Plackett–Burman designs are

experimental design The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...

s presented in 1946 by Robin L. Plackett and J. P. Burman while working in the British

Ministry of Supply The Ministry of Supply (MoS) 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. A separate ministry, however, was responsible for aircr ...

. Their goal was to find experimental designs for investigating the dependence of some measured quantity on a number of

independent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...

s (factors), each taking ''L'' levels, in such a way as to minimize the

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

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 to table). A complete

factorial design In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all s ...

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 Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to: *Scien ...

found in 1933 by

Raymond Paley Raymond Edward Alan Christopher Paley (7 January 1907 – 7 April 1933) was an English mathematician who made significant contributions to mathematical analysis before dying young in a skiing accident. Life Paley was born in Bournemouth, Engl ...

for generating

orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...

whose elements are all either 1 or −1 (

Hadamard matrices In mathematics, a 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. In geometric terms, this means that each pair of rows in ...

). 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 In statistics, fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design. The subset is chosen so as to exploit the sparsity-of-effects principle to ...

, 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 (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is par ...

and K. Kishen at the

Indian Statistical Institute Indian Statistical Institute (ISI) is a higher education and research institute which is recognized as an Institute of National Importance by the 1959 act of the Indian parliament. It grew out of the Statistical Laboratory set up by Prasanta C ...

. 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 In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when a ...

confounding In statistics, a confounder (also confounding variable, confounding factor, extraneous determinant or lurking variable) is a variable that influences both the dependent variable and independent variable, causing a spurious association. Con ...

Extended uses

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. The resulting matrix, minus that column, is a "supersaturated design"V K Gupta, Rajender Parsad, Basudev Kole and Lal Mohan Bhar,
Supersaturated Designs
,

Indian Agricultural Statistics Research Institute The Indian Agricultural Statistics Research Institute is an institute under the Indian Council of Agricultural Research (ICAR) with the mandate for developing new techniques for the design of agricultural experiments as well as to analyze data ...

, 18 October 2011 for finding significant first order effects, under the assumption that few exist. Box–Behnken designs can be made smaller, or very large ones constructed, by replacing the fractional factorials and incomplete blocks traditionally 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. Adding axial points allows 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 by ''a-1'' columns assigned to categorical variable ''A'' and following columns, where ''A'' = 1 + int(''a''·''i'' /(max(''i'') + 0.00001)), ''i'' = row number and ''a'' = ''A's'' number 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 like D-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 10⁴⁸⁴ matrices.

