Saturated factorial designs

But now p factor combinations available in a 2 full factorial design. The design does not have 100% efficiency (efficiency = pif). 

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]. 

The Plackett-Burman designs, as do the saturated fi actional factorial designs, only allow for estimating the main effects. One assumes that all interaction effects are negligible compared to main effects. 

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. 

The first case study we will consider is the assay of aspirin together with its major degradation product salicylic acid [19], This application study was selected as the HPLC assay of aspirin is well covered in the literature and we could select factors to test from the variety of HPLC conditions used in these published methods. This test was performed using a reflected saturated factorial design requiring a total of 15 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. 

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

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