Plackett–Burman design
(Redirected fromPlackett-Burman design)
Plackett–Burman designsareexperimental designspresented in 1946 byRobin L. PlackettandJ. P. Burmanwhile working in the BritishMinistry of Supply.[1] Their goal was to find experimental designs for investigating the dependence of some measured quantity on a number ofindependent variables(factors), each takingLlevels, in such a way as to minimize thevarianceof 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 whereeach combinationof levels for anypairof factors appears thesame number of times,throughout all the experimental runs (refer to table). A completefactorial designwould 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 themethodfound in 1933 byRaymond Paleyfor generatingorthogonal matriceswhose elements are all either 1 or −1 (Hadamard matrices). Paley's method could be used to find such matrices of sizeNfor mostNequal to a multiple of 4. In particular, it worked for all suchNup to 100 exceptN= 92. IfNis a power of 2, however, the resulting design is identical to afractional factorial design,so Plackett–Burman designs are mostly used whenNis a multiple of 4 but not a power of 2 (i.e.N= 12, 20, 24, 28, 36…).[3]If one is trying to estimate less thanNparameters (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 byRaj Chandra BoseandK. Kishenat theIndian Statistical Institute.[4] Plackett and Burman give specifics for designs having a number of experiments equal to the number of levelsLto some integer power, forL= 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 calledaliasingorconfounding.
Extended uses[edit]
In 1993, Dennis Lin described a construction method via half-fractions of Plackett–Burman designs, using one column to take half of the rest of the columns.[5]The resulting matrix, minus that column, is a "supersaturated design"[6]for finding significant first order effects, under the assumption that few exist.
Box–Behnkendesigns can be made smaller, or very large ones constructed, by replacing thefractional factorialsandincomplete blockstraditionally used for plan and seed matrices, respectively, with Plackett–Burmans. For example, a quadratic design for 30 variables requires a 30 column PB plan matrix of zeroes and ones, replacing the ones in each line using PB seed matrices of −1s and +1s (for 15 or 16 variables) wherever a one appears in the plan matrix, creating a 557 runs design with values, −1, 0, +1, to estimate the 496 parameters of a full quadratic model. Addingaxial pointsallows estimating univariate cubic and quartic effects.
By equivocating certain columns with parameters to be estimated, Plackett–Burmans can also be used to construct mixed categorical and numerical designs, with interactions or high order effects, requiring no more than 4 runs more than the number of model parameters to be estimated. Sort bya-1columns assigned to categorical variableAand following columns, whereA= 1 + int(a·i/(max(i) + 0.00001)),i= row number anda=A'snumber of values. Next sort on columns assigned to any other categorical variables and following columns, repeating as needed. Such designs, if large, may otherwise be incomputable by standard search techniques likeD-optimality.For example, 13 variables averaging 3 values each could have well over a million combinations to search. To estimate the 105 parameters in a quadratic model of 13 variables, one must formally exclude from consideration or compute |X'X| for well over 106C102,i.e. 313C105, or roughly 10484matrices.
4 to 48 runs, sorted to show half-fractions[edit]
P.B.4 + + + + – – – + – – – + P.B.8 + + + + + + + + + – – – – + + – + + – – – + – – – + + – – + + – + – – – + – + – + – – – + – – + + – – – + + – + P.B.12 + + + + + + + + + + + + + + + – – – + – – – + + – – – + – – + – + + – + – + + + – – – – + – – + – – + – + + – + – – – + – – + – + + – + + – – – + – – + + – + – + + + – – – + – – + – – + – + + + – – – – + + + – – – + – + – – + – – + – + + + – – – – + – + + + – – + P.B.16 + + + + + + + + + + + + + + + + + + – – – – – – – – + + + + + + – + – – – – + + + – – – + + + – – + + + + – – – – – – + + – + + + – – + + – – + – – – + – + – – + + – – + + + – – – + – – + – + + – + – – – + + – + – – – + – – + – + + – + + – – + + + – + – + – + – – + – – – + + – + – + – + – + – + – – – + – + + – + – – + – + – + – – + – – – + – + + – + + – + – – – + + – – + + – – + – – + + – – + – + + – – + + – – – + + – – – + + + – – – – + + + – + – – – – – – + + + + – + + – + P.B.20 + + + + + + + + + + + + + + + + + + + + + + – + – + – – – – + + – – + – – + + + – + – + 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– + + – + – – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + + – + – – + – + + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + + + – – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + – + – – + + + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + + – + – – + – + + + – – + – + + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + + – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + – + – – + + – – + + + – – – + + + + + + + + – – + + – – + + – – – – – – – – + + + + – – – – + + – – – – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – + + + + – – – – – – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – – – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – P.B.44 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + – + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – – + + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – – + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + – + + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – – + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + – + + + + 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+ + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – + – + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + + – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + + – + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – + – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + + – + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + + – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + + – – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + + – – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – + – – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – + – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – + – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – +
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This article incorporatespublic domain materialfrom theNational Institute of Standards and Technology
| Scientific method | |
|---|---|
| Treatment andblocking | |
| Models andinference | |
| Designs Completely randomized | |
