Full factorial designs three-level design

As for full factorial designs the levels of the variables are situated at the borders of the experimental interval for that variable. It is possible that the response function of that variable is curved with an optimum at an intermediate factor level (see Fig. 6.11). The effect calculated from the design can then be small and the variable may be incorrectly considered as non-significant 45. When such intermediate optima are considered possible, a solution can be to perform the screening at three levels by reflecting a... 

ANOVA in these chapters also, back when it was still called Statistics in Spectroscopy [16-19] although, to be sure, our discussions were at a fairly elementary level. The experiment that Philip Brown did is eminently suitable for that type of computation. The experiment was formally a three-factor multilevel full-factorial design. Any nonlinearity in the data will show up in the analysis as what Statisticians call an interaction term, which can even be tested for statistical significance. He then used the wavelengths of maximum linearity to perform calibrations for the various sugars. We will discuss the results below, since they are at the heart of what makes this paper important. 

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. 

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.

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). 

FIGURE 2.8 Coded levels for the experiments required by a two-level full factorial design with three factors. 

If there are only p=2, or p=3 variables then a full factorial design is often feasible. However, the number of runs required becomes prohibitively large as the number of variables increases. For example, with p=5 variables, the second-order model requires the estimation of 21 coefficients the mean, five main effects, five pure quadratic terms, and ten two-factor interactions. The three-level full factorial design would require... 

In a full factorial design all combinations between the different factors and the different levels are made. Suppose one has three factors (A,B,C) which will be tested at two levels (- and +). The possible combinations of these factor levels are shown in Table 3.5. Eight combinations can be made. In general, the total number of experiments in a two-level full factorial design is equal to 2 with /being the number of factors. The advantage of the full factorial design compared to the one-factor-at-a-time procedure is that not only the effect of the factors A, B and C (main effects) on the response can be calculated but also the interaction effects of the factors. The interaction effects that can be considered here are three two-factor interactions (AB,... 

A first possibility is the use of full factorial designs with three levels [31]. The disadvantage of the three-level factorial designs is that the number of experiments increase very rapidly for a larger number of factors. 

In terms of absolute magnitude the main effects tend to be higher than two factor interactions which in turn are higher than three factor interactions and so on. At some point it is true to say that after a certain order interaction effects become negligible and can thus be disregarded in the experimental design. To do this full factorial designs are fractionated to allow the estimation of only a certain level of interaction effects. 

If a method of analysis is fast or can be fully automated and requires the testing of few factors (three or less) then the larger designs can be considered. Good choices are central composite designs, or if a linear factor response is expected a full factorial design at two levels. 

Figure 3.9. Two-level, full factorial designs for (a) two and (b) three factors. The low value of each factor is designated by a contrast coefficient of— and the high value designated by +.

Figure 5.11 Full factorial design for three parameters at two levels. At each comer the level of the parameters is indicated. The centre of the cube forms the origin of the design (000 for parameter values). The arrows on the left illustrate the three different parameters. Figure taken from ref. [516]. Reprinted with permission.

Figure S.l 1 illustrates a full factorial design for three parameters at two levels. The eight experiments prescribed by eqn.(5.1) are positioned on the comers of a cube. The three parameters each take a high (+) and a low ( —) value. Each parameter can be assigned a direction in the cube. 

Three or More Levels in Full Factorial Designs.291... 

FIGURE 8.12 Two different presentations of a 12-point design for three mixture variables and two process variables, (a) The three-point 3,1 simplex-lattice design is constructed at the position of each of the 22 points of two-level full factorial design, (b) The 22 full factorial design is repeated at the position of each point of the 3,f simplex-lattice design. The way of representation is related to the order chosen for the variables. 

FIGURE 8.17 Two-level, three-factor full factorial design. 

Two-Level, Three-Factor Full Factorial Design... 

Three and four level full factorial designs... 

A full factorial design is one in which every level of each factor occurs for every combination of the levels of the other factors. If each of n factor has two levels, frequently denoted + for one level and for the other level, then the number of treatments is two raised to the nth power. FuU factorial designs when each factor has the same number of levels are referred to as factorial designs, where k is the number of levels and n is the number of factors, as is the number of treatments needed. If factor A has three levels, factor B has two levels, and factor C has three levels then a full factorial design has 3 x 2 x 3 = 18 treatments or experimental trials. 

The treatments of a design (or the experimental trials) can be represented in a design matrix. Each row of the matrix represents a treatment each column represents the level of each factor. A full factorial design for three factors, factor A,... 

Table 4 Full Factorial Design for Three Factors, Factors A, B, and C, Each with Two Levels, Low (-) and High (+)...

TABLE 2.5. Two-level full factorial design for three factors, and columns of contrast coefficients for the interactions... 

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