9.3 Several Categorical Factors: the Full Factorial Design

We frequently expect the response in an experiment to be affected by several factors. If so, it is interesting to find out which factor has the strongest effect on the response. Another important question is if the effects are separate or if the way in which the factors are combined is important. These questions can be answered by full factorial experiments.

We will begin by a case with only two factors, since this is convenient for understanding. The design is shown in Table 9.1, where the columns represent the two factors and the rows correspond to the investigated factor combinations, or treatments. We may note that all the possible combinations are present. To understand how the effects of the factors still can be separated, we will introduce the concept of orthogonality. This requires a brief mathematical digression, but we will soon be back on track again.

Table 9.1 Two-level full factorial design with two factors.

−1 −1
1 −1
−1 1
1 1

Mathematicians discern between two types of quantities: scalars and vectors. Numbers in general only have size and are called scalars. Scalars are used to describe things like the outside temperature or the atmospheric pressure. Vectors have both size and direction and are expressed in terms of coordinates. The wind velocity is one example. The plane in Figure 9.2 is defined by two vectors, x and y, each with a length of one unit. They represent the two principal ...

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