6.1.12. Plackett-Burman

Fractional factorials designs are useful for screening experimentation where the purpose is to identify the factors having large effects worthy of further experimentation.

The Plackett-Burman designs are fractional two-level designs 2^k with n = k+1 design points which should be a multiple of 4. Moreover, the Plackett-Burmann designs coincide with Reduced Factorial design when the number of runs is a power of 2.

Plackett-Burman designs can be constructed through the use of the Hadamard matrices (nxn matrices H of +1 and -1 such that HH'=nI). The matrix dimension should be the smallest multiple of four greater than the number of input variables.

These designs are useful in studying the main effects because they produce main effect estimates which are orthogonal.

Figure 6.21. Plackett-Burman Sequence Panel

In all cases, except for n = 28, the designs are specified by giving just the first row.

This row is then permuted cyclically the appropriate number of times to get an (n - 1)x(n - 1) matrix.

Finally, an extra row of all minus signs is added.

For example, let k = 7 so that n = 8 which is a multiple of 4 starting with the first row

The "-" sign corresponds to the minimum value and the "+" sign corresponds to the maximum value of the input variable range.

Rows 2 to 7 are then cyclically obtained shifting the signs one place to the right as shown below:

The above eight rows lead to the fitting of the linear model

y = b₀+b₁x₁+b₂x₂+b₃x₃+... +b₇x₇+e

with the 7 factor values defined as x₁, x₂, ..., x₇ and with 8 observations on the response.

modeFRONTIER^TM classify all the Plackett-Burman designs up to 60 designs.

The effect of a factor is the usual average response of the runs made at the higher level minus the average response of the runs made at the lower level. For example, the "effect" of a factor X is

and the "mean square" for X is found as:

Some of the designs suggested by Taguchi are Plackett-Burman designs which do not allow interactions to be estimated.

References