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Basic concepts of multi-factor experiments

The multiple factor experiments require special methods to design and analysis. The most of this chapter is reserved for two factor experiments, but when there are more than two factors, the methods are similar. We have to note that these factors must be independent, it means that we can change the value of a factor without any effect to other ones.

One strategy of experiment is one-factor-at-a-time, in which the experiment consists of several steps, each step studies only one factor. But by this way, we can not evaluate the interaction between factors.

Interaction between factors

To understand better the concept "interaction", let's consider an experiment of two factors A and B, and response `Y`. There are two levels for each factor: A₁ and A₂ for factor A, B₁ and B₂ for factor B. The results are shown in Table 1.

Table 1 Effect of A and B on `Y` in two cases

This result also can be illustrated in Fig. 1.

Fig. 1 Effect of A and B on `Y` in two cases.

From Table 1 and Fig. 1, we recognize that:

• In case a, response `Y` alway increases 30 units in both levels of factor A, regardless value of factor B. So factor B does not affect the effect of A on `Y`.
• In case b, effects of A on `Y` are not the same for two level of B: from increasing of 20 units in level B1, to decreasing of 40 units in level B2 (Figure 9.1). So B affects on the effect of A on `Y`. This phenomenon is known as interaction between factors.

The strategy "one-factor-at-a-time" cannot detect this interaction, and therefore can commit errors, sometimes serious.