8.4 - Plackett-Burman Designs

Download: FF2LevelCorr.mpx

The first is a fractional factorial, 4 factor design, Resolution IV with one generator ABCD or D = ABC. From this design, we get an alias structure that we are familiar with. Main effects are aliased with 3-way interactions which means that they are completely confounded with those factors. Two-way interactions are confounded with each other.

Let's look at the correlation among these factors, A, B, C and D, and then a couple of interaction columns.

This is just a simple Pearson correlation. What Minitab gives us is the coefficient and the p-value. We can ignore the p-values because we are not really interested in testing, however a correlation between A and B = 0, A and C = 0, A and D = 0, etc. The correlation between all these factors = 0 because of the orthogonality.

Look back up at the alias structure and you will see the D is confounded with ABC. As we look back at the correlation table the correlation between D and ABC = 1. The correlation between two factors that are confounded = 1. This is appropriate because they are completely correlated with each other. Therefore, in these \(2^{k-p}\) designs we can see through correlation that factors are either orthogonal (correlation = 0) or they are completely confounded (correlation = 1).

Next, let's look at the Plackett-Burman designs and see how this differs. Below, we have created a design for 9 factors, 12 runs and we are looking at the correlation among the main effects, A, B, C, D, and E.

These are the main factors themselves already set in orthogonal columns so these correlations = 0. If we look at the next design, however, in this case, we have the 12 runs and then we have created new 2-way interactions through multiplication of the factors already determined. Again, ignoring the p-values we produced a correlation matrix, (partially displayed below).

A is orthogonal to every other factor, the correlation value = 0. B is not correlated with all the other main effects, where correlation = 0, but with some of these two-way interactions the correlation = 0.333. This shows partial confounding with the two-way interaction. Likewise, C has partial confounding with AB and AD. D is partially confounded with AB and AC. F is partially confounded with AB and AC and AD, ... and so forth.

Plackett-Burman designs have partial confounding, not complete confounding, with the 2-way and 3-way and higher interactions. Although they have this property that some effects are orthogonal they do not have the same structure allowing complete or orthogonal correlation with the other two way and higher order interactions.

Like other Resolution II designs, these designs are also good for screening for important factors. But remember, in a Resolution II design a main effect might look important because some combination of interactions is important and the main effect itself might not be the important effect.

If you assume that your interactions = 0 or are not important these are great designs. If your assumption is wrong and there are interactions, then it could show up as influencing one or the other main effects. These designs are very efficient with small numbers of observations and useful but remember the caveat, you are assuming that the main effects are going to show up as larger effects than interactions so that they will dominate the interaction effects.

Using Minitab we can ask for up to 47 factors. In doing so you want to select a sufficient number of runs over the number of factors so that you have a reasonable number of degrees of freedom for error. At this stage a statistical test really isn't that important, you are just screening for a place to start.