Confounding pattern

Examination of the main effect of factor D reveals that it is identical to this main effect. This can be done for all two factor interaction effects to reveal the confounding pattern shown in Table 5.13. 

THE CONFOUNDING PATTERN FOR THE PLACKET-BURMAN DESIGN SHOWN IN TABLE 5.9... 

Table 5.13 shows the confounding pattern for the Plackett-Burman design shown in Table 5.9. The main effect for this design calculated for the plate count is shown in equation (4). The interaction effect between factor B with factor A is the difference between the main effect when B is at the method level and that when B is at its extreme level.

The Plackett-Burman designs are convenient for fitting linear screening models when the number of variables is large and when it is desirable to keep the number of necessary runs to a minimum. One disadvantage is that the confounding patterns... 

The generator can be used to identify the confounding pattern. To this end, we perform a transformation as follows ... 

From the set of generators, the following confounding pattern is derived ... 

The experiment with all variables at the (+l)-level has already been run, Exp. no 8. It will be necessary to run one complementary experiment. These two experiments define the smallest possible fractional factorial design, 2 . It is seen that the following confounding pattern is obtained... 

The first two are the independent generators, and 2345 has been obtained by multiplying the independent generators. As the sets of generators are different, the confounding patterns will be different. For clarity, only main effect and two—variable interaction effects are shown. 

As these generators are different from the generators of the parent design, they will give a different confounding pattern. The following estimates can be obtained. 

The consequences for the confounding patterns, if some variable turns out to be not significant, can be determined from the confounding pattern of the fractional design as all effects associated with the insignificant variables can be removed from the aliases obtained from each column. This technique is not entirely fool-proof and should not be routinely applied. It is sometimes a useful technique for obtaining new ideas. These ideas must then, of course, be checked through experiments. 

These designs will be Resolution III designs, and the estimated main effects will be confounded with two-variable interactions. However, the confounding patterns are not easy to elucidate directly from the design matrices. We do not have the... 

The moral of this is, that whenever there are any doubts as to the form of the model, it is always better to use a fractional factorial design for screening experiments. With fractional factorial designs, analysis of the confounding pattern may give clues to how the model should be refined. 

Appendix 7A Confounding pattern in Plackett-Burman designs... 

To analyze the confounding pattern in Plackett-Burman designs, a general method must be used. This method is to determine the alias matrix. 

The above method must be used to elucidate the confounding pattern of Plackett-Burman designs. It is seen, that a fold-over design would separate the main effects from confounding with the two-variable interactions. A fold-over design would switch the signs of the Xj matrix and hence switch the signs of the alias matrix... 

Fractional factorial designs The first choice when there are more than four variables, should always be to attempt a fractional factorial design. The confounding patterns are easily obtained firom the generators. It is also easy to append complementary runs to resolve any ambiguities. 

Plackett-Burman designs Do not consider a Plackett-Burman design as a first choice. These designs are Resolution III designs, but the confounding pattern in such designs is more difficult to analyze than that of fractional factorial designs. 

A twelve-run Plackett-Burman design can accomodate eleven variables. With 12 -15 variables, a fractional factorial design is better (confounding pattern can be analyzed). With more than 15 variables a Plackett-Burman is the preferred choice ... 

Step 8 If necessary, add more runs to the design matrix to eliminate aliases or confounding patterns. For example, if a fractional factorial experiment shows evidence of interactions between variables, it may be necessary to run the full factorial to determine which interactions are truly important. 

This design has all the desirable orthogonal properties of the full factorial. However, we have not gotten something for nothing. Although we will be able to determine up to 16 coefficients, they will all be confounded with others. We show the confounding pattern presently. Rather than list the coefficients, we merely list... 

In the extrusion-spheronization study described above (8) we recollect that the estimations of the 5 main effects contained aliased terms. These are listed below, on the left hand side (neglecting all interactions between more than two factors). It is clear that if we could carry out a second series of experiments with a different confounding pattern, with the estimations listed on the right hand side, then on combining the two series of experiments we would be able to separate main effects from first-order interactions. 

Variables and interactions Abbreviated confounding patterns Estimated effects upon responses -d(D/DnVdt DJDn sVx ... 

The relations between the contrasts calculated from the 2 halffraction and the effects obtained from the 2 full factorial (the so-called confounding patterns) are shown in the second column of Table 4.5. 

The confounding patterns are the same as those of Table 4.5, obtained from I = 1234. The following values are calculated for the contrasts ... 

All these relations introduced by means of generating functions are tedious to be calculated as such with pencil and paper, in particular in more complicated cases. However, a symbolic calculation program may serve as a practicable tool. In Table 23.8 a complete unabridged protocol to obtain the confounding pattern of a 2" factorial design is shown. 

Rearranging the last output in Table 23.8 and sorting gives the confounding patterns, as shown in Table 23.9. The variables of first order A, B, C, and D are confounded only with combinations of third order, BCD, ACD, ABD, and ABC. 

Finally, in Table 23.10 we show the confounding patterns in a 2" experiment. The data have been obtained just in the way as the data given in Table 23.8. Inspection of Table 23.8 shows that such a design is not reasonable, because D is confounded with the average. 

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