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Mathemagical musings

bradleyjones

Take a look at this matrix of symbols.

++++++++++++++++++++++++++++++++

+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−

++−−++−−++−−++−−++−−++−−++−−++−−

+−−++−−++−−++−−++−−++−−++−−++−−+

++++−−−−++++−−−−++++−−−−++++−−−−

+−+−−+−++−+−−+−++−+−−+−++−+−−+−+

++−−−−++++−−−−++++−−−−++++−−−−++

+−−+−++−+−−+−++−+−−+−++−+−−+−++−

++++++++−−−−−−−−++++++++−−−−−−−−

+−+−+−+−−+−+−+−++−+−+−+−−+−+−+−+

++−−++−−−−++−−++++−−++−−−−++−−++

+−−++−−+−++−−++−+−−++−−+−++−−++−

++++−−−−−−−−++++++++−−−−−−−−++++

+−+−−+−+−+−++−+−+−+−−+−+−+−++−+−

++−−−−++−−++++−−++−−−−++−−++++−−

+−−+−++−−++−+−−++−−+−++−−++−+−−+

++++++++++++++++−−−−−−−−−−−−−−−−

+−+−+−+−+−+−+−+−−+−+−+−+−+−+−+−+

++−−++−−++−−++−−−−++−−++−−++−−++

+−−++−−++−−++−−+−++−−++−−++−−++−

++++−−−−++++−−−−−−−−++++−−−−++++

+−+−−+−++−+−−+−+−+−++−+−−+−++−+−

++−−−−++++−−−−++−−++++−−−−++++−−

+−−+−++−+−−+−++−−++−+−−+−++−+−−+

++++++++−−−−−−−−−−−−−−−−++++++++

+−+−+−+−−+−+−+−+−+−+−+−++−+−+−+−

++−−++−−−−++−−++−−++−−++++−−++−−

+−−++−−+−++−−++−−++−−++−+−−++−−+

++++−−−−−−−−++++−−−−++++++++−−−−

+−+−−+−+−+−++−+−−+−++−+−+−+−−+−+

++−−−−++−−++++−−−−++++−−++−−−−++

+−−+−++−−++−+−−+−++−+−−++−−+−++−

Beautiful isn’t it? This matrix has several wonderful properties. Ignoring the first column composed of all pluses, each column has the same number of pluses and minuses. Also, each pair of columns has eight replicates of all possible pairs of symbols, (++, +−,−+,−−). That makes this matrix a so-called strength-two orthogonal array. Using this matrix as a design for an experiment, you can do a sensitivity analysis of the main effects of 31 factors using only 32 runs.

What is the probability of generating this matrix by chance?

Here is one possible answer. There are 32 rows and 32 columns in this matrix. That makes 32x32 = 1024 total symbols. Each symbol can take two values, so the number of possible matrices is 2¹⁰²⁴. If one generated a random 32 by 32 matrix of pluses and minuses, then the probability of generating the matrix above is 1/2¹⁰²⁴ – and that is a very small number. Since this way of generating the matrix by chance is not feasible in our lifetime (or very possibly the life of the universe), we might suppose that the matrix could not be generated by chance.

Could there be another way?

We have two symbols, + and −. Consider this rule for combining symbols. If I give you a symbol or matrix of symbols, H, you give me back a matrix that looks like this,

H H

H –H

For example, starting with + and applying our rule we get

+−

Applying the rule to the matrix above yields

++++

+−+−

++−−

+−−+

Three more iterations produce our elegant original matrix.

The rule is extremely simple. The probability of generating the rule by chance could be as large as ½⁴ = 1/16. So, to generate the original matrix all we need are the two symbols, our rule of combination and five repeats.

And your point is?

Our original probability assessment involved the assumption of 1,024 independent chance events. This assumption, this declaration of independence as it were, is not necessary.

The assumption of independence is extremely strong. Use it in statistical, or philosophical, arguments with care!

Footnote

The original matrix is called a Hadamard matrix after the 19th century French mathematician of that name. Hadamard also invented the simple rule that allows one to take one Hadamard matrix and turn it into another that has twice as many rows and columns.

Comments

WP_Comment

Manny Uy wrote:

Brad,

I am currently teaching a DOE course using Doug Montgomery's book "Design and Analysis of Experiments", and I normally do, utilize JMP to generate various designs and analysis within the textbook's examples. Then I came to the section on Plackett-Burman designs, where the first column generator for an 11 factor, 12 run design is + + - + + - - - + -. However, when I used JMP's secreening desigfn platform for a Plackett-Burman design with 11 factors and 12 runs, I was not able to generate the saqme design as one using the above generator (first column after the identity column). How does JMP generate its Plackett-Bruman design? I suspect with a Haadamard matrix like you mentioned in this article. Please help me answer this question posed by one of my students.

Manny

bradleyjones

Bradley Jones wrote:

The generator in the 7th edition of Montgomeryâ s book on page 326 is

+ + - + + + - - - + -

Since there are more plusses than minuses he ends with a row of minuses.

It is funny, however, that Table 8.24 on page 327 actually uses the generator

+ - + - - - + + + - +

The generator in JMP is

- + - + + + - - - + -

In this generator there is one more minus than plus, so you need a row of plusses.

Both designs are orthogonal and both have the same aliasing between main effects and two-factor interactions. I am not sure whether they are isomorphic in the sense of being able to obtain one from the other by switching rows, switching columns or interchanging plus and minus in a column. I think that there are multiple nonisomorphic designs for this case. However, for all practical purposes these designs are equivalent.

Table 8.24 can be obtained from the Plackett-Burman design using the Screening designer in JMP (and choosing Keep the Same for the row order) by interchanging the symbols for all the columns. Therefore Table 8.24 and the JMP screening design are isomorphic.

Using the Custom Designer for 11 factors in 12 runs you get another orthogonal design that has the same alias structure. It is not easy to see how to obtain one design from the other using the three allowed operations I reference above. Again, for all practical purposes the D-optimal design produced by the Custom Designer is equivalent to a Plackett-Burman design.