4 to 48 runs, sorted to show half-fractions

P.B.4
+ + + 
+ – – 
– + – 
– – + 
P.B.8
+ + + + + + + 
+ + – – – – + 
+ – + + – – – 
+ – – – + + – 
– + + – + – – 
– + – + – + – 
– – + – – + + 
– – – + + – + 
P.B.12
+ + + + + + + + + + + 
+ + + + – – – + – – – 
+ + – – – + – – + – + 
+ – + – + + + – – – – 
+ – – + – – + – + + – 
+ – – – + – – + – + + 
– + + – – – + – – + + 
– + – + + + – – – + – 
– + – – + – + + + – – 
– – + + + – – – + – + 
– – + – – + – + + + – 
– – – + – + + + – – + 
P.B.16
+ + + + + + + + + + + + + + + 
+ + + – – – – – – – – + + + + 
+ + – + – – – – + + + – – – + 
+ + – – + + + + – – – – – – + 
+ – + + + – – + + – – + – – – 
+ – + – – + + – – + + + – – – 
+ – – + – + + – + – – – + + – 
+ – – – + – – + – + + – + + – 
– + + + – + – + – + – – + – – 
– + + – + – + – + – + – + – – 
– + – + + – + – – + – + – + – 
– + – – – + – + + – + + – + – 
– – + + – – + + – – + – – + + 
– – + – + + – – + + – – – + + 
– – – + + + – – – – + + + – + 
– – – – – – + + + + – + + – + 
P.B.20
+ + + + + + + + + + + + + + + + + + + 
+ + + – + – + – – – – + + – – + – – + 
+ + – + – + – – – – + + – – + – – + + 
+ + – + – – – – + + – – + – – + + + – 
+ + – – – – + + – – + – – + + + + – – 
+ – + + + + – + – + – – – – + + – – – 
+ – + – + – – – – + + – – + – – + + + 
+ – + – – + + + + – + – + – – – – + – 
+ – – + – – + + + + – + – + – – – – + 
+ – – – + + – – + – – + + + + – + – – 
– + + + + – + – + – – – – + + – – + – 
– + + + – + – + – – – – + + – – + – + 
– + + – – + – – + + + + – + – + – – – 
– + – – + + + + – + – + – – – – + + – 
– + – – + – – + + + + – + – + – – – + 
– – + + – – + – – + + + + – + – + – – 
– – + – – – – + + – – + – – + + + + + 
– – – + + + + – + – + – – – – + + – + 
– – – + + – – + – – + + + + – + – + – 
– – – – – + + – – + – – + + + + – + + 
P.B.24
+ + + + + + + + + + + + + + + + + + + + + + + 
+ + + – + – + + – – + + – – + – + – – – – – + 
+ + + – – + + – – + – + – – – – – + + + + – – 
+ + – + + – – + + – – + – + – – – – – + + + – 
+ + – + – + + – – + + – – + – + – – – – – + + 
+ + – – – – – + + + + – + – + + – – + + – – – 
+ – + + – – + – + – – – – – + + + + – + – + – 
+ – + – + + – – + + – – + – + – – – – – + + + 
+ – + – + – – – – – + + + + – + – + + – – + – 
+ – – + + – – + – + – – – – – + + + + – + – + 
+ – – + – + – – – – – + + + + – + – + + – – + 
+ – – – – + + + + – + – + + – – + + – – + – – 
– + + + + – + – + + – – + + – – + – + – – – – 
– + + + – + – + + – – + + – – + – + – – – – + 
– + + – – + – + – – – – – + + + + – + – + + – 
– + – + – – – – – + + + + – + – + + – – + + – 
– + – – + + – – + – + – – – – – + + + + – + + 
– + – – + – + – – – – – + + + + – + – + + – + 
– – + + + + – + – + + – – + + – – + – + – – – 
– – + + – – + + – – + – + – – – – – + + + + + 
– – + – – – – – + + + + – + – + + – – + + – + 
– – – + + + + – + – + + – – + + – – + – + – – 
– – – – + + + + – + – + + – – + + – – + – + – 
– – – – – – + + + + – + – + + – – + + – – + + 
P.B.28
+ + + + + + + + + + + + + + + + + + + + + + + + + + – 
+ + + – – + + + – + + – – + + + + – – – – – – – – + + 
+ + + – – – – – – – – + + + + – – + + + – + + – – + + 
+ + – + + – – + + + + – – – – – – – – + + + + – – + + 
+ + – + – – + + – – – + – – + + – + – – + – + – + – – 
+ + – + – – + – + – + – + + – + – – + + – – – + – – – 
+ + – – – + – – + + – + – – + – + – + – + + – + – – – 
+ – + + – + – – + + – – – + – – + + – + – – + – + – – 
+ – + – + + – + – – + + – – – + – – + + – + – – + – – 
+ – + – + – + + – + – – + + – – – + – – + + – + – – – 
+ – + – + – + – + – + – + – + – + – + – + – + – + – + 
+ – – + + + + – – – – – – – – + + + + – – + + + – + + 
+ – – + + + – + + – – + + + + – – – – – – – – + + + + 
+ – – – – – – – – + + + + – – + + + – + + – – + + + + 
– + + + + – – + + + – + + – – + + + + – – – – – – – + 
– + + + + – – – – – – – – + + + + – – + + + – + + – + 
– + + + – + + – – + + + + – – – – – – – – + + + + – + 
– + + – – + + + + – – – – – – – – + + + + – – + + + + 
– + – – + + – + – – + – + – + – + + – + – – + + – – – 
– + – – + + – – – + – – + + – + – – + – + – + – + + – 
– + – – + – + – + – + + – + – – + + – – – + – – + + – 
– – + + – + – – + – + – + – + + – + – – + + – – – + – 
– – + + – – – + – – + + – + – – + – + – + – + + – + – 
– – + – + – + – + + – + – – + + – – – + – – + + – + – 
– – – + + + + – – + + + – + + – – + + + + – – – – – + 
– – – + – – + + – + – – + – + – + – + + – + – – + + – 
– – – – – + + + + – – + + + – + + – – + + + + – – – + 
– – – – – – – + + + + – – + + + – + + – – + + + + – + 
