Resolution IV designs

Some protection against the effect of biases in the estimation of the first-order coefficients can be obtained by running a resolution IV fractional factorial design. With such a design the two-factor interactions are aliased with other two-factor interactions and so would not bias the estimation of the first-order coefficients. In fact the main effects are aliased with three-factor interactions in a resolution IV design and so the first-order effects would be biased if there were third-order coefficients of the form xxx, in... 

A resolution IV design has main effects clear of (not aliased with) two-factor interactions, but at least some two-factor interactions are aliased with each other so the 24 1 design in Table 1 is a resolution IV design (often denoted 24IV]). 

Note that this choice of generators results in 15 of the 21 two-factor interactions being aliased with each other across seven alias chains. However, if instead we choose F = ABC D and G = ABDE as the generators, then another resolution IV design results, but in this design the two-factor interactions are aliased with each other as follows. 

The sparsity of effects principle (see Box and Meyer, 1986) makes resolution III and IV fractional factorial designs particularly effective for factor screening. This principle states that, when many factors are studied in a factorial experiment, the system tends to be dominated by the main effects of some of the factors and a relatively small number of two-factor interactions. Thus resolution IV designs with main effects clear of two-factor interactions are very effective as screening... 

Resolution III designs can be constructed to accomodate three variables in four runs, up to seven variables in eight runs etc. Such designs can be augmented to Resolution IV designs by fold-over. Resolution IV designs are useful, since they make it possible to detect presence of strong interaction effects. 

It is, of course, possible to construct Resolution IV designs directly, by using three-variable interaction column to define the "extra" variables. This is advantageous with eight variables, which by this technique can give a resolution IV design in 16 runs, 2 " (I = 1235 = 1246 = 1347 = 2348). 

D-optimal designs are discussed more fully in the final section of chapter 8. All the designs of resolution V and higher for 7 or more factors require a large number of experiments and the reader s attention is drawn, in particular, to the resolution IV designs which may provide a viable alternative, though further experimentation may be required where interactions are found active. 

Resolution IV designs are easily constructed from saturated designs of resolution III. For example, starting from our first 2jh fractional, we can... 

With resolution IV designs, we can completely separate all the main effects from the two-factor interactions, as shown in Table 4.16. The contrasts corresponding to the actual columns of the design estimate the main effects of the eight variables, while the contrasts defined by products of two columns estimate combinations of two-factor interactions. ... 

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