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14-6 FACTORIAL EXPERIMENTS WITH RANDOM FACTORS (CD ONLY) 14-7 2k FACTORIAL DESIGNS
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Factorial designs are frequently used in experiments involving several factors where it is necessary to study the joint effect of the factors on a response. However, several special cases of the general factorial design are important because they are widely employed
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CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS
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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 k factors, 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 factor, or perhaps the presence and absence of a factor. A complete replicate of such a design requires 2 2 2 2k observations and is called a 2k factorial design. k The 2 design 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 k factors 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.
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22 Design
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The simplest type of 2k design is the 22 that is, two factors A and B, each at two levels. We usually think of these levels as the low and high levels of the factor. The 22 design is shown in Fig. 14-13. Note that the design can be represented geometrically as a square with the 22 4 runs, or treatment combinations, forming the corners of the square. In the 22 design it is customary to denote the low and high levels of the factors A and B by the signs and , respectively. This is sometimes called the geometric notation for 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 corresponding 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 a indicates that factor A is at the high level and factor B is at the low level. The treatment combination with both factors at the low level is represented by (1). This notation is used throughout the 2k design series. For example, the treatment combination in a 24 with A and C at the high level and B and D at the low level is denoted by ac. The effects of interest in the 22 design are the main effects A and B and the two-factor interaction AB. Let the letters (1), a, b, and ab also represent the totals of all n observations 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 A is at the high level, and subtract from this the average of the observations on the left side of the square, where A is at the low level, or
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Figure 14-13 The 22 factorial design.
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Similarly, the main effect of B is found by averaging the observations on the top of the square, where B is at the high level, and subtracting the average of the observations on the bottom of the square, where B is at the low level: