# When and How to Use Plackett-Burman Experimental Design

Plackett-Burman experimental design is used to identify the most important factors early in the experimentation phase when complete knowledge about the system is usually unavailable. Developed in 1946 by statisticians Robin L. Plackett and J.P. Burman, it is an efficient screening method to identify the active factors using as few experimental runs as possible.

In Plackett-Burman designs, main effects have a complicated confounding relationship with two-factor interactions. Therefore, these designs should be used to study main effects when it can be assumed that two-way interactions are negligible.

In practical use, two-level full or fractional factorial designs, and Plackett-Burman designs are often used to screen for the important factors that influence process output measures or product quality. These designs are useful for fitting first-order models (which detect linear effects) and can provide information on the existence of second-order effects (curvature) when the design includes center points.

### Full Factorial Experiment

To demonstrate the effectiveness of Plackett-Burman design, an experiment was conducted to compare a full factorial experiment with a Plackett-Burman design. Let’s start with the full factorial experiment, which consists of five factors with two levels for each factor.

In this case the total number of experiments is 25 = 32 (Table 1). The characteristic of the response, Y, is the larger the better.

 Table 1: Five Factor Analyses with Two Levels (Full Factorial Design) A B C D E Y -1 -1 -1 -1 -1 75.87 1 -1 -1 -1 -1 76.14 -1 1 -1 -1 -1 109.95 1 1 -1 -1 -1 109.55 -1 -1 1 -1 -1 80.17 1 -1 1 -1 -1 80.3 -1 1 1 -1 -1 114.07 1 1 1 -1 -1 114.05 -1 -1 -1 1 -1 68.77 1 -1 -1 1 -1 68.63 -1 1 -1 1 -1 102.41 1 1 -1 1 -1 102.27 -1 -1 1 1 -1 72.95 1 -1 1 1 -1 72.68 -1 1 1 1 -1 106.98 1 1 1 1 -1 106.65 -1 -1 -1 -1 1 75.79 1 -1 -1 -1 1 75.71 -1 1 -1 -1 1 110.08 1 1 -1 -1 1 110.04 -1 -1 1 -1 1 80.49 1 -1 1 -1 1 80.86 -1 1 1 -1 1 113.49 1 1 1 -1 1 114.06 -1 -1 -1 1 1 68.22 1 -1 -1 1 1 68.6 -1 1 -1 1 1 102.5 1 1 -1 1 1 102.3 -1 -1 1 1 1 73.03 1 -1 1 1 1 73.3 -1 1 1 1 1 106.08 1 1 1 1 1 106.7
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Normal Plot

The primary goal of screening designs is to identify the vital few factors or key variables that influence the response. A normal plot is one of the graphs that help identify these influential factors.

In the normal probability plot of the effects, points that do not fall near the line have measured values that are significantly beyond the observed variation, and usually signal important effects. Important effects are larger and generally further from the fitted line than unimportant effects. Unimportant effects tend to be smaller and centered around zero.

Figure 1: Normal Plot of the Effects (Full Factorial Design)

Figure 2: Analysis of Variance (Full Factorial Design)

As shown in the normal plot (Figure 2) and the analysis of variance (ANOVA, Figure 3), the factors B, C and D are significant to the response. Additionally, there are two-way interactions between factors B and C and three-way interactions among B, C and E.

Main Effects Plot

In experimental design, a main effects plot is used in conjunction with ANOVA to examine differences among level means for one or more factors. It graphs the response mean for each factor level connected by a line. A main effect is present when different levels of a factor affect the response differently (shown as a slope on a two-level plot).

Some general patterns to look for with main effects plots include the following:

• When the line is horizontal (parallel to the x-axis), then there is no main effect present. Each level of the factor affects the response in the same way, and the response mean is the same across all factor levels.
• When the line is not horizontal, then a significant main effect may be present. Different levels of the factor affect the response differently. The greater the slope, the greater the likelihood that a main effect is statistically significant.

