Introduction to design

Olav M. Kvalheim

Content• Making your data work twice• Effect of correlation on data interpretation• Effect of interaction on data interpretation

Chemometrics/Infometrics

Design of information-rich experiments and use of

multivariate methods for extraction of maximum relevant information

from data

Making your data work twice

What is Information?

A

B

C

•A - mean value, no standard deviation given

•B - mean value with standard deviation given, large value of stand. dev.

•C - mean value, low standard deviation

A B

Hotelling (1944) Ann. Math. Statistics 15, 297-306

Measurement strategy?

Unknowns Calibration Weights

The univariate weighing design

Weigh A and B separately

mA ± A

mB ± B

A = B=

Precision is for both A and B

The multivariate design

Weigh A and B jointly to determine sum and difference:

mA+ mB =S

mA- mB =D

mA = ½S + ½D

mB = ½S - ½D

7.02

1

4

1

4

1 22 BA

Precision is 0.7 for both A and B

Precision for S

Precision for D

Precision in mAand mB

Univariate Design Bivariate Design 0.7

Precision is improved by 30% by using a multivariate design with the same number of measurementsas for the univariate!

Univariate vs Bivariate strategy

With N masses to weigh, a multivariate design provides an estimate of each mass with a precision

N

1

The larger the number of unknowns, the larger the gain in precision using a multivariate weighing design.

Univariate vs Multivariate weighing

Effect of correlation on data interpretation

X1 X2

Example

• Process output is function of temperature and amount of catalyst

Correlation between amount of catalyst and amount produced

• Strong positive correspondence

Correlation between Temperature and Produced amount

• Weak positive correspondence

Conclusion from correlation analysis

• Increase amount of catalyst and temperature to increase production

Result of test

• Produced amount was lowered!

Bivariate Regression Model

• Produced amount = 300

• + 2.0 * Catalyst

• - 0.5 * Temperature

Correlation between temperature and amount of catalyst

• Strong positive correspondence

Solution to correlation problem

• Multivariate Design - Change many process variables simultaneously according to experimental designs

Effect of interaction on data interpretation

X1X2

The yield of a chemical reaction is a function of temperature (t) and concentration (c).

y = f (t,c)

The task

Optimise the yield for the reaction!

Concentration, M

Temperature, ºC

0.1 0.2

140

160

150

170 756070 50 4045

Response surface in the presence of interaction

Univariatedesign(COST)

Multivariatedesign

Information

Number of experiments

Efficiency of information extraction

Multivariate Designvs.

Univariate Design

• Correct Models Possible (Interactions)

• Efficient Experimentation

• Improved Precision/Information quality