21.4.3.5. Plackett and Burman Design

The original Plackett-Burman (1946) designs are two-level fractional factorial designs for studying \(p=N-1\) factors in \(N\) runs, where \(N\) is a multiple of 4. If \(N\) is a power of 2, these designs are identical to those of \({{2}^{k-p}}\) fraction factorial. However, for other cases, the Plackett-Burman designs are sometimes of interest.

As an example, the Plackett-Burman design for \(N=12\) and \(p=11\) is derived. The element of \(GF{{\left( 11 \right)}^{*}}\) are 0, 1, 2, … , 10 with arithmetic being carried out mod 11. The quadratic residues are \(1={{1}^{2}}={{10}^{2}}\), \(4={{2}^{2}}={{9}^{2}}\), \(9={{3}^{2}}={{8}^{2}}\), \(5={{4}^{2}}={{7}^{2}}\), \(3={{5}^{2}}={{6}^{2}}\). Then \(\chi {{\left( z \right)}^{*}}=+1\) if \(z=1,3,4,5,9\) and \(\chi \left( z \right)=-1\) if \(z=2,6,7,8,10\). These designs are shown in the following table.

../_images/image172.png

In AutoDesign, the Plackett-Burman design is extended to overcome the lack of balance in some factors. As an example, the Plackett-Burman design for \(N=16\) and \(p=15\) give imbalanced sampling, which is only due to the quadratic residue of \(GF\left( {{p}^{m}} \right)\). In this case, RcurDyn/AutoDesign automatically increase the number of trials until satisfying the balance of sampling. Also, for the mixed-level design such as 2, 3 and 4 levels, the contractive replacement method of Addelman and Kempthorne (1961) is used. As an example, for the mixed-level such as two 2-level factors, two 3-level factors and two 4-level factors, the extended Plackett-Burman design is given in the following table.

../_images/image173.png

Note

Galois Field: Let z be any nonzero element of the finite field \(GF\left( {{p}^{m}} \right)\); \(z\) is said to be a quadratic residue of the field if there is an element \(w\) in the field such that \({{w}^{2}}=z\); otherwise, \(z\) is a nonquadratic residue. Then, the Legendre symbol, \(\chi \left( z \right)\), is defined as follows: \(\chi \left( 0 \right)=0\);:math:chi left( z right)=+1, if \(z\) is a quadratic residue; \(\chi \left( z \right)=-1\) if \(z\) is a non-quadratic residue.

Reference

  1. Plackett, R.L. and Burman, J.P., 1946, “The Design of Optimum Multi-factorial Experiments”, Biometrika, Vol. 33, pp. 305~325.

  2. Addelman, S. and Kempthorne, O. 1961, “Some Main Effects Plans and Orthogonal Arrays of Strength Two”, Ann. Math. Statist., Vol. 32, pp. 1167~1176.

  3. John, P.M.J., 1998, Statistical Design and Analysis of Experiments, SIAM, pp.185~190.