Screening experiments with many variables

When an experimental procedure is analyzed as was discussed in Chapter 4, it is rather common that more than a handful of variables need to be considered. Determination of the influence of k variables through a complete two-level factorial design calls for 2 individual experimental runs. Seven variables give 128 runs ten variables give 1024 runs fifteen variables give 32,768 runs. It is evident that many variables would result in a prohibitively large number of runs if analyzed by a factorial design. 

In this chapter, it is discussed how to select a subset of experiments from a complete factorial design in such a way that it will be possible to estimate the desired parameters through a limited number of experimental runs. We shall see that it is very easy to construct designs which are 1/2, l/4, 1/8, 1/16. 1/2P fractions of a complete factorial design. This will give a total of 2 P experimental runs, where k is the number of variables, and p is the size of the fraction. 

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Let us now have a look at general screening experiments with many variables. Assume that k variables (xj, X2,..., xJ have been studied by a fractional factorial design and that a response surface model with linear and cross-product interaction terms has been determined. 

The results for the RF screening study are shown in Table 3. The most striking result to come out of this experiment was that there appears to be a strong correlation between the low level of catalyst concentration and gel formation. The low level was outside the range of what had previously been tried. This has been confirmed in many subsequent experiments. Another important conclusion was that the chemistry appears to dominate the process, so it was reasonable to proceed with an RSM which dealt only with the formulation variables. Although the oven time was significant at the 90% confidence level, it was decided to optimize the chemistry first and deal with this as part of the processing conditions in later experiments. 

Having carried out as many experiments as there are coefficients in the model equation and not having any independent measurements or estimation of the experimental variance we cannot do any of the standard statistical tests for testing the significance of the coefficients. However all factorial matrices, complete or fractional, have the fundamental property that all coefficients are estimated with equal precision and, like the screening matrices, all the coefficients have the same unit, that of the response variable. This is because they are calculated as contrasts of the experimental response data, and they are coefficients of the dimensionless coded variables X. They can therefore be compared directly with one another. 

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