Factorial designs three-level

For n = 30, we must examine 330 - 1 = 2.0588 X 1014 values of /(x) if a three-level factorial design is to be used, obviously a prohibitive number of function evaluations. 

Common coding systems for two- and three-level factorial designs. 

In some applications, Latin square designs can be thought of as fractional three-level factorial designs that allow the estimation of one main factor effect while... 

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... 

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. 

Figure 3.8. Possible configurations of nine sets of factor values (experiments) that could be used to discover information about the response of a system. Dotted ellipses indicate the (unknown) response surface, (a) A two-factor, three-level factorial design (b) a two-factor central composite design.

The extremely versatile full second-order polynomial model in Equation 3.32 can also be fitted when at least three-level factorial designs are used and 3 experiments are run. Alternatively, a central composite design may be used effectively (see Figure... 

The most evident design would appear to be a three-level factorial design. An example of a three-level factorial design is shown in Fig. 6.1.. A full three-level factorial design, 3, can be used to obtain quadratic models. However, unless / is small (/ = 2) the design requires a number of experiments (= V) that is not often feasible. An example of the use of a 3" design can be found in Ref. 51]. The two factors, the pH and the percentage acetonitrile were examined to evaluate their influence on the retention and resolution of three isoxazolyl penicillin antibiotics. The centre point experiment was triplicated to evaluate experimental error. 

In a two-level factorial design, each variable can take two values. For continuous variables, this will signify a low level and a high level. For discrete variables, this wiU signify the alternatives. In a three-level factorial design, three different levels of each variable is investigated, etc. 

With two experimental variables, a three-level factorial design, 3, is convenient. One example is shown below. With more than two variables, it is much more convenient to use a composite design. Such designs are discussed after the example below. 

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. 

T. J. Mitchell and C. K. Bayne, D-optimal fractions of three-level factorial designs. Technometrics, 20, 369-383 (1978). 

These equations show that the Box-Behnken and the full three-level factorial designs result in essentially the same models for both yield and Young s modulus as functions of time, oxidant concentration and particle size. 

Examples of spherical designs are the central composite designs (except the face-centered central composite design), the Box-Behnken, and the Doehlert design. The three-level factorial design and the face-centered central composite design are cubic designs. 

Decreasing the length of the air gap used during the spinning process caused the wall thickness of the resultant fibre to increase in most cases, but did not effect the wall thickness for others cases. It is likely the effect of the length of the air gap is not a linear one, a three level factorial design would confirm if this was the case. 

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