between factors 1 and 2

Interaction between factors 1 and 2

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Temp (x1) pH (x2) %Yield (y)

40 3 55

40 8 61

60 3 82

60 8 75

Temperature (x1)

pH

(x 2

)3-

8-

40 60

Model and Design Matrices

x1: Temperature x2: pH y: Yield

Interaction term

2112 xxb22110ˆ xbxbby

Response

Constant

Linear terms Quadratic terms

2

222

2

111 xbxb

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Response Surface & Contour Plot

Response surface is a graph of the response versus the

factors and contour plot is the image of the response factor

on a lower dimension space.

pHTemp

Response

Global

maximum

Local

maximum

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Why not optimize by

“changing one factor at a time”

Because it does not always give the correct optimum if

there are local optimums or if the factors interact.

Factor 1 (Temp.)

Fa

cto

r 2

(pH

)

F2 is kept constant, F1 is optimized

F1 is kept constant at its optimum,

F2 is optimized

One at a time means:

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30°C

Global

maximum

Local

maximum

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Coding the data

Temp (x1) pH (x2) %Yield (y)

40 3 55

40 8 61

60 3 82

60 8 75

Temp (x1) pH (x2) %Yield (y)

-1 -1 55

-1 +1 61

+1 -1 82

+1 +1 75

34055 210 bbb

22110ˆ xbxbby

Temperature (x1)

pH

(x 2

)

3-

8-

40 60

- -

- +

+ -

+ +

)1()1(55 210

Coefficient in the model of coded data:

Magnitude: The larger the coefficient, the

greater its significance.

Sign: A positive coefficient means that the

response becomes larger as the factor

goes from -1 to +1.

High Level: +1

Low Level: -1

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Full Factorial Design

N=LK

N: number of experiments

L: number of levels (usually 2)

K: number of factors

Examples: 2 levels, 2 factors 4 experiments

2 levels, 3 factors 8 experiments

3 levels, 4 factors 81 experiments

For Screening L=2 then N=2K

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Full Factorial Design Contrast Coefficient Tables

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Full Factorial Design

Example: HPLC analysis of Phenols

Determine the effect of acetic acid, methanol and citric acid by

measuring the chromatographic response factor (CRF)

2 level design is chosen first

Value at

low level (-)

Value at

high level (+)

Acetic acid 4 mM 10 mM

Methanol 70 % 80 %

Citric acid 2 g/L 6 g/L

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Full Factorial Design Example

Run Intercept A M C A×M A×C M×C A×M×C CRF

1 + - - - + + + - 10.0

2 + + - - - - + + 9.5

3 + - + - - + - + 11.0

4 + + + - + - - - 10.7

5 + - - + + - - + 9.3

6 + + - + - + - - 8.8

7 + - + + - - + - 11.9

8 + + + + + + + + 11.7

Value at - Value at +

Acetic acid 4 mM 10 mM

Methanol 70 % 80 %

Citric acid 2 g/L 6 g/L

A= 10 Mm

M= 80%

C= 2 g/L

Contrast Coefficient Table:

Main factors Interaction between factors

AMCbMCbACbAMbCbMbAbbCRF AMCMCACAMCMA 0

Number of experiments=

LK=23=8

L levels K factors

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Calculation of effects in a full factorial design

For each TERM (main effects, interactions,

quadratic, etc.) the effect is calculated by summing

the responses multiplied by their contrast

coefficients, then dividing by the number of runs/2.

375.04

7.119.118.83.97.100.115.90.10

A

925.14

7.119.118.83.97.100.115.90.10

M

125.04

7.119.118.83.97.100.115.90.10

C

125.04

7.119.118.83.97.100.115.90.10

MA

025.0

825.0

025.0

CMA

CM

CA

It means that:

On average CRF is

lower by -0.375 when

the concentration of

acetic acid increases

from 4 to 10mM

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Significance of effects

Rankit Plot

0.4

0.5

0.6

0.7

0.8

-1 0 1 2 3

Effect

Pro

ba

bilit

ies MC

M

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Experimental Designs

1. Factorial Designs

1.1. Full Factorial design

1.2. Fractional Factorial Design

1.3. Plackett-Burman and Taguchi Designs

1.4. Calibration Design (Partial Factorial at several Levels)

2. Central Composite or Response Surface Designs

3. Mixture Designs

4. Simplex Optimisation

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Fractional Factorial Design

To study K factors 2k experiments are needed in a two

level full factorial design.

