Talk:Plackett–Burman design

Edit war

The section "4 to 48 runs, sorted" has now been added twice and removed twice. I would like to request that the parties involved discuss the issue here before making further changes. Accusations of sock-puppetry should not be made lightly. My personal opinion is that the added data are likely to be useful to someone, but that Wikipedia may not be the place for them. Their presence here certainly does not enhance the readability of the article; with that section added one has, for example, to do a lot of scrolling in order to reach the references at the end of the article.

I also have some technical questions.

1. The article states that Plackett and Burman used the method of Paley to construct their designs, but this does not unambiguously specify the design. For example, if N=80, one can use Paley I directly, since 80−1=79 is prime. But one could also use Sylvester's doubling construction twice on an N=20 design. The latter could, in turn, be constructed using Paley I (since 20−1=19 is prime) or using Paley II (since 20/2−1=9 is a prime power. Similarly, if N=32, one could use doubling five times on the N=1 design, or one could use Paley I, since 32−1=31 is prime. These designs are not equivalent. On the other hand, if N=16, one could use doubling four times on the N=1 design, or one could use doubling once on the N=8 design obtained by Paley I (since 8−1=7 is prime), but these designs are equivalent. If "Plackett–Burman design" refers to a specific design, the article should specify what the rule is for choosing among the various possibilities. (In reference to the discussion above, I think that an algorithm would be preferable to a list of matrices.)
2. Is it really the case that "Plackett–Burman design" refers only to designs obtained from Hadamard matrices constructed by Paley's methods? If so, is this merely for historical reasons, or is there some reason why Paley's methods are preferred to other Hadamard-matrix construction methods? In other words, would one say that there is no Plackett–Burman design for N=92 even though there is a Hadamard matrix of that size?
3. The article makes the statement "If N is a power of 2, however, the resulting design is identical to a fractional factorial design, so Plackett–Burman designs are mostly used when N is a multiple of 4 but not a power of 2 (i.e. N = 12, 20, 24, 28, 36 …)" Why would a Plackett–Burman design being "identical to" a fractional factorial design be a reason not to use a Plackett–Burman design?

Will Orrick (talk) 03:56, 16 August 2013 (UTC)

Hi, I put those here, because the algorithms to generate them are not simple, nor are the software packages cheap, and there simply is no listing on the Internet, yet, of these designs. I hope moving them to the end makes the article sufficiently readable.

1. I agree but leave that to someone else to produce a compact code.
2. SAS at least can generate all P-B designs up to 448, and many above that.
3. For 2-level designs, NIST seems to have come to that conclusion.

Attleboro (talk) 18:00, 20 August 2013 (UTC)

Thanks for your reply. Putting the references before the listing of designs was a good idea.

I've been doing some reading, and it seems that one can always obtain a Plackett–Burman design with N runs by taking any N × N Hadamard matrix, normalizing rows so that column 1 consists entirely of 1s, normalizing columns 2 through N so that their first elements are all −1, and then removing the first column. Free online sources of Hadamard matrices are listed in the External links section of the Hadamard matrix article, so one should be able to construct a design for every N up to 664 using those resources.

It appears, however, that both the choice of Hadamard matrix and the choice of column to play the role of column 1 make a difference. The behavior of the design with respect to interactions among factors will depend on these choices. It is not clear to me yet how the designs in the commercial packages are selected and whether any attempt is made to optimize the behavior with respect to interactions. Some articles I've seen like to choose designs where all rows are cyclic permutations of one another, but this is not possible for every N. I will continue to try to find out more.

In regard to question 3, the Minitab website contains a discussion of the issue that sheds some light. In particular, it seems that for powers of 2, the fractional factorial designs have higher resolution than the Plackett–Burman designs. Given that they have different properties, it doesn't make sense to me to call the two types of design "identical". Statistics is not my area, so I don't feel competent to address this issue, but I will try to learn more.

Will Orrick (talk) 16:38, 21 August 2013 (UTC)

Plackett–Burmans all have Resolution III, which is to say that, under the assumption no higher order effects exist, all first order effects are estimable. For powers of 2, the P-B and Res.III fractional factorial can be the same. Fractional factorials can be and often are constructed at Res.IV and Res.V, but these are much larger than P-Bs. Res.IV assumes 2nd order effects exist and must not be confounded with first order ones so all first order effects remain estimable. Res.V assumes 2nd order effects exist and higher order ones are negligible, and thus all 1st and 2nd order effects are estimable. The confounding patterns for P-B and Res.III fractional factorials can be different in that specific 2nd order effects are confounded with specific first order ones for R.III.FFs but for P-Bs can be distributed over multiple first order effects. Attleboro (talk) 21:02, 27 August 2013 (UTC)