Resolution of a fractional factorial design

Degree of resolution, or shortly, the resolution of a fractional factorial design is defined by the length of the shortest word in the set of generators.[l] The resolution is commonly specified by roman numeral characters. 

Resolution V, main effects are confounded with four-variable interaction effects, two-variable interaction effects are confounded with three-variable interaction effects. 

Fractional factorial designs with higher resolution than V are rarely used in sceening experiments. 

A procedure for the synthesis of semicarbazone from phenylglyoxalic acid was studied.[2] 

The reaction is one of the steps in a multi-step synthesis of azauracil, which is a anti-leucemic cytostatic agent. The experiments were run to determine suitable conditions for scale-up synthesis in the pilot plant. 

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The resolution of a fractional factorial design is a convenient way to describe the alias relationships ... 

Note that X5X5 has a power of 2, which becomes 0 and hence drops out. Since the length of the word is 5, the resolution is V. It is always true that the resolution of a fractional factorial design is equal to the length of the smallest word in the defining relationship. 

Table 2.9, shows the minimum number of runs for a single replicate of a fractional factorial design with the desired resolution for p variables, /7=3,...,11. 

The main criterion for quality in the choice of a fractional factorial design is that it should be of maximum resolution. There could however be several fractional designs, each of the same resolution, but with apparent differences in quality. We take for example two 16 experiment designs for 6 factors (2 ), both of resolution... 

An alternative approach to constructing designs for estimating second-order models is to consider building a design from those constructed for the first-order model. In Section 2.2.1, we discussed the use of fractional factorial designs to estimate the coefficients of the first-order model. It was noted that a fractional factorial design of resolution V would yield... 

In general the cube portion might be replicated times and the star portion might be replicated times. Also, it might be possible to use a fractional factorial design of resolution less than V if the experimenter is prepared to assume that certain interactions are negligible. A central composite design in four variables is shown in Table 2.6. In this table, runs 1-16 are the cube portion, runs 17-24 are the star portion, and runs 25-27 are the center points. 

To construct the central composite design to estimate the coefficients of the second-order model (equation (14)), usually a fractional factorial design of at least resolution V is used. In this case, if the model is valid, then all of the estimates of the main effect coefficients, p., and the interaction coefficients, p. are imbiased. An alternative to the central composite designs for estimating the coefficients of the second-order model are the Box-Behnken designs or the designs referenced in Section 2.2.5. 

As far as the above example is concerned, there is also a 2 design of resolution IV, with independent generators 1235 and 1246 which is normally to be preferred to either of the two designs given, neither of which appears in the table of optimum fractional factorial designs (table 3.21). 

There are 8 coefficients in the model. The design must therefore contain not less than 8 experiments. Is it possible to constmct a fractional factorial design of only 8 (2 ) experiments allowing the resolution of the problem ... 

In chapter 3 we showed that a factorial design of resolution V is needed to determine the constant term, the main effects and the first order interactions in the model. We may thus refer to any design as being of resolution V, should it allow these effects to be estimated, even if it is not a fractional factorial design. 

Example 3.3. For the following equation, create a fractional factorial design with a resolution of three and define three generator functions ... 

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 reflected half-fraction factorial design for three factors (2 ) was performed. The influence of the factors on the responses recovery (%), resolution between peaks R) and Ry-value was calculated. Approximate critical values were obtained using the method given by Youden and Steiner (Ecritical MSE)e)- The standard error was estimated from... 

Three-level fractional factorial designs are also very useful, and charting the effects can be very helpful especially where there are more than three factors. The Plackett-Burman designs are often used to confirm (or otherwise ) the robustness of a method from the set value. Figure 17 shows some results from a ruggedness study for an HPLC method for salbutamol where the resolution factor, between it and its main degradation product is critical. 

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... 

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