P.B.32
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 
+ + + + – – – – – – – – – – – – – – – – + + + + + + + + + + + 
+ + + – + – – – – – – – – + + + + + + + – – – – – – – + + + + 
+ + + – – + + + + + + + + – – – – – – – – – – – – – – + + + + 
+ + – + + – – – – + + + + – – – – + + + – – – – + + + – – – + 
+ + – + – + + + + – – – – + + + + – – – – – – – + + + – – – + 
+ + – – + + + + + – – – – – – – – + + + + + + + – – – – – – + 
+ + – – – – – – – + + + + + + + + – – – + + + + – – – – – – + 
+ – + + + + – – + + – – + + – – + + – – + – – + + – – + – – – 
+ – + + – – + + – – + + – – + + – – + + + – – + + – – + – – – 
+ – + – + – + + – – + + – + – – + + – – – + + – – + + + – – – 
+ – + – – + – – + + – – + – + + – – + + – + + – – + + + – – – 
+ – – + + – + + – + – – + – + + – + – – – + + – + – – – + + – 
+ – – + – + – – + – + + – + – – + – + + – + + – + – – – + + – 
+ – – – + + – – + – + + – – + + – + – – + – – + – + + – + + – 
+ – – – – – + + – + – – + + – – + – + + + – – + – + + – + + – 
– + + + + – + – + – + – + – + – + – + – – + – + – + – – + – – 
– + + + – + – + – + – + – + – + – + – + – + – + – + – – + – – 
– + + – + + – + – + – + – – + – + – + – + – + – + – + – + – – 
– + + – – – + – + – + – + + – + – + – + + – + – + – + – + – – 
– + – + + + – + – – + – + + – + – – + – + – + – – + – + – + – 
– + – + – – + – + + – + – – + – + + – + + – + – – + – + – + – 
– + – – + – + – + + – + – + – + – – + – – + – + + – + + – + – 
– + – – – + – + – – + – + – + – + + – + – + – + + – + + – + – 
– – + + + – – + + – – + + – – + + – – + – – + + – – + – – + + 
– – + + – + + – – + + – – + + – – + + – – – + + – – + – – + + 
– – + – + + + – – + + – – – – + + – – + + + – – + + – – – + + 
– – + – – – – + + – – + + + + – – + + – + + – – + + – – – + + 
– – – + + + + – – – – + + + + – – – – + + + – – – – + + + – + 
– – – + – – – + + + + – – – – + + + + – + + – – – – + + + – + 
– – – – + – – + + + + – – + + – – – – + – – + + + + – + + – + 
– – – – – + + – – – – + + – – + + + + – – – + + + + – + + – + 
P.B.36
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + – 
+ + + + – + + + + – – + + – – – – – – + + + + – – – – – – + + – – + + 
+ + + – – + + – – – – – – + + + + – – – – – – + + – – + + + + + – + + 
+ + + – – – – – – + + – – + + + + + – + + + + – – + + – – – – – – + + 
+ + – + + + + – – + + – – – – – – + + + + – – – – – – + + – – + + + + 
+ + – + – – + – + – + + – – + + – + – – – + – + – – + + – – + – + – – 
+ + – + – – – + – + – – + + – – + – + – + + – + – – + – + – + + – – – 
+ + – – + + – + – – – + – + – – + + – – + – + – + + – + – – + – + – – 
+ + – – + – + – + + – + – – + – + – + + – – + + – + – – – + – + – – – 
+ – + + – + – – + – + – + + – – + + – + – – – + – + – – + + – – + – – 
+ – + + – – + + – + – – – + – + – – + + – – + – + – + + – + – – + – – 
+ – + – + + – + – – + – + – + + – – + + – + – – – + – + – – + + – – – 
+ – + – + + – – + + – + – – – + – + – – + + – – + – + – + + – + – – – 
+ – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + 
+ – – + + + + + – + + + + – – + + – – – – – – + + + + – – – – – – + + 
+ – – + + – – – – – – + + + + – – – – – – + + – – + + + + + – + + + + 
+ – – – – – – + + + + – – – – – – + + – – + + + + + – + + + + – – + + 
+ – – – – – – + + – – + + + + + – + + + + – – + + – – – – – – + + + + 
– + + + + + – + + + + – – + + – – – – – – + + + + – – – – – – + + – + 
– + + + + – – + + – – – – – – + + + + – – – – – – + + – – + + + + + + 
– + + + + – – – – – – + + – – + + + + + – + + + + – – + + – – – – – + 
– + + – – + + + + + – + + + + – – + + – – – – – – + + + + – – – – – + 
– + + – – – – – – + + + + – – – – – – + + – – + + + + + – + + + + – + 
– + – + – – + + – – + – + – + + – + – – + – + – + + – – + + – + – – – 
– + – – + + – – + – + – + + – + – – + – + – + + – – + + – + – – – + – 
– + – – + – + – + + – – + + – + – – – + – + – – + + – – + – + – + + – 
– + – – – + – + – – + + – – + – + – + + – + – – + – + – + + – – + + – 
– – + + – + – – – + – + – – + + – – + – + – + + – + – – + – + – + + – 
– – + + – – + – + – + + – + – – + – + – + + – – + + – + – – – + – + – 
– – + – + – + + – + – – + – + – + + – – + + – + – – – + – + – – + + – 
– – + – + – + + – – + + – + – – – + – + – – + + – – + – + – + + – + – 
– – – + + + + – – – – – – + + – – + + + + + – + + + + – – + + – – – + 
– – – + + – – + + + + + – + + + + – – + + – – – – – – + + + + – – – + 
– – – + – + – – + + – – + – + – + + – + – – + – + – + + – – + + – + – 
