Incomplete factorial design

A sensory study based on an incomplete factorial design allowed to demonstrate that astringency of procyanidins was reduced in the presence of rhamnogalaturonan II added at levels encountered in wine but was modified neither by anthocyanins nor by the other wine polysaccharides (mannoproteins and arabinogalactan proteins). Increase in ethanol level resulted in higher bitterness perception but had no effect on astringency. 

Table 23.5 Incomplete factorial design with three variables and four experiments...

Table 23.6 Incomplete factorial design with four vtiriables and eight experiments (2 " )...

When the seven differences for factors A-G have all been calculated in this way, it is easy to identify any factors that have a worryingly large effect on the results. It may be shown that any difference that is more than twice the standard deviation of replicate measurements is significant and should be further studied. This simple set of experiments, technically known as an incomplete factorial design, has the disadvantage that interactions between the factors cannot be detected. This point is further discussed in Chapter 7. 

Sparse matrix screens These screens are based on the factorial or incomplete factorial statistical approach for designing the screening experiment (Jancarik and Kim, 1991). An extended version of this approach, called sparse matrix sampling has been developed to cover a very large number of conditions for initial screening. 

Another alternative to the 3 full factorial is the Box-Behnken design (Box and Behnken [19]). These designs are a class of incomplete three-level factorial designs that either meet, or approximately meet, the criterion of rotatability. A Box-Behnken design for p=3 variables is shown in Table 2.7. This design will estimate the ten coefficients of the second-order... 

When interaction of factors z and z2 is negligible it is even possible to create an incomplete or fractional factorial design 2 "1. Obviously this saves 22 = 4 measurements or experiments, the drawback is, as already mentioned, that the estimate of the major effect of z3 is confounded with the (minor) effect of interaction of z, and z2. 

FIGURE 8.13 Mixture+process factor space for incomplete cubic simplex-lattice design combined with a 33 full factorial design. 

In the introduction to this chapter, it was said that complete three-level factorial designs with more than two variables would give too many runs to be convenient for response surface modelling. It is, however, possible to select a limited number of runs from such designs to obtain incomplete 3 designs which can be used to fit quadratic models. 

A complete two-step factorial design with n experimental variables requires 2" of experiments. With a large number of variables, the experiment becomes large. We can circumvent this drawback by preparing incomplete attempt factorial designs. In the extreme case, we lose the information from interaction effects. 

The Box-Behnken is considered an efficient option in response surface methodology and an ideal alternative to central composite designs. It has three levels per factoi but avoids the corners of the space, and fills in the combinations of center and extreme levels (Figure 5). It combines a fractional factorial with incomplete block designs in such a way as to avoid the... 

Analysis of variance becomes more complicated with more complex designs. It is often the case with incomplete block and ffactional factorial designs that some or all interactions cannot be tested. In the case of more complex designs, it is advisable to consult an experienced statistician for assistance, both in the design of the experiment and in setting up the analysis. 

A much more complicated example for a mixture-process space is shown in Figure 8.13, where an incomplete cubic simplex-lattice design is combined with a 33 full factorial experimental design. The matrix of the design for Figure 8.13 is shown in Table 8.3. 

Mixture+Process Design Constructed from a Full Factorial 33 Design and Incomplete 3,3 Lattice... 

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