Figure 3: Main Effects Plot for Y (Full Factorial Design)

The factors B, C and D are shown to be significant to the response Y as shown in Figure 3.

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### Response Optimization

The Response Optimizer function in Minitab helps to identify the combination of input variable settings that jointly optimize a single response or a set of responses. (Note: Minitab requires the user create a starting point for the optimization and will find the first best solution based on the requested optimization objective.) This function provides an optimal solution for the input variable combinations and an optimization plot. The optimization plot is interactive – input variable settings on the plot can be adjusted to search for more desirable solutions.

Figure 4: Optimal Solution Using Response Optimizer (Full Factorial Design) (where D = optimal response)

The nature of the response, Y, is “the larger the better” – and the maximum value that can be achieved in this experiment is 114.07, as shown in Figure 4.

That means to achieve the maximum value of Y, the process variables should be set as shown in Table 2.

 Table 2: Levels of Process Variables (Full Factorial Design) Variable Optimal Level Notes A -1 A is insignificant and might not create high impact if it is changed to +1 B +1 C +1 D -1 E -1 E is insignificant and might not create high impact if it is changed to +1

### Plackett-Burman Experiment

Having completed the two-level full factorial experiment, let’s turn to the Plackett-Burman experiment for comparison. In this experiment five factors with two levels are considered. The total number of experiments is selected to be the minimum from the Plackett-Burman design – 12 experiments (Table 3). Remember that for Y, the larger the better.

 Table 3: Five Factor Analysis with Two Levels (Plackett-Burman Design) A B C D E Y 1 -1 1 -1 -1 75.67 1 1 -1 1 -1 102.4 -1 1 1 -1 1 113.71 1 -1 1 1 -1 72.95 1 1 -1 1 1 102.32 1 1 1 -1 1 113.85 -1 1 1 1 -1 106.51 -1 -1 1 1 1 72.66 -1 -1 -1 1 1 68.62 1 -1 -1 -1 1 75.54 -1 1 -1 -1 -1 109.28 -1 -1 -1 -1 -1 75.69

Figure 5: Normal Plot of Effects (Plackett-Burman Design)

Figure 6: Analysis of Variance (Plackett-Burman Design)

As shown in the normal plot in Figure 5, factors B and D are significant. As shown in Figure 6 with ANOVA, factors B, C and D are significant to the response Y. It is important to note that the significance of interactions is not displayed as Plackett-Burman design is interested only in main effects.

The main effects plot (Figure 7) shows that the factors B, C and D are significant to the response Y.

Figure 7: Main Effects Plot for Y (Plackett-Burman Design)

Figure 8: Optimal Solution Using Response Optimizer (Plackett-Burman Design) (where D = optimal desirability)

The nature of the response, Y, is the larger the better, and the maximum value that can be achieved in this experiment, according to the Response Optimizer (Figure 8), is 116.46.

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Thus, to achieve the maximum value of Y, the process variables should be set as shown in Table 4.

 Table 4: Levels of Process Variables (Plackett-Burman Design) Variable Optimal Level Notes A -1 A is insignificant and might not create high impact if it is changed to +1 B +1 C +1 D -1 E -1 E is insignificant and might not create high impact if it is changed to +1

### Comparison of the DOE Results

A comparison of the results obtained from the both the full factorial design of experiments and the Plackett-Burnam design of experiments are shown in Table 5.

 Table 5: Comparison of Results Full Factorial Design Plackett-Burman Design Number of experiments 32 12 Significant factors B, C, D B, C, D Significant levels B: +1, C: +1, D: -1 B: +1, C: +1, D: -1 Significant interactions (B, C) and (B, C, E) Non-detectable Optimized value of Y 114.07 116.46

There is no difference in significant factors or significant levels, but there is a slight difference in the optimized value of Y. The factor settings and conclusion remains the same when using either design, but there is a significant difference in the number of experiments that need to be conducted to achieve these results.

