Three factor, two level

In Chapter 8, we looked at some experiments that involved two parameters (factors), each at two levels. In Chapter 10, we briefly looked at a three-factor, two-level design, with attention to how it could be represented geometrically. The use of the term three factor, two level to describe the design means that each factor was present at two levels, that is, the corresponding parameters were each permitted to assume two values. 

Three-factor two-level factorial designs like that shown in Figure 14.2 can be shown on the printed page as cube plots ... 

Figure 14.2 A three-factor two-level full factorial design in factors A, B, and C. The open circle locates the center of the design.

Table A10 Design and Analysis Matrix for Three-Factor, Two-Level Experiment Design Matrix...

To determine the influence of the ingredients of a recipe on the stability of an emulsion of a cosmetic cream, an experimental plan is proposed with three factors (two levels) the nature of the cream (oil in water for a positive effect and water in oil for a negative one), an emulsifier (dilute or very dilute), and fatty acid concentration (high or low). The indices of the emulsion stabilities obtained are 38, 37, 26, 24, 30, 28, 19, and 16, with an experimental error of 2. Lamine... 

Figure 10.2 Graphical representation of a three factor, two level design.

Table 10.1 Coding for a three factor, two level design with three centre points. Coding for the effect of interactions is also shown.

In a first attempt to derive characterization factors with QSARs, the entire dataset of plastics additives was included, and aquatic ecotoxicity was predicted for two different trophic levels. This generated characterization factors that did not correspond well with the ones derived from experimental data [30]. Hardly surprising, but a clear indication that two trophic levels are unsufficient. A second attempt to derive characterization factors with QSARs are currently being performed [31]. In this second attempt, substances that are difficult to model in QSAR models have been removed from the dataset and the ecotoxicity has been predicted for three different trophic levels instead of two. However, results have not yet been obtained from this second attempt. If the results show that it is possible to derive reliable characterization factors by the use of QSARs, the current data gap regarding characterization factors for human toxicity and ecotoxicity could be... 

In the case that interactions prove to be insignificant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 

Table 5.9. Design matrix for three factors at two levels (+ and — stand for +1 and —1) ...

Fig. 5.2. Geometrical representation of a complete two level factorial matrix (three influence factors) with experiments in the centre point (0,0,0)... 

FIGURE 7 Experimental domains when examining (a) two and (b) three factors with either the OVAT or an experimental design approach. ( ) Nominal level Ex, effect of factor X. 

Full factorial designs allow the estimation of all main and interaction effects, which is not really necessary to evaluate robusmess. They can perfectly be applied when the number of examined factors is maximally four, considering the required number of experiments. In references 69 and 70, four and three factors were examined at two levels in 16 and 8 experiments, respectively. When the number of factors exceeds four, the number of experiments increases dramatically, and then the full factorial designs are not feasible anymore. 

In reference 69, results were analyzed by drawing response surfaces. However, the data set only allows obtaining flat or twisted surfaces because the factors were only examined at two levels. Curvature cannot be modeled. An alternative is to calculate main and interaction effects with Equation (3), and to interpret the estimated effects statistically, for instance, with error estimates from negligible effects (Equation (8)) or from the algorithm of Dong (Equations (9), (12), and (13)). Eor the error estimation from negligible effects, not only two-factor interactions but also three- and four-factor interactions could be used to calculate (SE)e. 

Factorial designs are usually discussed in terms of coded factor spaces. Table 14.1 shows some of the common coding systems for two- and three-level designs. Our emphasis in this chapter will be on the two-level designs. 

Calculate the grand average (MEAN), the three classical main effects (A, B, and C), the three two-factor interactions (AB, AC, and AD), and the single three-factor interaction (ABC) for the 2 full factorial design shown in the cube plot in Section 14.1. (Assume coded factor levels of -1 and +1). 

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