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Forms minimum aberration factorial or fractional-factorial designs.


PRINT = string tokens Controls printed output (summary, keyblocks, keydefining, monitoring); default *
NTIMES = scalar Number of designs to try in a random search; default 0 does the full search
SEED = scalar Seed for the random number generator used to search the designs randomly; default 0


LEVELS = scalars Number of levels of the treatment factors, must be a power of a prime number
NTREATMENTFACTORS = scalars Number of treatment factors
NUNITS = scalars Number of units in each block of a block design or in the principal block of a fractional factorial
NSUBUNITS = scalars Number of units in each (sub-)block
KEYBLOCKS = matrices Design key for the blocks and sub-blocks
KEYDEFINING = matrices Design key specifying the defining contrasts
RESOLUTION = scalars Saves the resolution of the design
ABERRATION = scalars Saves the aberration of the design
SUBRESOLUTION = scalars Saves the resolution of the sub-design
SUBABERRATION = scalars Saves the aberration of the sub-design
NDESIGN = scalars Saves or defines the design number
NSUBDESIGN = scalars Saves or defines the sub-design number


The concept of minimum aberration provides an effective way of selecting either a full factorial design where treatment contrasts are confounded with blocks, or a fractional factorial. (Essentially, these are equivalent – the fractional factorial design is formed by taking only one block of the full factorial.) The resolution of the design is defined as the largest integer r such that no interaction term with r factors is confounded with blocks (or aliased). The aberration of the design is the number of interaction terms with r+1 factors that are confounded (or aliased). A minimum aberration design is a design with the smallest aberration out of the designs with the highest available resolution. It is thus a design that is closest to the next level of resolution.

AFMINABERRATION searches for minimum aberration designs using the algorithm of Laycock & Rowley (1995), and we gratefully acknowledge Patrick Laycock’s assistance with the implementation into Genstat. The number of treatment factors is specified by the NFACTORS parameter. Their number of levels is specified by the LEVELS parameter. This must be an integer power of a prime number. The number of units in each block (or the number of plots in the equivalent fractional factorial) is specified by the NUNITS parameter, and must be a power of LEVELS.

AFMINABERRATION can also form a sub-blocking factor that can be used to define blocks if the design is to be used to form a fractional factorial. The number of units in each sub-block is defined by the NSUBBLOCKS parameter (and again must be a power of LEVELS).

If there are very many designs to search, you may prefer to examine only a random selection. The NTIMES option sets the number of designs to try; its default of zero requests the standard (full) search. The SEED option sets the seed for the random numbers that are used to select the designs randomly; the default of zero continues the existing sequence or (if none) initializes the seed automatically. (Note that this version of the random number generator is shared with other design construction algorithms, such as FKEY.)

Printed output is controlled by the PRINT option, with settings:

    summary summarizes the design properties;
    keyblocks prints a design key to generate the block and sub-block factors from the treatment factor (or pseudo-factors to generate them if they have more than p levels);
    keydefining prints a design key specifying the defining contrasts i.e. all the treatment contrasts confounded with blocks or sub-blocks;
    monitoring prints monitoring information about the design construction.

You can save the design keys using the KEYBLOCKS and KEYDEFINING parameters. In addition, the NDESIGN parameter can save a unique “design number” for the design, and the NSUBDESIGN parameter can save a unique number for the sub-design of the design. You can input these with NDESIGN and NSUBDESIGN later, along with the same settings for NTREATMENTFACTORS, LEVELS, NUNITS and NSUBUNITS, to obtain the design keys without repeating the design search. The RESOLUTION and ABERRATION parameters can save the resolution and aberration of the (main) design, and the SUBRESOLUTION and SUBABERRATION parameters can save the resolution and aberration of a sub-design.




Laycock, P.J. & Rowley, P.J. (1995). A method for generating and labelling all regular fractions or blocks for qn-m designs. Journal of the Royal Statistical Society, Series B, 57, 191-204.

See also



Commands for: Design of experiments, Analysis of variance.

Updated on March 11, 2019

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