In Design of Experiments DOE, Plackett-Burman design is one of the screening designs which is used to detect the most significant factors out of large number of factors. The idea behind this design is to make number of level combinations of any two factors is the same through the whole design. To achieve this criteria, massive iterations are required; for this reason statisticians decided to calculate this design offline and save it instead of calculating it each time.
Plackett-Burman design is based on 2-level factors and has a resolution of III.
To construct Plackett-Burman design we first calculate number of runs. Number of runs is a multiplier of 4: e.g. 4,8,12,16,.... where number of runs should be larger than number of factors. For example, 9 factors will require 12-runs Plackett-Burman design. Simply, number of runs of this design can be calculated by the simple following equation:
Number of runs=ceiling((Number of factors+1)/4)*4
Since the treatments of this design are pre-calculated, there are two ways to get these treatments. In this article we will go through an example of 12-runs Plackett-Burman design.
The first method is to use what is called the "generating vector". The generating vector is a pre-calculated vector of all levels of the first factor (factor A) except for the last level. For example, the generating vector of 12-runs design will has the length of 11. In this method, treatments will be created by writing all levels of each factor column one by one.
To create treatments, we start with the first factor and assign the following vector to its levels:
++-+++---+-
Where "+" denotes the high level and "-" denotes the low level. Reading this vector from left to right represent factor levels from top to bottom in the design matrix.
After that, each vector of each factor will be derived from the previous vector. For example, vector B will be derived from vector A, vector C will be derived from vector B and so on. Simply, the last element of vector A will be the first element of vector B. This is better imagined by assuming that the generating vector forms a closed circle and each factor vector will start with the last value of the previous vector. The following image can explain how this may look like.
At the end, after generating all vectors, the last level of each factor will be set to low level "-". Or in other words, the last treatment in the table will be with all factors set at low level.
The following .GIF animation shows how the generation of each vector is derived from the previous vector.
Alternatively, the second method, some books prefer to give the entire table of factor levels for all treatments. Programming wise, the single generating vector has the advantage of small memory consumption and the disadvantage of extra processing time. The following table shows all treatments for 11 factors. It is very noticeable how diagonal the matrix looks like because of shifting explained before.
References:
https://www.itl.nist.gov/div898/handbook/pri/section3/pri335.htm
https://en.wikipedia.org/wiki/Plackett%E2%80%93Burman_design
Key words:
Plackett-Burman design
Minitab Plackett-Burman design
Plackett-Burman design matrix
Plackett-Burman design pattern
Plackett-Burman design table
Plackett-Burman design algorithm
Plackett-Burman design step by step
Plackett-Burman design is based on 2-level factors and has a resolution of III.
To construct Plackett-Burman design we first calculate number of runs. Number of runs is a multiplier of 4: e.g. 4,8,12,16,.... where number of runs should be larger than number of factors. For example, 9 factors will require 12-runs Plackett-Burman design. Simply, number of runs of this design can be calculated by the simple following equation:
Number of runs=ceiling((Number of factors+1)/4)*4
Since the treatments of this design are pre-calculated, there are two ways to get these treatments. In this article we will go through an example of 12-runs Plackett-Burman design.
The first method is to use what is called the "generating vector". The generating vector is a pre-calculated vector of all levels of the first factor (factor A) except for the last level. For example, the generating vector of 12-runs design will has the length of 11. In this method, treatments will be created by writing all levels of each factor column one by one.
To create treatments, we start with the first factor and assign the following vector to its levels:
++-+++---+-
Where "+" denotes the high level and "-" denotes the low level. Reading this vector from left to right represent factor levels from top to bottom in the design matrix.
After that, each vector of each factor will be derived from the previous vector. For example, vector B will be derived from vector A, vector C will be derived from vector B and so on. Simply, the last element of vector A will be the first element of vector B. This is better imagined by assuming that the generating vector forms a closed circle and each factor vector will start with the last value of the previous vector. The following image can explain how this may look like.
At the end, after generating all vectors, the last level of each factor will be set to low level "-". Or in other words, the last treatment in the table will be with all factors set at low level.
The following .GIF animation shows how the generation of each vector is derived from the previous vector.
Alternatively, the second method, some books prefer to give the entire table of factor levels for all treatments. Programming wise, the single generating vector has the advantage of small memory consumption and the disadvantage of extra processing time. The following table shows all treatments for 11 factors. It is very noticeable how diagonal the matrix looks like because of shifting explained before.
References:
https://www.itl.nist.gov/div898/handbook/pri/section3/pri335.htm
https://en.wikipedia.org/wiki/Plackett%E2%80%93Burman_design
Key words:
Plackett-Burman design
Minitab Plackett-Burman design
Plackett-Burman design matrix
Plackett-Burman design pattern
Plackett-Burman design table
Plackett-Burman design algorithm
Plackett-Burman design step by step
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