Plackett-Burman Extreme Screening Experiment

June 24, 2009 by Gregg Larson

The Plackett-Burman is an Extreme Screening Experiment in which the researcher wants to screen out the critical main effects from the significant or trivial effects.  This design is very effective if time and cost are an issue, if each main effect is equally probable of affecting the criterion measures, and the main effects have either slight or no interactions . The Plackett-Burman Design’s strengths are that it is relatively easy to construct as well as being easy to analyze.  It also allows for the study of K number of factors in only K+1 runs.  This means that the seven factors in this experiment can be tested in eight runs. The weakness of the design is that if there are significant interactions, they will be inextricably confounded with the main effects.  Lastly, all factors to be studied must be restricted to only two levels. There are two major categories for Plackett-Burman designs: Geometric and non-geometric.  Geometric designs are based on the requirement that the number of runs must be a power of two .  This result is a set of designs that consist of runs in lengths of eight, sixteen, or thirty-two.  The geometric designs can be constructed with a useful alias structure. Non-geometric designs are based on a number of runs that are a multiple of four .  The resulting set of designs consists of runs in lengths of 12, 20, 24, and 28.  The drawback with non-geometric designs is that the alias structures are too complex to indentify within a standard Alias Structure, in that the confounded interactions are approximately distributed across the entire array. The construction of a Plackett-Burman design starts by selecting a standard pattern based on the number of factors you are studying.  Table 1 lists common patterns for both geometric and non-geometric run counts.  Each row represents the first run of the design.  The subsequent runs are constructed by performing a cyclical shift on the previous row.  This means moving the signs one place to the right and shifting the final sign to the first sign.  This is repeated for each of the runs in the design.  Table 2 shows the factorial design used to determine the critical factors. Factors    Pattern    PB -Design N=8    + + + − + − −    Geometric N=12    + + − + + + − − − + −    Non-geometric N=16    + + + + − + − + + − − + − − −    Geometric N=20    + + − − + + + + − + − + − − − − + + −     Non-geometric N = 24    + + + + + − + − + + − − + + − − + − + − − − −     Non-geometric N = 32     − − − − + − + − + + + − + + − − − + + + + + − + + − + − − +     Geometric Table 1: Plackett-Burman Patterns Run    A    B    C    D    E    F    G 1    +    +    +    −    +    −    − 2    −    +    +    +    −    +    − 3    −    −    +    +    +    −    + 4    +    −    −    +    +    +    − 5    −    +    −    −    +    +    + 6    +    −    +    −    −    +    + 7    +    +    −    +    −    −    + 8    −    −    −    −    −    −    − Table 2: Plackett-Burman N=8 design Being a Resolution III design, the eight run Plackett-Burman design will have an alias structure made up of the main effects confounded with the two-way interactions.  The Alias Structure for this design shown in Table 2 is listed in Table 3: Column Effect    Confounded Two Way Interactions A    BF    CD    EG B    AF    CG    DE C    AD    BG    EF D    AC    BE    FG E    AG    BD    CF F    AB    CE    DG G    AE    BC    DF Table 3: Alias structure for Plackett-Burman N=8 One of the unique features of geometric Plackett-Burman designs is that they provide the researcher with the ability to “Fold-Over” the design that allows us to “de-alias” the effects studied.  What this means is that by creating a mirror image of the original design, and performing two independent tests each with eight samples, you can understand the relative importance of the main effects without confounding the two-way interactions.  This technique was used in my experiments, and the “mirror image” design is shown in Table 4. Comparing the signs in the array to the previous N=8 design shown, it can be easily seen that the number of runs and columns are identical, but all of the signs have been reversed. Run    A    B    C    D    E    F    G 1    −    −    −    +    −    +    + 2    +    −    −    −    +    −    + 3    +    +    −    −    −    +    − 4    −    +    +    −    −    −    + 5    +    −    +    +    −    −    − 6    −    +    −    +    +    −    − 7    −    −    +    −    +    +    − 8    +    +    +    +    +    +    +

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