PROC FACTEX: Types of Factors

The FACTEX Procedure

Types of Factors

The factors of a design are variables that an experimenter can set at several values. In general, experiments are performed to study the effects of different levels of the factors on the response of interest. For example, consider an experiment to maximize the percentage of raw material that responds to a chemical reaction. The factors might include the reaction temperature and the feed rate of the chemicals, while the response is the yield rate. Factors of different types are used in different ways in constructing a design. This section defines the different types of factors.

Block factors are unavoidable factors that are known to affect the response, but in a relatively uninteresting way. For example, in the chemical experiment, the technician operating the equipment might have a noticeable effect on the yield of the process. The operator effect might be unavoidable, but it is usually not very interesting. On the other hand, factors whose effects are directly of interest are called design factors. One goal in designing an experiment is to avoid getting the effects of the design factors mixed up, or confounded, with the effects of any block factors.

When constructing a design by orthogonal confounding, all factors formally have the same number of levels $\text{[math]}$ , where $\text{[math]}$ is a prime number or a power of a prime number. Usually, $\text{[math]}$ is two, and the factor levels are chosen to represent high and low values.

However, this does not mean, for example, that a design for two-level factors is restricted to no more than two blocks. Instead, the values of several two-level factors can be used to index the values of a single factor with more than two levels. As an example, the values of three two-level factors ( $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ ) can be used to index the values of an eight-level factor ( $\text{[math]}$ ), as follows:

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
0	0	0	0
0	0	1	1
0	1	0	2
0	1	1	3
1	0	0	4
1	0	1	5
1	1	0	6
1	1	1	7

The factors $\text{[math]}$ are used only to derive the levels of the factor $\text{[math]}$ ; thus, they are called pseudo-factors, and $\text{[math]}$ is called a derived factor. In general, $\text{[math]}$ $\text{[math]}$ -level pseudo-factors give rise to a single $\text{[math]}$ -level derived factor. Block factors can be derived factors, and their associated formal factors (the $\text{[math]}$ factors) are called block pseudo-factors.

The method for constructing an orthogonally confounded design for $\text{[math]}$ -level factors in $\text{[math]}$ runs distinguishes between the first $\text{[math]}$ factors and the remaining factors. Each of the $\text{[math]}$ different combinations of the first $\text{[math]}$ factors occurs once in the design in an order similar to the preceding table. For this reason, the first $\text{[math]}$ factors are called the run-indexing factors.

Table 7.7 summarizes the different types of factors disCussed in this section.

Table 7.7 Types of Factors
Block factor	Unavoidable factor whose effect is not of direct interest
Block pseudo-factor	Pseudo-factor used to derive levels of a block factor
Derived factor	Factor whose levels are derived from pseudo-factors
Design factor	Factor whose effect is of direct interest
Pseudo-factor	Formal factor combined to derive the levels of a real factor
Run-indexing factors	The first $\text{[math]}$ design factors, whose $\text{[math]}$ combinations index the runs in the design

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