Matrix of Factors

The situation of having three variables, each at two levels, can be represented pictorially, as shown in Figure 8.1. Each factor combination of the eight combinations we have dealt with as “designed factors” is represented by a vertex of the orthorhom­bic volume; the space represented by this shape is often referred to as the experimental space. If each factor has more than two levels, the experimental space can be extended, and the various factor combinations can be visually represented. If more than three factors are involved, however, then we need a space of more than three dimensions, which cannot be represented visu­ally. That is the limitation of the experimental space. Statisti­cians have devised several other ways of compiling the factor combinations, which do not suffer such limitations; these are often referred to as matrices. The experiments involved are referred to as matrix-design experiments, also more popularly known as “factorial-design.”

1. When only two levels of factors are involved, the words “high” and “low” can be used to represent any number of factors. Thus, the eight combina- tions we dealt with earlier can be represented as they are in Table 8.1.

2. The arithmetic symbols “+” and “—” are often used in place of “high” and “low,” respectively, making the matrix look more crisp, as shown in Table 8.2.

If we adapt this symbolism, the experimental space shown in Fig­ure 8.1 can be represented as shown in Figure 8.2.

3. A slight modification of the above method is to use +1 in place of + and —1 in place of — ; this makes it possible to determine the interaction offactors given by such arithmetic relations as (+ 1) x (+ 1) = + 1, (-1) x (-1) = +1, (+1) x (-1) = -1, and (-1) x (+ 1) = -1. We will see the convenience of this symbolism further in this chapter.
4. Another way of symbolizing the experimental space is to identify each combination with only those letters that are “high,” and to leave out those that are “low” in the combination. According to this symbolism, referring to Figure 8.1,

The combination a^q, instead of being blank, is repre­sented by “1.” In graphical form, it can be represented as shown in Figure 8.3. In tabular form in Table 8.3, we can notice that this matrix offers an interesting relation, which will be even more obvious when dealing with a larger number of factors.

1. More Than Three Factors

Suppose there are four variables: a, b, c, and d. Using this new symbolism, we may list the combinations as shown in Table 8.4.

However, the reader should note that the case of four factors at two levels each cannot be represented graphically as an experi­mental space because it requires four dimensions; nonetheless, it can be expressed as a matrix. Further, the matrix method, on the same lines as shown in Table 8.4, can be used for five, six, indeed, any number of variables. It is also capable of providing the number of combinations under each category of one, two, three, or more symbols representing the combinations of vari­ables. It is obvious that all these forms of matrices are suitable for situations wherein factors are tried at only two levels. Matri­ces suitable for three or more levels for each factor will be dealt with in Chapter 9.

We will conclude this section with some more terminology used in the context of designing factors. Both the eight-trial experiment with three variables and the sixteen-trial experiment with four variables, wherein each factor was listed for its effect at two levels, are known as complete (or full) factorial experiments, implying that all possible combinations of factors are tested. It is also to be noted that the three-factor, two-level experiment has eight possible combinations. These numbers are related as 2³ = 8. This experiment, hence, is referred to as a 2 experiment. Along similar lines, the four-factor, two-level situation is referred to as a 2 experiment and has 2⁴ = 16 possible combinations. Further, this relation can be generalized for any complete (or full) fac­tional experiment: the number of combinations of factors, possi­ble = l^f, wherein

l = number of levels at which each factor is tried

f = number of factors involved

A full-factorial experiment with four factors, each at five levels, according to the above relation, requires 5⁴ = 625 replications. In such situations, a more reasonable, smaller number of replications can usually be justified and is accomplished by eliminating some factors, some levels, or some of both. Such multilevel-multifactor experiments, with replications less than full factorial, are referred to as partial (or fractional) factorial experiments.

Source: Srinagesh K (2005), The Principles of Experimental Research, Butterworth-Heinemann; 1st edition.