Contrast coefficient

To calculate the interaction effects following the reasoning given above would require a lot of work. An easier way exists, namely by using the columns of so-called contrast coefficients. The contrast coefficients for the... 

COLUMNS OF CONTRAST COEFFICIENTS FOR A 2 FULL FACTORIAL DESIGN... 

THE COLUMNS OF CONTRAST COEFFICIENTS FOR THE THREE-FACTOR INTERACTIONS OF THE HALF-FRACTION FACTORIAL DESIGN FOR FOUR FACTORS SELECTED FROM TABLE 3.8... 

In Table 3.16 the columns of contrast coefficients for the two-factor interactions are given. They were obtained using the above stated rules. The contrast coefficients for three- and higher-order interactions can be... 

PLACKETT-BURMAN DESIGN FOR 7 FACTORS (N=8) AND THE COLUMNS OF CONTRAST COEFFICIENTS FOR TWO-FACTOR INTERACTIONS AND FOR A THREE-FACTOR INTERACTION... 

Figure 3.9. Two-level, full factorial designs for (a) two and (b) three factors. The low value of each factor is designated by a contrast coefficient of— and the high value designated by +.

Table 3.2. Contrast coefficients for a two-level, three-factor, full-factorial experimental design...

There is a very easy way to perform the calculations. Multiply the response (y) by the contrast coefficient of the effect to be estimated, sum the values,... 

Each experiment is performed at one of two levels of each factor, just as for the two-level factorial designs described above. The contrast coefficients are a consequence of by the method and are generated from the conditions of a seed experiment by advancing each designated level around the fac-... 

The value of an effect and its standard deviation are calculated in the same way as for factorial designs. Multiply the responses by their contrast coefficients for a given factor, sum them, and divide this number by half the number of experiments (2n, for 4n experiments) to give the value of the effect. With no high-order interactions available in the design, either an independent estimate of repeatability or the use of dummy variables is essential. For m dummy variables, the effect standard deviation is calculated using equation 3.14, where ) is the measured effect of the Jth dummy factor. The significance of the effect is then determined by a Student s t test described earlier. 

Table 3.5. The designated levels as contrast coefficients for a seven-factor, eight-experiment Plackett-Burman design.

Table 3.6. Seed contrast coefficients for Plackett-Burman experimental designs.

Table 3 Experimental design matrix with contrast coefficients and experimental values...

In order to decode the effects and interactions the full design matrix with all the contrast coefficients (columns) is needed. This is shown in Table 4. The T column contains the data of all the CRF values and is used to calculate the overall mean effect. 

The normal way of carrying out such decoding calculations is by use of specialised software or a customised spreadsheet. However, for the purposes of illustration, the calculation process will be described. Basically, all that is required is to superimpose the sign convention of the contrast coefficients onto the experimental responses and perform some simple arithmetic. For this example, the calculations are shown in Table 5. 

Contrast coefficients for the Plackett-Burman experimental design used for interference studies. +1 represents high concentration and —1 represents low concentration of the interfering ions. All runs also contain the target metal ion... 

The nonregular nature of the fractional factorial design makes it possible to consider interaction effects as well as main effects see also Chapter 7. An interaction between two factors, each with three levels, has four degrees of freedom which can be decomposed into linear x linear, linear x quadratic, quadratic x quadratic, and quadratic x linear effects. The contrast coefficients for these effects are formed by multiplying the coefficients of the corresponding main effect contrasts. 

The interaction effects are calculated similarly to the main effects using Eq. (6.5) by using the so-called contrast coefficients. These coefficients are determined from the columns for the main factors. The column of contrast coefficients for the interaction AB, for instance, is obtained by multiplying the corresponding values in the main factor columns for A and B. In Table 6.3 the columns of contrast coefficients for all possible interactions are shown. The interaction effects obtained using Eq. (6.5) for the example of Table 6.1 are also shown in Table 6.2. 

COLUMNS OF CONTRAST COEFFICIENTS FOR THE INTERACTIONS OCCURRING IN THE DESIGN OF TABLE 6.1... 

TABLE 2.5. Two-level full factorial design for three factors, and columns of contrast coefficients for the interactions... 

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