Saturated fractional factorial design

Most screening designs are based on saturated fractional factorial designs. The firactional factorial designs in Section 14.8 are said to be saturated by the first-order factor effects (parameters) in the four-parameter model (Equation 14.27). In other words, the efficiency E = p/f = 4/4 = 1.0. It would be nice if there were 100% efficient fractional factorial designs for any number of factors, but the algebra doesn t work out that way. 

Notice that this is an orthogonal design in coded factor space (-1 and +1) any one column multiplied by any other column will give a vector product of zero. Other saturated fractional factorial designs may be found in the literature [Box and Hunter (1961a, 1961b), Anderson and McLean (1974), Barker (1985), Bayne and Rubin (1986), Wheeler (1989), Diamond (1989)]. 

The saturated fractional factorial designs are satisfactory for exactly 3, or 7, or 15, or 31, or 63, or 127 factors, but if the number of factors is different from these, so-called dummy factors can be added to bring the number of factors up to the next largest saturated fractional factorial design. A dummy factor doesn t really exist, but the experimental design and data treatment are allowed to think it exists. At the end of the data treatment, dummy factors should have very small factor effects that express the noise in the data. If the dummy factors have big effects, it usually indicates that the assumption of first-order behavior without interactions or curvature was wrong that is, there is significant lack of fit. 

As an example of the use of dummy factors with saturated fractional factorial designs, suppose there are 11 factors to be screened. Just add four dummy factors and... 

Now suppose there are 16 factors to be screened. We would have to add 15 dummy factors and use the 2 " saturated fractional factorial design, but this would give an efficiency of only 17/32 = 53%. This is not very efficient. Most researchers would rather eliminate one of their original 16 factors to give only 15 factors. There is a saturated fractional factorial design that will allow these factors to be screened in only 16 experiments. 

A seven-factor saturated fractional factorial design. 

Plackett-Burman and saturated fractional factorial designs. 

Using row and column operations, convert the following 7-factor Plackett-Burman design to the saturated fractional factorial design shown in Table 14.7. 

Hint row 8 of this design and row 8 of the saturated fractional factorial design in Table 14.7 suggest that the reflection or foldover must be carried out first. Repetitions of switching one row with another, and switching one column with another, will eventually yield the desired result. Retain the identities of the rows and columns. 

For this reason one prefers to apply an experimental design. In the literature a number of different designs are described, such as saturated fractional factorial designs and Plackett-Burman designs, full and fractional factorial designs, central composite designs and Box-Behnken designs [5]. 

A Plackett-Burman design with N experiments can examine up to N-1 factors. This is a difference with fractional factorial designs. Some saturated fractional factorial designs however contain also N-1 factors (e.g. the design of Table 3.14) but this is not always the case. The saturated design for 5 factors, for example, is the 2 design. In this design only 5 factors are examined in 8 experiments. 

M. Mulholland and J. Waterhouse, Investigation of the Limitations of Saturated Fractional Factorial Designs with Confounding Effects for a HPLC Ruggedness Test, Chromatographia, 25(9) (1988) 769-774. 

The smallest fraction of a full factorial still able to estimate the factor effects needs at least one expen ment more than the number of factors considered. Such a design is called a saturated fractional factorial design. Detailed guidelines to create a specific fractional factorial design can be found in 127.35]. The number of experiments N in... 

The most important alternatives for the saturated fractional factorial designs are the Plackett-Burman designs. The number of experiments for these designs is a multiple of four. They too allow the evaluation of maximally N — 1 factors. This means that it is for instance possible to study 11 factors with 12 experiments which is not possible for the fractional factorial designs. An example is shown in Table 6.6. 

A saturated fractional factorial design with eight runs, to evaluate how seven factors affect the accuracy of the serves of an amateur tennis player... 

A saturated fractional factorial design with eight runs that, together with the fractional of Table 4.10, permits separating the main effect for factor 5 from the two-factor interactions... 

When studying a new process one does not know which factors are relevant and many possible factors may affect the response. In such a case, the first step is to screen the candidate factors to select those that are relevant. Therefore the smallest possible fraction of a two-level design (a saturated fractional factorial design) or the related Plackett-Burman designs can be used (see Section 2.2.5.2). 

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