PLACKETT and BURMAN

Plackett and Burman [1946] have developed a special fractional design which is widely applied in analytical optimization. By means of N runs up to m = N — 1 variables (where some of them may be dummy variables which can help to estimate the experimental error) can be studied under the following prerequisites and rules ... 

Plackett and Burman [14] give the method of design construction for all values of n that are multiples of 4 up to 100 except n=92. 

Especially the full and fractional factorial designs are best performed in a random order to avoid the influence of systematic errors because they are constructed so that one factor is at one level in the first N/2 experiments and at the other in the last N/2 experiments. The Plackett-Burman designs can be considered as randomized when they are performed in the sequence that is described in the original papers of Plackett and Burman. 

The influence of water components on the flame photometric determination of potassium and sodium can be detected by factorial experiments. By application of multifactorial plans according to PLACKETT and BURMAN the qualitative determination of the influence of various variables is possible with relatively few experiments [SCHEFFLER, 1986]. For mathematical fundamentals see Chapter 3. 

An approach similar to that of Plackett and Burman called "Ruggedization" was introduced by Youden (4) for the screening of only seven variables with eight experiments. The combination of the high and low levels of the variables in Youden s design is... 

Lin, D. K. J. and Draper, N. R. (1991). Projection properties of Plackett and Burman designs. Technical Report 885, Department of Statistics, University of Wisconsin. 

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. 

We see that this applies to the model matrices from factorial designs and fractional factorial designs these matrices are Hadamard matrices. It is possible to construct Hadamard matrices by other principles, and it was shown by Plackett and Burman[l] how such matrices can be obtained for zi = 4, 8, 12, 16, 20, 24, 28, 32,..., i.e. when n is a multiple of four. 

Plackett and Burman have shown the construction of all matrices for n up to 100, with one exception, (92 x 92) matrix, which they failed to construct. This matrix has been reported later.[2] The Plackett-Burman paper[l] discusses the use of Hadamard matrices to define screening experiments, in which (n —1) variables can be tested in n runs. 

The construction of a design by a Hadamard matrix is simple. Plackett and Burman have determined how the first row in the design matrix should be constructed so that the remaining rows can be obtained by cyclic permutations of the first row. An example to show this is given below. For a given n there can be many different n xn matrices which are Hadamard matrices. The Plackett-Burman matrices are by no means the unique solutions. 

Sign sequences for experimental designs with n runs for n = 8, 12, 16, 20, 24, 28 are given here. It is rare that more than 27 variables are involved in synthetic procedures. Readers who are interested in constructing larger designs, should consult the paper by Plackett and Burman. 

The Plackett and Burman design of 8 experiments is given below (in reality the values -1 and +1 have been inverted with respect to those of the original design given in appendix II). 

Table 2.3 Plackett and Burman Screening Design for Extrusion-Spheronization Study Coded Variables...

In the previous section we saw how factors at two levels may be screened using a Plackett-Burman design. In their paper (2), Plackett and Burman constructed experimental designs for factors at 2, 3, 5, and 7 levels. Such designs are termed symmetric, because all factors have the same number of levels. The designs may... 

We have just discussed designs with which we can study the influence of up to 7, 15, 31,...,2 "- factors using 8, 16, 32,...,2" experimental nms. Another class of fractional factorial designs employs a total of 12, 20, 24, 28,... runs to simultaneously investigate up to 11, 19, 23, 27,... factors. With these designs, proposed by R.L. Plackett and J.P. Burman, it is possible to estimate all k = n-l main effects (where n represents the number of runs) with minimum variance (Plackett and Burman, 1946). Table 4.17 presents a Plackett-Burman design for n = 12. 

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