Multifactor experimental designs

For many method development problems, three or four factors are often the norm. The message is clearly that a simple approach to experimental design can be a crucial tool in ascertaining those factors which need to be controlled in order to maximise method robustness. In this example, the level of citric acid will have to be tightly controlled, as well as the methanol concentration, if consistent and high values of CRF are to be regularly obtained. 

Note how in this instance the column-to-column variability is so large that the suitability for use must certainly be questioned. 

Optimisation methods may also be used to maximise key parameters, e.g. resolution, but are beyond the scope of this handbook. Miller and Miller s book on Statistics for Analytical Chemistry provides a gentle introduction to the topic of optimisation methods and response surfaces as well as digestible background reading for most of the statistical topics covered in this handbook. For those wishing to delve deeply into the subject of chemometric methods, the Handbook of Chemometrics and Qualimetrics in two volumes by Massart et al., is a detailed source of information. 

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In previous chapters, many of the fundamental concepts of experimental design have been presented for single-factor systems. Several of these concepts are now expanded and new ones are introduced to begin the treatment of multifactor systems. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

In this chapter we discuss the multifactor concepts of confounding and randomization. The ideas underlying these concepts are then used to develop experimental designs for discrete or qualitative variables. 

Factorial designs are a popular class of experimental designs that are often used to investigate multifactor response surfaces. The word factorial does not have its usual mathematical meaning of an integer multiplied by all integers smaller than itself (e.g. 5 5x4/3/2 / 1) instead, it simply indicates that many... 

Response Surfaces. 3. Basic Statistics. 4. One Experiment. 5. Two Experiments. 6. Hypothesis Testing. 7. The Variance-Covariance Matrix. 8. Three Experiments. 9. Analysis of Variance (ANOVA) for Linear Models. 10. A Ten-Experiment Example. 11. Approximating a Region of a Multifactor Response Surface. 12. Additional Multifactor Concepts and Experimental Designs. Append- ices Matrix Algebra. Critical Values of t. Critical Values of F, a = 0.05. Index. 

DOE is an acronym for design of experiments, also called experimental design or multifactor experiments. These experimental approaches provide rich veins of data that can be mined for information that cannot be found any other way. 

In the present example, we have examined only one factor the component concentration. In the case of studying several factors, randomization is similarly necessary. In multifactor experiments, the experimental design must be run in a randomized order, as we will see. 

S. N. Deming and S. L. Morgan, in Experimental Design A Chemometric Approach, 2nd ed., Elsevier, Amsterdam, 1993, pp. 227-274. Approximating a Region of a Multifactor Response Surface. 

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