– – – – – + + + + – – – – – – + + – – + + + + + – + + + + – – + + – + 
– – – – – + + – – + + + + + – + + + + – – + + – – – – – – + + + + – + 
P.B.40
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + – 
+ + + + + + + – – + + – – + + – – – – – – – – + + + + – – – – + + – – – – + – 
+ + + + + – – + + – – + + – – – – – – – – + + + + – – – – + + – – – – + + + – 
+ + + – – + + – – + + – – – – – – – – + + + + – – – – + + – – – – + + + + + – 
+ + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – + – 
+ + – + – + – + – – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + 
+ + – + – – + – + + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + 
+ + – – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + – + – – + 
+ + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + + – + – – + – + 
+ + – – + – + – + – + + – + – – + – + + – – + – + + – + – + – + – – + + – – + 
+ – + + – + – + – + – – + + – – + + – – + – + – + – + + – + – – + – + + – – + 
+ – + + – + – – + – + + – – + – + + – + – + – + – – + + – – + + – – + – + – + 
+ – + + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + + – + – – + 
+ – + – + + – + – – + – + + – – + – + + – + – + – + – – + + – – + + – – + – + 
+ – + – + – + + – + – – + – + + – – + – + + – + – + – + – – + + – – + + – – + 
+ – – + + – – + + – – – – – – – – + + + + – – – – + + – – – – + + + + + + + – 
+ – – + + – – – – – – – – + + + + – – – – + + – – – – + + + + + + + + – – + – 
+ – – – – + + + + + + + + – – + + – – + + – – – – – – – – + + + + – – – – + – 
+ – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – + + + – 
+ – – – – – – – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – + – 
– + + + + + + + + – – + + – – + + – – – – – – – – + + + + – – – – + + – – – – 
– + + + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – 
– + + – – + + – – – – – – – – + + + + – – – – + + – – – – + + + + + + + + – – 
– + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – + + + + – – – – 
– + + – – – – – – – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – 
– + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + – + + 
– + – + – + – – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + + + 
– + – + – – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + + – + + 
– + – – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + + 
– + – – + – + + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + + + 
– – + + – – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + – + + 
– – + + – – + – + – + – + + – + – – + – + + – – + – + + – + – + – + – – + + + 
– – + – + + – + – + – + – – + + – – + + – – + – + – + – + + – + – – + – + + + 
– – + – + + – – + – + + – + – + – + – – + + – – + + – – + – + – + – + + – + + 
– – + – + – + – + + – + – – + – + + – – + – + + – + – + – + – – + + – – + + + 
– – – + + + + + + + + – – + + – – + + – – – – – – – – + + + + – – – – + + – – 
– – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – 
– – – + + – – – – + + + + + + + + – – + + – – + + – – – – – – – – + + + + – – 
– – – – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – – – 
– – – – – – – + + + + – – – – + + – – – – + + + + + + + + – – + + – – + + – – 
P.B.44
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 
+ + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + – 
+ + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + 
+ + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – – 
+ + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – – 
+ + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + 
+ + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + 
+ + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + – 
+ + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – – 
+ + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + 
+ + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + 
+ – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + 
+ – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – – 
+ – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + 
+ – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – – 
+ – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + – 
+ – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – – 
+ – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + – 
+ – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + 
+ – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + 
+ – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + – 
+ – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + 
– + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + + 
– + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – 
– + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – + 
– + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – 
– + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + + 
– + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – 
– + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – + 
– + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – 
– + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – 
– + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + + 
– + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – 
– – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – + 
– – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – 
– – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + + 
– – + – + + + – – – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – 
– – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – + 
– – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – 
– – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – – + – – + + 
– – – + – + – – + + + – + + + + + – – – + – + + + – – – – – + – – – + + – + – + + – + 
– – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – 
– – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – + 
– – – – + – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – 
P.B.48
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 
+ + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + – 
+ + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – – 
+ + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + 
+ + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + 
+ + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + – 
+ + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – – 
+ + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + 
+ + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – – 
+ + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – – 
+ + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + 
+ + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + 
+ – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + 
+ – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – – 
+ – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + 
+ – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + – 
+ – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + – 
+ – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + 
+ – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + – 
+ – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + 
+ – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + 
+ – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + – 
+ – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – – 
+ – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + 
– + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – 
– + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – + 
– + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + + 
– + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – 
– + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + + 
– + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – + 
– + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – 
– + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – 
– + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – 
– + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + + 
– + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + + 
– + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – 
– – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – 
– – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + + 
– – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + + 
– – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – + 
– – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – + 
– – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – 
– – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – 
– – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – – 
– – – + + – + + – – – + – + – + + – – – – + – – – – – + + + + – + + + + – – + – + – + + + – + 
– – – + – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – 
– – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – – + – 
– – – – – – + + + + – + + + + – – + – + – + + + – – + – – + + – + + – – – + – + – + + – – – +

References

{{DEFAULTSORT:Plackett-Burman Design Design of experiments