Dummy factor

The two-factor interaction effects and the dummy factor effects in FF and PB designs, respectively, are often considered negligible in robustness testing. Since the estimates for those effects are then caused by method variability and thus by experimental error, they can be used in the statistical analysis of the effects. Requirement is that enough two-factor interaction or dummy factor effects (>3) can be estimated to allow a proper error estimate (see Section VII.B.2.(b)). 

In reference 22, the effects of eight factors are examined in 12 experiments, which means that three dummy factors were included in the design, e.g., in columns I, J, and K. On the other hand, in reference 23, seven factors are evaluated in eight experiments, leaving no room for dummies. 

After selection of the experimental design, the experiments can be defined. For this purpose, the level symbols, —1 and +1, as given in Tables 6 and 7, are replaced by the real factor values, as for instance shown in Tables 2 and 3, respectively, yielding the factor level combinations to be performed. Dummy factors in PB designs can be neglected during the execution of the experimental work. 

Another way to estimate (SE)e is using effects that are a priori considered negligible, such as two-factor interaction effects and dummy factor effects " in EE and PB designs, respectively (Equation (8)). Such effects are considered solely due to the experimental error of the method. ... 

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. 

Each Plackett-Burman design contains a fixed number of factors (a multiple of four minus one). After determination of the number of factors to be examined (this factors will be called real factors), the remaining number of potential factors in the design are defined as dummy factors. A dummy factor is an imaginary factor. A fractional factorial design on the other hand will be constructed depending only on the number of real factors. Normally no dummies are entered in those designs. 

The difference can be seen from the following example. If there are six factors, one can perform a Plackett-Burman design with 8 experiments, containing six real and one dummy factor. Another possibility would be to perform the twelve experiment design that would contain five dummies. 

AB and F = AC). This is also a design with 8 experiments as the Plackett-Burman design but containing no dummy factors. 

Using dummy factors in Plackett-Burman designs [24,26,27]. The effect of a dummy factor is considered to be due to experimental error. 

A possible consequence of using the dummy factor effects or the two-factor interactions is that significant interactions will increase the (SE)g... 

One can also use the dummy factor effects in a similar way for the interpretation of Plackett-Burman designs. One then uses a formula analogous to equation (20). Instead of using the multiple-factor effects in the formula one uses the dummy factor effects ... 

The experimental scheme for a three level reflected saturated fractional design for seven factors is shown in Table 5.15 ( note that one factor was retained as a dummy factor to be used as an additional error check). The experimental order of the scheme was sorted on acid type as this required long equilibration times, this ordering loses some of the features of the initial design but is a compromise that can be justified on the fact that... 

The value of an effect and its standard deviation are calculated in the same way as for factorial designs. Multiply the responses by their contrast coefficients for a given factor, sum them, and divide this number by half the number of experiments (2n, for 4n experiments) to give the value of the effect. With no high-order interactions available in the design, either an independent estimate of repeatability or the use of dummy variables is essential. For m dummy variables, the effect standard deviation is calculated using equation 3.14, where ) is the measured effect of the Jth dummy factor. The significance of the effect is then determined by a Student s t test described earlier. 

Construct a two-level Plackett-Burman experimental design [6] as illustrated in Table 13.1. Choose six metal ions for interference studies and include a dummy factor (one in which nothing is changed). 

Table 2. Important factors identified in sequential bifurcation for the Old supply chain N/A denotes a dummy factor and denotes an important factor...

We label the factors such that all factors have the same meaning in the three simulation models. To achieve this, we introduce dummy factors for the Current and the Next Generation models that represent those factors that are removed as the supply chain is changed. Such dummy factors have zero effects but simplify the calculations and interpretations of the sequential bifurcation results. 

The aggregated effects of the Old supply chain exceed those of the Next Generation supply chain, because the former aggregates more (positive) individual effects. For example, the Current simulation model has 14 dummy factors (which have zero effects), so the first sequential bifurcation step gives a smaller main (group) effect for the Current model this effect is 7,101,983, whereas it is 15,016,102 for the Old model. 

Another very simple approach is to include one or more dummy factors. These can be built into a design, and might, for example, be the colour of shoes worn by the experimenter, some factor that is not likely to have a real effect on die experiment level —1 might correspond to black shoes and level +1 to brown shoes. Madiematical models can be built including diis factor, and effects smaller dian diis factor ignored (remembering as ever to ensure diat die scaling of the data is sensible). 

The design, including an eleventh dummy factor, is as follows, with the observed yields ... 

Why is a dummy factor employed Why is a Plackett-Burman design more desirable titan a two level fractional factorial in this case ... 

A simple method for reducing the number of experimental conditions for further study is to look at the size of the factors and eliminate those that are less than the dummy factor. How many factors remain and what are they ... 

At this point, the required experiments can be defined. For this purpose, the levels (e.g., -a, -1,0, -f1, H-a) in the theoretical experimental design (e.g., Tables 2.8, 2.14, and 2.9) are replaced by the real factor levels (e.g., Tables 2.2-2.4, respectively).This results in the experimental conditions for each experiment. The dummy factor columns in PB designs can be ignored at this point. Often a number of replicated experiments at nominal or center point conditions are added to the setup (see above). 

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