Applied Statistics and Probability for Engineers

524 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

in research work and because they form the basis of other designs of considerable practical

value.

The most important of these special cases is that of kfactors, each at only two levels.

These levels may be quantitative, such as two values of temperature, pressure, or time; or they

may be qualitative, such as two machines, two operators, the “high’’and “low’’levels of a fac-

tor, or perhaps the presence and absence of a factor. A complete replicate of such a design

requires 2 2  2  2 kobservations and is called a 2 kfactorial design.

The 2kdesign is particularly useful in the early stages of experimental work, when many

factors are likely to be investigated. It provides the smallest number of runs for which kfac-

tors can be studied in a complete factorial design. Because there are only two levels for each

factor, we must assume that the response is approximately linear over the range of the factor

levels chosen.

14-7.1 2^2 Design

The simplest type of 2kdesign is the 2^2 —that is, two factors Aand B, each at two levels. We

usually think of these levels as the low and high levels of the factor. The 2^2 design is shown in

Fig. 14-13. Note that the design can be represented geometrically as a square with the 2^2  4

runs, or treatment combinations, forming the corners of the square. In the 2^2 design it is cus-

tomary to denote the low and high levels of the factors Aand Bby the signsand, respec-

tively. This is sometimes called the geometric notationfor the design.

A special notation is used to label the treatment combinations. In general, a treatment

combination is represented by a series of lowercase letters. If a letter is present, the corre-

sponding factor is run at the high level in that treatment combination; if it is absent, the factor

is run at its low level. For example, treatment combination aindicates that factor Ais at the

high level and factor Bis at the low level. The treatment combination with both factors at the

low level is represented by (1). This notation is used throughout the 2kdesign series. For ex-

ample, the treatment combination in a 2^4 with Aand Cat the high level and Band Dat the low

level is denoted by ac.

The effects of interest in the 2^2 design are the main effects Aand Band the two-factor in-

teraction AB. Let the letters (1), a, b, and abalso represent the totals of all nobservations taken

at these design points. It is easy to estimate the effects of these factors. To estimate the main

effect of A, we would average the observations on the right side of the square in Fig. 14-13

where Ais at the high level, and subtract from this the average of the observations on the left

side of the square, where Ais at the low level, or

Low

(–)

High

(+)

(1)

A

B

b

a

ab

Low

(–) High

(+)

Treatment

(1)

a

b

ab

A

B

Figure 14-13 The 2^2

factorial design.

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