It is reasonable to ignore the third and higher order

interactions to be able to reduce the number of experiments

for screening purposes

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Fractional Factorial Design

Two level fractional factorial designs with 2k-p experiments are

used to reduce the number of experiments by 1/2, 1/4, 1/8, …, 1/2p.

K: number of factors

2p: size of fraction No. of factors (K) Fraction (P)No. of experiments

Fractional Factorial (N=2K-P)

No. of experiments

Full Factorial (N=2K)

2 1 22-1=2 4

3 1 23-1=4 8

4 1

2

24-1=8

24-2=416

5 1

2

25-1=16

25-2=832

6 1

2

3

26-1=32

26-2=16

26-3=8

64

7 1

2

3

4

27-1=64

27-2=32

27-3=16

27-4=8

128

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How to reduce the number of experiments

Full factorial design

Fractional factorial design

(x1X x2)(x1X x3)(x2X x3)

The interaction between

factor 1 and 2 is

confounded with factor 3

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What does confounding mean?

For more details about the rules of confounding refer to:

T. Lundstedt et al.; Chem. Int. Lab. Syst, 42 (1998) 3-40

Using fractional factorial designs many factors can be

studied (screened) with few experiments, but less

information is gained compared to the full factorial designs.

The price to be paid is that the main effects are

confounded. It means that the main effects are

“contaminated” with interaction effects.

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Plackett-Burman and Taguchi Designs

When the number of factors are large, the rule of N=2k-P can be

restrictive, i.e. for 19 factors, 32 experiments are needed.

Plackett and Burman overcame this problem by introducing a design

for N-1 factors in which N (number of experiments) is a multiple of four,

for example: 4, 8, 12, 16, …

N=k+1

Design matrices are

built using generators

Factors

Exp

erim

ents

1 2 3 4 5 6 7 8 9 10 11

1 - - - - - - - - - - -

2 + - + - - - + + + - +

3 + + - + - - - + + + -

4 - + + - + - - - + + +

5 + - + + - + - - - + +

6 + + - + + - + - - - +

7 + + + - + + - + - - -

8 - + + + - + + - + - -

9 - - + + + - + + - + -

10 - - - + + + - + + - +

11 + - - - + + + - + + -

12 - + - - - + + + - + +

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Central Composite or Response Surface Designs

It is used to study the system in more detail

for optimisation

for quantitative modeling

321123322331132112

2

333

2

222

2

1113322110ˆ xxxbxxbxxbxxbxbxbxbxbxbxbby

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Experimental design process

1. Select the factors you are interested in to study

2. Choose a proper design

3. Decide number of replicates

4. Randomise the design

5. Perform the experiments

6. Use statistics to interpret the effect of factors

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Simplex Optimisation

Simplex optimization is a model free approach

It is a step-wise method in which the result from the previous

simplex is used to build a new simplex and it continues this way.

Simplex is a geometric figure with k+1 corners where k is the

number of factors.

When k=2 then simplex is a triangleTe

mp.

pH

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Simplex Optimisation

Procedure: two factors

Three experiments are performed at coordinates of 3 corners of

an initial simplex (triangle) and the three responses are

measured.

The corner with the lowest response is mirrored through the

geometrical midpoint of the other two corners.

An experiment at the new coordinate is performed and the same

procedure is repeated for the new simplex.

If the new coordinate is the worst amongst three, then the

second lowest corner of the last simplex is mirrored.

The process is continued until simplex encircles, i.e. it has

reached the optimum.

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Simplex Optimisation

Te

mp.

pH

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References:

1- “Chemometrics, Data Analysis for the Laboratory and Chemical

Plant”, chapter two; Richard G. Brereton; John Wiley & Sons Ltd,

2003

2- T. Lundstedt et al.; Chemom. Intell. Lab. Syst., 42 (1998) 3-40

3- Statistics for experimenters, Box; Hunter; Hunter; John Wiley &

Sons Ltd, 2005

4- Chemometrics: Experimental Design, Ed Morgan; ACOL, Wiley,

1991

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