Coded factor

Example of Uncoded and Coded Factor Levels and Responses for a 2 Factorial Design... 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

To check the result we substitute the coded factor levels for the first run into the coded empirical model, giving... 

Another, often major, advantage of using coded factor levels is that the numerical values involved in matrix manipulations are smaller (especially the products and sums of products), and therefore are simpler to handle and do not suffer as much from round-off errors. 

It is to be stressed, however, that the geometric interpretation of the parameter estimates obtained using coded factor levels is usually different from the interpretation of those parameter estimates obtained using uncoded factor levels. As an illustration, pj (the intercept in the coded system) represents the response at the center of the experimental design, whereas Pq (the intercept in the uncoded system) represents the response at the origin of the original coordinate system the two estimates (Pq and Pq) are usually quite different numerically. This difficulty will not be important in the remainder of this book, and we will feel equally free to use either coded or uncoded factor levels as the examples require. Later, in Section 11.5, we will show how to translate coded parameter estimates back into uncoded parameter estimates. 

The data and the least squares straight line relationship are shown in Figure 11.2. It is to be remembered that the parameter estimates are those for the coded factor levels (see Section 11.2) and refer to the model... 

Many workers do not transform the parameter estimates back to the original coordinate system, but instead work with the parameter estimates obtained in the coded factor space. This can often lead to surprising and seemingly contradictory results. As an example, the fitted model in the coded factor space was found to be... 

Thus, the fitted model in coded factor space is... 

The effects of visual scaling and numerical coding are very similar. However, the distortions caused by numerical coding are not always as readily apparent as some distortions caused by visual scaling. For example, rotatable designs in coded factor spaces might not produce rotatable designs in uncoded factor spaces (see Section 12.9). We simply warn the reader that concepts such as rotatable , circular . 

The design in Figure 13.4 is similar to the design in Figure 13.3, but the center point has been replicated a total of eight times, not four. This makes the design not only rotatable but also orthogonal in the coded factor space that is, the estimate of one factor effect (i.e pj, Pj, P , P22, or P j) is independent of the estimates of all other factor effects (see Section 12.10). 

Factorial designs are usually discussed in terms of coded factor spaces. Table 14.1 shows some of the common coding systems for two- and three-level designs. Our emphasis in this chapter will be on the two-level designs. 

The first column of Table 14.2 lists the experiment numbers (1-8). The next three columns list the abbreviated coded factor levels (- and +) for factors A, B, and C. Note that these three columns are equivalent to the abbreviated coded D matrix ... 

Thus, in modem research using interval and ratio scales the 5x usually shouldn t be ignored. Let s add 8x, to the calculation to obtain b, as would be done with regression analysis. Because Xj went from a coded level of -1 to a coded level of -t-1, 5x, = 2. Thus, b (the factor effect in the coded factor space) = 8y,/8xJ = -i-3.6% per 2 coded units = -i-1.8% per coded unit. The fact that 8x is equal to 2 with this system of coding is why regression analysis of coded data gives results that are smaller by Vi from the results obtained from the classical approach ... 

Notice that this is an orthogonal design in coded factor space (-1 and +1) any one column multiplied by any other column will give a vector product of zero. Other saturated fractional factorial designs may be found in the literature [Box and Hunter (1961a, 1961b), Anderson and McLean (1974), Barker (1985), Bayne and Rubin (1986), Wheeler (1989), Diamond (1989)]. 

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). 

In this example, orthogonality of all factor effects has been achieved by including additional center points in the coded rotatable design of Equation 11.81. Orthogonality of some experimental designs may be achieved simply by appropriate coding (compare Equation 11.26 with Equation 11.20, for example). Because orthogonality is almost always achieved only in coded factor spaces, transformation of... 

Since the relation between actual and coded factors is given by expressions ... 

After choosing the simplex matrix of design of experiments with coded factors, we should switch to an operational matrix with real factor values, taking into account factor-variation intervals and coordinates of the center of the experiment. The general formula for transfer from coded to real values (2.59) is also valid in this case ... 

After the choice of design of experiments matrix with coded factor values, one should switch to an operational matrix or to a matrix with real factor values. This shift from coded to real factor values is done by the known formula (2.59). Accord-... 

It should be noted that pseudocomponents or coded factors appear in the regression model. A check of lack of fit of the regression model in control points has shown that the regression model is adequate with 95 % confidence. 

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

Connections between real and coded factors are given by these relations ... 

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