Confounding Fractional Factorial Designs

Note that as the 2 full factorial designs needs 8 runs, it is also possible to select the other half fraction of the experiments, with the complementary signs in column C. In this case the generator would be denoted C = - AB. Furthermore, the product ABC renders a column, denoted I, with all elements showing positive signs. So we call 1 = ABC the defining relation for this design. 

Using the BH algorithm to estimate the effects, we note that column AB has the same signs as column C, column AC has those of column B, column BC those of column A and column ABC those of I. Hence the linear combination of observations in column A, 1a, can be used to estimate not only the main effect of A but also the BC interaction (1a = A + BC 1 a = A — BC if we select C = —AB as generator). Two (or more) effects that share this type of relationship are termed aliases . As a consequence, aliasing is a direct result of fractional replication. In many practical situations, it will be possible to select a fraction of the experiments so that the main effect and the low-order interactions that are of interest become aliased (confounded) only with high-order interactions, which are probably negligible. 

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Fractional replication causes confusion among the factor effects. This confusion is called confounding or aliasing . To see this, compare the signs in the columns in the fractional factorial design ... 

In terms of absolute size, main effects tend to be larger than two-factor interactions, which in turn tend to be larger than three-factor interactions, and so on. In the half-fraction factorial design of Table 3.9 the main effects are expected to be significantly larger than the three-factor interactions with which they are confounded. As a consequence it is supposed that the estimate for the main effect and the interaction together is an estimate for the main effect alone. 

To determine whether these interactions are important one can create a fractional factorial design in which the two factor-interactions of interest are not confounded with each other nor with main effects. In such a design the interaction effects can be directly estimated (see also Section 3.4.10). 

Analogous results are found from the statistical analyses of the Plackett-Burman and the fractional factorial design in spite of the different level of confounding in these designs and of the different ways of estimating (SE). ... 

Merck also proposed recently an expert system called Ruggedness Method Manager for ruggedness tests of chromatographic assay methods. The system uses fractional factorial designs. Besides the factors to be examined, interactions that possibly also could be relevant have to be defined by the user. The system then calculates a design in which the main effects are not confounded with one of the specified interactions. The interpretation criterion to identify statistically significant effects is not known to the authors of this chapter. 

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. 

When interaction of factors z and z2 is negligible it is even possible to create an incomplete or fractional factorial design 2 "1. Obviously this saves 22 = 4 measurements or experiments, the drawback is, as already mentioned, that the estimate of the major effect of z3 is confounded with the (minor) effect of interaction of z, and z2. 

Setting up a fractional factorial design and determining which terms are confounded is relatively straightforward and will be illustrated with reference to five factors. 

Of course, the information on each variable which can be obtained by a fractional factorial design is less than what can be obtained from a complete factorial design. There is a price to pay for being lazy, and the price is confounding. 

To illustrate what is meant by confounding, we shall first treat a simple case three variables in a 2 fractional factorial design. It is seen that over the set of... 

To analyze which effects will be confounded in a fractional design, we shall introduce a new concept, the generator of a fractional factorial design.[1] As an example to illustrate the principles, we shall use a fractional factorial design To understand why the generators are practical to use we shall write down the complete variable matrix of a 2 factorial design. 

The experiment with all variables at the (+l)-level has already been run, Exp. no 8. It will be necessary to run one complementary experiment. These two experiments define the smallest possible fractional factorial design, 2 . It is seen that the following confounding pattern is obtained... 

This illustrates a great advantage of fractional factorial designs, namely, that confounded effects can be separated by doing complementary runs. For more details, see Box, Hunter and Hunter.[4]... 

When a fractional factorial design has been employed, there is always a possibility to run complementary fractions to resolve confounded effects. 

A 2 " fractional factorial design (I = 12345) was used to estimate the model parameters. The design has a Resolution V and the desired parameters can be estimated free from confoundings with each other. The design matrix and the yields (%), y, obtained are given in Table 6.7. 

A more convenient design in three blocks can be obtained by using a fractional factorial design (I = 123) to vary the experimental conditions. In such design, main effects will be confounded with the two-variable interactions, b, = (6, + B23)— etc. 

The moral of this is, that whenever there are any doubts as to the form of the model, it is always better to use a fractional factorial design for screening experiments. With fractional factorial designs, analysis of the confounding pattern may give clues to how the model should be refined. 

Fractional factorial designs The first choice when there are more than four variables, should always be to attempt a fractional factorial design. The confounding patterns are easily obtained firom the generators. It is also easy to append complementary runs to resolve any ambiguities. 

Plackett-Burman designs Do not consider a Plackett-Burman design as a first choice. These designs are Resolution III designs, but the confounding pattern in such designs is more difficult to analyze than that of fractional factorial designs. 

A twelve-run Plackett-Burman design can accomodate eleven variables. With 12 -15 variables, a fractional factorial design is better (confounding pattern can be analyzed). With more than 15 variables a Plackett-Burman is the preferred choice ... 

Step 3 Choose the design for your experiment. Fractional factorial designs or low-resolution designs are best for process development work where there are several (say four or more) factors to consider. Full factorial designs are used when it is necessary to eliminate all confounding or aliases between main effects and interactions. 

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