When to Use Plackett-Burman Design

It is particularly helpful to use Plackett-Burman design:

• In screening
• When neglecting higher order interactions is possible
• In two-level multi-factor experiments
• When there are more than four factors (if there are between two to four variables, a full factorial can be performed)
• To economically detect large main effects
• For N = 12, 20, 24, 28 and 36 (where N = the number of experiments)

Drawbacks of Plackett-Burman Design

There are also some drawbacks to using Plackett-Burman designs that practitioners should be aware of:

• They do not verify if the effect of one factor depends on another factor.
• If you run the smallest design you can, it does not follow that enough data has been collected to know what those effects are precisely.

### Conclusion

Plackett-Burman design is helpful if complete knowledge about the system is unavailable or in the case of screening with a higher number of factors. But once the significant factors are available and the interactions between the factors are required, it is better to go with full factorial design as it takes the combinations of all the levels between the factors and provides the interaction details.

##### Manikandan Jayakumar

1. Mike Carnell

It is common knowledge that you use a fractional design to screen factors prior to running full factorial designs. Is the point you are trying to make that Plackett-Burman designs are better suited for screening than other fractional factorial designs?

2. Charlie Carpenter

It is good to see that someone is trying the Plackett-Burman Fractional Factorials. You unfortunately were restricted by Minitab not allowing you to use the 8 Run design which would have been even more efficient. MTB removed the 8 run design when they released version 14. The only concern with the use of the PB designs is the fact that you cannot study interactions. These effects have been evenly distributed across all columns by the nature of the design. Generally, if you see a large unexplained, also know as error, you can plot the effects to see if any interactions exist. Then you would have to use a Yates Algorithm fractional factorial design. So the rule of thumb for the use of PD designs is if you don’t think that any of the factors will have interaction effects, then go for it. I have found these designs to be of great value over my 28 years of applications of DOE on hundreds of products, processes, and also promotional marketing effectiveness. Here is the string for the 8 run design, column #1 + + + – + – – the last row is always -. Then you take the value for row 7 in column 1 and place it as the first value in column 2 and then put in the rest of the string starting with row 1 of column 1. Keep rotating through the layout to complete the design. You will see how the pattern evolves. The 8 and 12 run PBs are very efficient, but beware of interaction effects. I successfully used a 24 Run PB with 23 Factors as we tracked the production of tires from the Banbury Mixing rubber, extrusion, 1st and 2nd stage tire building, and then curing. Saved well over 100k in cost! I teach and practically apply DOE methods.

3. Chris Seider

I would have loved to see a discussion of why you should use a PB instead of a fractional factorial design. I personally use fractional factorials with success and haven’t understood why some love PB.

As Mike stated well, I’d take a resolution V design with 16 experimental runs and get an idea about main effects separately from the 2-way interactions which a 12 experimental PB design can’t do.

Curious for a response to Mike Carnell’s question. Why Plackett – Burman design? If it is better than fractional factorial DOE – then how?

5. mahdi elahi

dear sir:
do you have any example of Plackett-Burman Design in SAS.
I want tr analyze the dfata with SAS commands.
thanks

6. Chuck

If interactions are to be ignored, I can run a One Factor At a Time (OFAT) on 5 factors in 10 runs (+/- for each of the 5 factors). PB ignores interactions with 12 runs, whats the point?

The whole point of a DOE is to study interactions, which is why I prefer resolution V DOEs. If interactions are unimportant, there is no need for a DOE.

If PB can actually be done in 8 runs, as mentioned below by Charlie, that would beat my 10 run OFAT. I might be interested in that.

• Lakisrit

It’s not only about interactions: the prediction variance is minimal with an orthogonal array of n runs. Now, you can go for near-orthogonal experiments using a custom doe algorithm if the number of run doesn’t suit you. OFAT is not optimal, you will loose information with OFAT.

7. varsha tamil

Hello,
I used plackett Burman for media optimization, and upon the analysis of data with minitab 16, i found out that partial confounding is present. What do i conclude with that ? Please help.

8. SKS

In hplc if i have optimised method through experimental design, say by central composite design, is it advisable to check roubstness through say, plackette burman? Or we can defend the roubstness through data from CCD itself?