Plackett-Burman design
Encyclopedia
Plackett–Burman designs are experimental designs presented in 1946 by Robin L. Plackett and J. P. Burman while working in the British Ministry of Supply
.
Their goal was to find experimental designs for investigating the dependence of some measured quantity on a number of independent variable
s (factors), each taking L levels, in such a way as to minimize the variance
of the estimates of these dependencies using a limited number of experiments. Interactions between the factors were considered negligible. The solution to this problem is to find an experimental design where each combination of levels for any pair of factors appears the same number of times, throughout all the experimental runs (refer table). A complete factorial design would satisfy this criterion, but the idea was to find smaller designs.
For the case of two levels (L=2), Plackett and Burman used the method found in 1933 by Raymond Paley
for generating orthogonal matrices
whose elements are all either 1 or -1 (Hadamard matrices
). Paley's method could be used to find such matrices of size N for most N equal to a multiple of 4. In particular, it worked for all such N up to 100 except N = 92. If N is a power of 2, however, the resulting design is identical to a fractional factorial design, so Plackett–Burman designs are mostly used when N is a multiple of 4 but not a power of 2 (i.e. N = 12, 20, 24, 28, 36 …). If one is trying to estimate less than N parameters (including the overall average), then one simply uses a subset of the columns of the matrix.
For the case of more than two levels, Plackett and Burman rediscovered designs that had previously been given by Raj Chandra Bose
and K. Kishen at the Indian Statistical Institute
.
Plackett and Burman give specifics for designs having a number of experiments equal to the number of levels L to some integer power, for L = 3, 4, 5, or 7.
When interactions between factors are not negligible, they are often confounded in Plackett–Burman designs with the main effects, meaning that the designs do not permit one to distinguish between certain main effects and certain interactions. This is called aliasing or confounding
.
Ministry of Supply
The Ministry of Supply was a department of the UK Government formed in 1939 to co-ordinate the supply of equipment to all three British armed forces, headed by the Minister of Supply. There was, however, a separate ministry responsible for aircraft production and the Admiralty retained...
.
Their goal was to find experimental designs for investigating the dependence of some measured quantity on a number of independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
s (factors), each taking L levels, in such a way as to minimize the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of the estimates of these dependencies using a limited number of experiments. Interactions between the factors were considered negligible. The solution to this problem is to find an experimental design where each combination of levels for any pair of factors appears the same number of times, throughout all the experimental runs (refer table). A complete factorial design would satisfy this criterion, but the idea was to find smaller designs.
| Run | X1 | X2 | X3 | X4 | X5 | X6 | X7 | X8 | X9 | X10 | X11 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | |||||||||||
| 2 | − | − | − | − | − | − | |||||
| 3 | − | − | − | − | − | − | |||||
| 4 | − | − | − | − | − | − | |||||
| 5 | − | − | − | − | − | − | |||||
| 6 | − | − | − | − | − | − | |||||
| 7 | − | − | − | − | − | − | |||||
| 8 | − | − | − | − | − | − | |||||
| 9 | − | − | − | − | − | − | |||||
| 10 | − | − | − | − | − | − | |||||
| 11 | − | − | − | − | − | − | |||||
| 12 | − | − | − | − | − | − |
For the case of two levels (L=2), Plackett and Burman used the method found in 1933 by Raymond Paley
Raymond Paley
Raymond Edward Alan Christopher Paley was an English mathematician. Paley was born in Bournemouth, England. He was educated at Eton. From there he entered Trinity College, Cambridge where he showed himself the most brilliant student among a remarkable collection of fellow undergraduates...
for generating orthogonal matrices
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
whose elements are all either 1 or -1 (Hadamard matrices
Hadamard matrix
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal...
). Paley's method could be used to find such matrices of size N for most N equal to a multiple of 4. In particular, it worked for all such N up to 100 except N = 92. If N is a power of 2, however, the resulting design is identical to a fractional factorial design, so Plackett–Burman designs are mostly used when N is a multiple of 4 but not a power of 2 (i.e. N = 12, 20, 24, 28, 36 …). If one is trying to estimate less than N parameters (including the overall average), then one simply uses a subset of the columns of the matrix.
For the case of more than two levels, Plackett and Burman rediscovered designs that had previously been given by Raj Chandra Bose
Raj Chandra Bose
Raj Chandra Bose was an Indian mathematician and statistician best known for his work in design theory and the theory of error-correcting codes in which the class of BCH codes is partly named after him. He was notable for his work along with S. S. Shrikhande and E. T...
and K. Kishen at the Indian Statistical Institute
Indian Statistical Institute
Indian Statistical Institute is a public research institute and university in Kolkata's northern outskirt of Baranagar, India founded by Prasanta Chandra Mahalanobis in 1931...
.
Plackett and Burman give specifics for designs having a number of experiments equal to the number of levels L to some integer power, for L = 3, 4, 5, or 7.
When interactions between factors are not negligible, they are often confounded in Plackett–Burman designs with the main effects, meaning that the designs do not permit one to distinguish between certain main effects and certain interactions. This is called aliasing or confounding
Confounding
In statistics, a confounding variable is an extraneous variable in a statistical model that correlates with both the dependent variable and the independent variable...
.