Regarding the Plackett-Burman screening design

Question

I am starting to implement screening designs in my optimization work. The one I am most familiar with is the Plackett-Burman design, so I am starting there. I have written my own scripts in MATLAB but I am trying to reference them with minitab.

MATLAB suggests that users uses the hadamard command to generate a hadamard matrix for the desired number of factors. For instance, the call hadamard(8) produces an 8 by 8 hadamard matrix, where the second to last columns form the Plackett-Burman design. However, minitab only allows the user a few set number of runs: 12, 20, and so on (8 isn't an option). Is there something inherent to the Plackett-Burman design I am missing?

Second - one oft cited metric I have run into is that the effect of each factor can be calculated according to

$$ \text{Effect} = \frac{2 \left[\sum (y+) - \sum(y-) \right]}{N} $$

where $y(+)$ is the response of an experiment at the high leve, $y(-)$ the response at the low level, and $N$ the number of experiments. I have performed this calculation in MATLAB and I get the same result as Minitab produces. However, I am a bit curious where this equation derive from ? what I am more concerned about is the following: from what I can tell, minitab compares these effects to the $t$-value at the specified number of DOF. However, I don't see how these two are comparable? Why should the significance of an "effect" be equivalent to a $t$-value?

Any help is appreciated.

Answer 1

It's not easy to generate Plackett-Burman designs/Hadamard matrices in full generality, so software tends to have a few pre-calculated ones and perhaps the ability to expand them into families of larger matrices.

It's easy to calculate Hadamard matrices of order $n=2^k(p+1)$, where $p$ is prime and $p+1$ is a multiple of 4. Also, given matrices of orders $m$ and $n$ it is easy to make one of order $2n$ or $mn$, so if you store a few examples you can make bigger ones. (There are other families of Hadamard matrices, eg of order $n=2^k(p^r+1)$ for $r>1$, but these are less easy to generate).

Matlab says it provides Hadamard matrices of order $n$ where $n$, $n/12$ or $n/20$ is a power of 2. (Matlab is thinking of these as Hadamard matrices)

Minitab has Plackett-Burman designs for 12-48 levels (except for $n=16$). The documentation says 8 and 16 are left out because there are factorial designs with better resolution (Minitab is clearly thinking of these as designs, not as matrices with other potential uses)

The survey package for R has a hadamard function that gives more possible values.