# Picking interaction effects for PB design and Hamada-Wu analysis

I am trying to use Plackett-Burman design to construct a design matrix for $$OA(12, 2^{11})$$. However, my experiment has only $$6$$ main effect factors. Similar to the well-known Cast Fatigue experiment, this means I would need to add 5 columns that correspond to the 5 unknown two-level interaction factors such that each of those 5 columns are orthogonal to any other column in the design matrix.

Question: do we have a particular way to pick which of the two-level interaction factors (there are $$6C2 = 15$$ of those factors) for these $$5$$ columns? Also, if we are only interested in finding out the significant main effects, would it be valid to pick the first 6 columns of the design matrix $$OA(12, 2^{11})$$, even though the dimension of $$OA(12, 2^{11})$$ is $$12\times 11$$?

On a related topic to PB design: does anyone happen to know of a concrete example that uses Hamada and Wu's analysis for PB design that you can share with me, particularly if such example has the accompanying code? I would love to see an example that uses R to conduct this analysis, as the theoretical description of their strategy is still abstract to me.

• The interactions will be uniquely determined by the main effect levels, so you don't choose them separately from your choice of columns for main effects. Are you wanting to choose 6 of the 11 columns from the PB-12 run design? Also, what does $\mathrm{OA}(12,2^11)$ mean? Commented Apr 13, 2021 at 6:27
• Is it just a typo of $\mathrm{OA}(12, 2^{11})$? Commented Apr 13, 2021 at 6:35
• @neverKnowsBest Thank you very much for your help. How do we know which interactions would be chosen by the main effect levels for column $7 - 11$ though? And yes, it was a typo. I fixed it. Commented Apr 19, 2021 at 6:41
• There might still be some typos ... Commented Apr 19, 2021 at 12:13

You don't want to pick interactions first because you can choose AB, AC, and BC columns that are impossible (no A, B, and C columns can produce them so you can't run that experiment in the real world). Also, if you have 6 main effects than there are 6 choose 2 (so 15) two-factor interactions.

The canonical approach is to pick the columns for the main effects first and then determine the properties of your estimators (including the estimators for interaction). But even here we have a problem, since the full model with two-factor interactions has 22 terms but you're only collecting data from 12 distinct points, and thus no orthogonal design can exist. But we can choose the least bad design in some sense. Let's continue.

Plackett-Burman designs are nongeometric designs that have partially aliased terms. See section of 8.6.3 of Design and Analysis of Experiments, 9ed (2019) by Montgomery for discussion. Also section 8.4 of Experiments: Planning, Analysis and Optimization, 2ed (2009) by Wu and Hamada is also a good resource. I'm sure Box, Hunter, and Hunter is also a good resource but it's stuck in my garage right now.

For concreteness I'll use some R code to show the design and the resulting properties of the 2 factor interaction estimates. Plackett-Burman designs can be generated with the FrF2 package. As in the comment I'll choose the last 6 columns for my main effects. The design matrix below gives the conditions for each of the 12 runs.

> library(FrF2)
> design <- pb(12)[,6:11]
> print(design)
F  G  H  J  K  L
1   1 -1  1  1 -1  1
2  -1 -1 -1 -1 -1 -1
3  -1 -1  1 -1  1  1
4  -1 -1 -1  1 -1  1
5   1  1 -1 -1 -1  1
6   1 -1 -1 -1  1 -1
7  -1  1  1  1 -1 -1
8   1  1  1 -1 -1 -1
9   1 -1  1  1  1 -1
10 -1  1  1 -1  1  1
11  1  1 -1  1  1  1
12 -1  1 -1  1  1 -1


We can build the model matrix as follows

X <- model.matrix(
~ .:.,            # This tells R to look at all main effects and 2 factor interactions
data=design
)


There are 462 possible choices of 6 columns from the 11 columns in the PB design. There are many ways to look at these designs in order to pick one. Let's look at "$$D$$-optimality" which seeks to minimize the determinant of the covariance matrix for the parameter estimates of the given design.

The determinant of the covariance matrix has the property that: $$\left|\left(\mathbf{X}^\mathrm{T} \mathbf{X}\right)^{-1}\right| = \frac{1}{ \left|\mathbf{X}^\mathrm{T} \mathbf{X}\right| }$$ so a reasonable choice of columns are those that maximize $$\left|\mathbf{X}^\mathrm{T} \mathbf{X}\right|$$ and hence minimize the determinant of the covariance matrix.

In R I can simply enumerate these and choose the best ones.

 design <- pb(12)
columns <- combn(1:11, 6)
d.criteria <- apply(
columns,            # Look at all 462 combinations of 6 columns
2,
function(ix){
X <- model.matrix(
~ .:.,
data=design[,ix]
)
return(det(t(X) %*% X))
}
)


and then select the best (there are 12 of them)

> ix <- which(d.criteria == max(d.criteria))
> good.columns <- columns[,ix]
> print(good.columns)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
[1,]    1    1    1    1    1    1    1    2    2     2     2     3
[2,]    2    2    2    3    3    4    5    3    4     4     5     5
[3,]    3    4    4    4    5    5    7    5    5     6     6     6
[4,]    5    5    8    6    7    6    8    6    7     8     7     8
[5,]    6    7   10    8    8    7    9    8    9     9     8    10
[6,]   11   10   11    9   10   10   11   11   10    11    11    11


One such good design is to choose columns 1, 2, 3, 5, 6, and 11. This design is about 21% more efficient than just choosing the last 6 columns (in terms of $$D$$-criterion).

In the above cited books I'd suggest reading about aliasing and the Alias matrix so you can see how the design will affect the expected value of your estimators. My advice is to read at least one of the above sources and play around - hopefully the R code can help with that.