Classical factorial design

In the classical factorial design literature, a factor effect is defined as the difference in average response between the experiments carried out at the high level of the factor and the experiments carried out at the low level of the factor. Thus, in a 2 full factorial design, the main effect of A would be calculated as ... 

Taylor expansion 1s that of obtaining an efficient experimental design. The classical factorial designs depend upon the experimenter s ability to choose levels of his independent variables to hold them at these prescribed values. By defining the independent variables to be In with the intention of calculating from from... 

Moreover in a classical factorial design, in the case of linear relationships, only two conditions for each independent variable should be tested at the lowest and highest possible setting. If the dependent variable is a function of three independent variables (A, B, and C), each of which can take on two possible values or levels, then to test all possible variable combinations the number of experiments is equal to the number of levels, N, raised to the power of the number of independent variables, n, e.g. N" = 23 = 8 (Figure 10.5 and Table 10.2). 

Table 10.2, Experimental program for classical factorial design.

This is a novel feature of factorial design when compared with the classical laboratory procedure which excludes indications of the interaction of tire variable. The method of airalysis of the data, due to Yates, which is commonly used to evaluate these effects, requires tlrat tire uials are conducted in the sequence shown above, and proceeds as follows. 

Classical mathematical treatment of a full 2 factorial design... 

To illustrate this classical approach to the calculation of factor effects, consider the following 2 full factorial design ... 

Traditional tabular presentation of data for calculating the classical factor effects in a 2 factorial design. 

Table 14.4 shows a typical regression analysis output for the 2 factorial design in Table 14.3. Most of the output is self-explanatory. For the moment, however, note the regression analysis estimates for the parameters of the model given by Equation 14.5 and compare them to the estimates obtained in Equations 14.8-14.15 above. The mean is the same in both cases, but the other non-zero parameters (the factor effects and interactions) in the regression analysis are just half the values of the classical factor effects and interaction effects How can the same data set provide two different sets of values for these effects ...

Calculate the grand average (MEAN), the two classical main effects (A and B), and the single two-factor interaction (AB) for the two-factor two-level full factorial design shown in the square plot in Section 14.1. (Assume coded factor levels of -1 and +1). 

When few factors (/ from two to four) are studied, the full factorial design is the most common approach. The full factorial scheme is the basis for all classical experimental designs, which may be used in more complex situations. For a general two-level full factorial design, each factor has to be considered at a low level (coded as —1) and a high level... 

The first designs for mixture experiments were described by Scheffe [3] in the form of a grid or lattice of points uniformly distributed on the simplex. They are called q, i j simplex-lattice designs. The notation q, v implies a simplex lattice for q components used to construct a mixture polynomial of degree v. The term mixture polynomial is introduced to distinguish it from the polynomials applicable for mutually independent or process variables, which are described later in our discussion of factorial designs (section 8.4). In this way, we distinguish mixture polynomials from classical polynomials. 

There are several ways to study the effect of various experimental factors on an analytical method. The classical method of studying one variable at a time while holding others constant is extremely inefficient. Other approaches such as regression analysis and complete factorial designs involve a large number of experiments and are also inefficient. For example, a factorial design of seven factors at two levels requires 2 or 128 experiments. Therefore, alternate approaches which reduce the experimental work are very attractive. 

Plackett and Burman published their classical paper in 1946, which has been much cited by chemists. Their work originated from the need for war-time testing of components in equipment manufacture. A large number of factors influenced the quality of these components and efficient procedures were required for screening. They proposed a number of two level factorial designs, where the number of experiments is a multiple of four. Hence designs exist for 4, 8, 12, 16, 20, 24, etc., experiments. The number of experiments exceeds the number of factors, k, by one. 

The first advantage of our factorial design is that our main effects are obtained as the difference between the mean of one set of four observations and the mean of another set of four observations. In the classical design our main effects were the differences between means of two observations. We have thus obtained double the accuracy for the same number of experiments. If this experiment is being carried out on the full industrial scale at considerable cost and trouble, this factor can be of the greatest economic importance. 

The third type of experimental design is the factorial design, in which there are two or more clearly understood treatments, such as exposure level to test chemical, animal age, or temperature. The classical approach to this situation (and to that described under the latin square) is to hold all but one of the treatments constant and at any one time to vary just that one factor. Instead, in the factorial design all levels of a given factor are combined with all levels of every other factor in the experiment. When a change in one factor produces a different change in the response variable at one level of a factor than at other levels of this factor, there is an interaction between these two factors which can then be analyzed as an interaction effect. 

Three variables will give eight different combinations of the variable settings. These experiments were run and are summarized in the classic representation of a factorial design shown in Table 5.2. 

Chapter 1 introduces experimental design, describes the book s plan, and elucidates some key concepts. Chapter 2 describes screening designs for qualitative factors at different levels, and chapter 3 covers the classic two-level factorial design for studying the effects of factors and interactions among them. Chapter 4 summarises some of the mathematical tools required in the rest of the book. 

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