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Fractional Factorial Designs

With a larger number of factors to consider, the number of experiments needed for a full factorial design quickly explodes. One approach is to assume that some interactions are not important (in general, higher-order interactions of more than two factors are seldom significant), and to use the `-' and `+' signs of that interaction for an extra factor. This yields the following table:

Factor 1Factor 2Factor 3Result
Experiment 1+++...
Experiment 2+--...
Experiment 3-+-...
Experiment 4--+...

This table is exactly equal to the full factorial table in the previous subject, but now the third column determines the settings of the third factor instead of the two-factor interaction. Again, these settings are determined by the settings of the first two factors! Note that it is never possible to distinguish the effect of the third factor from the two-factor interaction; the only reason why this works is that we assume the interaction to be negligible. If all interactions are replaced by new factors, as in the - trivial - example above, such a design is called saturated.

There is also another way of looking at it: a saturated design for three factors and four experiments is just half of a full-factorial design for three factors (and eight experiments). Two graphical representations are shown in the figure above. The first set of experiments, corresponding with the red colour is a tetra-edric subset of the full factorial design table. Another, equally good subset is indicated in blue. The colours correspond to the experiments in the following tables.

Experiments shown in red
Experiment 1 - - -
Experiment 2 + + -
Experiment 3 + - +
Experiment 4 - + +
Experiments shown in blue
Experiment 5 + + +
Experiment 6 - + -
Experiment 7 - - +
Experiment 8 + - -

Notice that both fractional designs, when added together, will form the complete 8-experiment full factorial design. The blue fractional experiment corresponds with the example at the top of the page. Note also that the plusses and minusses of the red design are seemingly incorrect in that the third column is not equal to the product of the first two; this can easily be solved by relabelling the levels (plus becomes minus and vice versa).

If n is the number of factors in a full factorial design, the corresponding saturated fractional design can investigate at most 2n - 1 factors; seven factors need eight experiments, 15 need 16 experiments, and so on.

Plackett-Burman designs

Plackett-Burman designs are another, very useful, class of screening designs. Again, they assume that interactions can be ignored. As in saturated fractional factorial designs, the number of factors is the number of experiments minus one. However, an important advantage is that the number of experiments is a multiple of four, rather than a power of two. With Plackett-Burman designs, it is therefore possible to investigate 11 factors in 12 experiments, or 19 factors in 20 experiments.

They are so-called cyclic designs. They are generated by one line of plusses and minusses; the next line is the same sequence but shifted by one position. The final line consists of plusses only. The following table shows the Plackett-Burman design for investigating seven factors in eight experiments.

Experiment 1++- + - - -
Experiment 2+ - + - - - +
Experiment 3- + - - - + +
Experiment 4+ - - - + + -
Experiment 5- - - + + - +
Experiment 6- - + + - + -
Experiment 7- + + - + - -
Experiment 8+ + + + + + +

You may want to check in an earlier exercise that this design indeed has orthogonal columns.

Now what if you have only six factors to vary, and want to do a Plackett-Burman design? The solution is to introduce a so-called 'Dummy' variable which does not influence the outcome of the experiment. (You may change your t-shirt or use a different mantra, if you insist on actually changing something.) Of course, the main effect of a dummy variable should be zero; and of course, it never really is. Therefore, the main effect of a dummy variable is a measure for the error in estimating main effects. The same principle can be used in fractional factorial designs for interactions: if we are willing to assume that an interaction is negligible, its size can be used as an indication of the standard deviation.

Designs at more levels

As we have seen, factorial designs at more than two levels quickly become impractical when the number of factors increases. Several other types of design are available in that situation. The idea in all cases is to fit a polynomial model with quadratic and interaction terms, and sometimes even higher-order terms. Such a model can be used to estimate optimal conditions, for example. The effects are calculated with multiple linear regression. One example for two factors is shown in the figure on the right: a central composite design. This combines a two-level factorial design with a star design, resulting in five levels for both factors.

Many more types of experimental design exist, for several different applications. A current example from our own research (in collaboration with the department of Organic Chemistry) can be found here.