Forms minimum aberration factorial or fractional-factorial designs.

### Options

`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 |

### Parameters

`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 |

### Description

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.

Options: `PRINT`

, `NTIMES`

, `SEED`

.

Parameters: `LEVELS`

, `NTREATMENTFACTORS`

, `NUNITS`

, `NSUBUNITS`

, `KEYBLOCKS`

, `KEYDEFINING`

, `RESOLUTION`

, `ABERRATION`

, `SUBRESOLUTION`

, `SUBABERRATION`

, `NDESIGN`

, `NSUBDESIGN`

.

### Reference

Laycock, P.J. & Rowley, P.J. (1995). A method for generating and labelling all regular fractions or blocks for *q ^{n-m}* designs.

*Journal of the Royal Statistical Society, Series B*, 57, 191-204.

### See also

Directives: `AFRESPONSESURFACE`

, `FKEY`

, `FPSEUDOFACTORS`

.

Procedures: `AGFACTORIAL`

, `AKEY`

, `ARANDOMIZE`

, `ASAMPLESIZE`

, `FACPRODUCT`

.

Commands for: Design of experiments, Analysis of variance.