Placket–Burman Designs

Plackett–Burman (PB) designs are a class of fractional factorial designs first developed by two mathematicians/statisticians: R.L. Plackett and J.P. Burman, at the University of Newcastle in northeast England in 1946 (1). Looking back to an original article published by Plackett and Burman, the authors note:

Lynn D. Torbeck

"A problem that often occurs in the design of an experiment in physical or industrial research is that of determining suitable tolerances for the components of a certain assembly; more generally of ascertaining the effect of quantitative or qualitative alterations in the various components upon some measured characteristic of the complete assembly. It is sometimes possible to calculate what this effect should be; but it is to the more general case when this is not so that the methods given below apply" (1).

This statement is interesting for several reasons. First, there was little published in 1946 using experimentation for industrial research. The researchers were ahead of their time. Second, they were studying the tolerances for a mechanical assembly. Most experimentation at the time was for crops and livestock. Third, they were empirically determining tolerance stack up in the mechanical assembly rather than trying to calculate it. Lastly, they published this industry oriented paper in a biology journal. Needless to say, the paper didn't get a lot of attention at the time.

The most popular text for designed experiments at the time didn't include PB designs (2). Even the most famous of books on experimentation, by Box, Hunter and Hunter, devoted less than a page to the topic and then only noted the statistical characteristics without any suggestions for application (3). (The latter edition in 2005 included a chapter on PB.)

PB designs can be at two, three, five, and seven levels. In practice, the two and three levels are primarily used. The numbers of factors to be evaluated are one less than the number of runs or trials in the study. Designs do not exist for all number of runs. The original paper published 8, 12, 16, 20, 24 ... 96 and 100 runs. Thus, it is possible to study 7 factors in 8 runs, 11 factors in 12 runs, or even 99 factors in 100 runs.

One could make the case that PB designs are the step-child of designed experiments. They are recognized as being an excellent way to study many factors in very few runs or experiments. But, historically, statisticians are taught to find and estimate the main effects and the two-factor interaction in a given experiment. PB designs, however, are sparse (an advantage), and confound the main effects and two-factor interactions (a disadvantage), such that neither can be estimated separately. At which point, some say, why do this?

PB designs were originally proposed for testing tolerances of simple mechanical assemblies. Using these designs, many combinations could be studied in a relatively few number of runs or trials. This mechanical situation did not have any interactions between parts to be considered. However, once published, other industrial applications were found.

PB designs did not enjoy a lot of popularity until the early 1980s when a new emphasis on quality swept the nation under the term "Total Quality Management" (TQM). As part of that movement, PB designs were recognized as a way to screen many factors in a few number of trials. Some statisticians were still bothered by the inability to study the interactions but, pragmatically, it was a reasonable tradeoff.

In recent years, the designs have been used in the early stages of quality by design to define design space, as indicated in the International Conference on Harmonization's Q8 Pharmaceutical Development guideline. It is also reasonable to use PB designs for these assumptions and situations:

• When there are a large number of factors to screen to eliminate unimportant ones. This use assumes that factors that seem to be important are due to a large main effect and not interactions.

• When it is reasonable to assume that most or all of the two factor interactions do not exist or are small.

• When one wants to demonstrate ruggedness or robustness of equipment or processes (4). This use assumes that the factors used are not critical and the study is to demonstrate that.

• When one wants to find a set of conditions that performs better than those being used and when empirical relationships are not a concern.

Additional reading

See Chap. 7, "Statistics for Experimenters," for an introduction to PB applications (5).

Lynn D. Torbeck is a statistician at PharmStat Consulting, 2000 Dempster, Evanston, IL 60202, tel. 847.424.1314, LDTorbeck@PharmStat.com, www.PharmStat.com.

References

1. R.L.Plackett and J.P. Burman, Biometrika 33, 305–325 (1946).

2. W.G. Cochran and G.M. Cox, Experimental Designs (Wiley, New York, 1950).

3. G.E. Box, W.G.Hunter, and J.S. Hunter, Statistics for Experimenters (Wiley, New York, 1978).

4. L.D. Torbeck, Pharm. Technol. 21 (3) 1996.

5. G.E. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters, 2nd edition (Wiley, New York, 2005).