factors in 
runs, where is a multiple of 4. If is a power of 2, these designs are
identical to those of 
fraction factorial. However, for other
cases, the Plackett-Burman designs are sometimes of interest.
The original Plackett-Burman (1946) designs are
two-level fractional factorial designs for studying factors in runs, where is a multiple of 4. If is a power of 2, these designs are
identical to those of fraction factorial. However, for other
cases, the Plackett-Burman designs are sometimes of interest.
As an example, the Plackett-Burman design for 
and 
is derived. The element of 
are 0, 1, 2, … , 10 with
arithmetic being carried out mod 11. The quadratic residues are 
, 
, 
, 
, 
. Then 
if 
and 
if 
. These designs are shown in the following
table.
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 
and 
give imbalanced sampling, which is
only due to the quadratic residue of 
. In this case,
RecurDyn/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.
NOTE
Galois Field: Let z be any nonzero element of the finite
field ; 
is said to be a quadratic residue of
the field if there is an element 
in the field such that 
; otherwise, is a nonquadratic residue. Then, the
Legendre symbol, 
, is defined
as follows: 
; 
, if is a quadratic residue; if 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.