## Abstract

A method of restricted randomization, which avoids bad patterns in the treatments but retains validity in the analysis, is demonstrated on an example that exhibits many features of real agricultural experiments, such as untidy treatment structure and uncertainty about blocking. The method uses permutation groups.

Modern farming methods force agricultural experimenters to use long thin lines of plots for blocking, rather than more compact areas. Ordinary randomization of a long thin block can produce a plan with, say, all low levels of nitrogen at one end of the block. The first section of the article discusses why simple rerandomization is not a satisfactory method of avoiding poor layouts. Restricted randomization has been developed as a method of avoiding badly patterned plans while ensuring validity of the usual analysis.

One method of restricted randomization uses special groups of permutations of the plots. Catalogs of good restricted randomizations have been given for symmetric factorial designs.

The example described here is fairly typical of agricultural experiments in that the treatment structure is a full factorial plus control and the treatment factors do not have equal numbers of levels. In addition, there is some confounding of treatments with subblocks. The general method of restricted randomization can still be used, even though the catalogs contain nothing relevant.

The method uses group theory in two distinct ways. For subblocks of size 13 the first step is the choice of a suitable group G of permutations of 13 objects. Then the factorial structure of the treatments in a subblock is elucidated. Lattice diagrams prove particularly helpful and show how to match the noncontrol treatments to the elements of an abstract group C of order 12.

The experimenter specifies which layouts he would like to avoid. An initial arrangement of the 13 treatments is sought with the property that no permutation in G transforms it into any of the specified bad plans. In principle each possible initial arrangement would have to be tested on every permutation in G: the purpose of the previous matching of the treatments to C is to reduce the number of tests by a factor of 12.

Typically, the first specification is too stringent and there is no solution. Alternatively, an initial arrangement that satisfies the original specification may exhibit some other pattern that the experimenter then decides he wishes to avoid. In the light of the experience gained during the searching procedure, the specification is relaxed or strengthened until a satisfactory initial arrangement is found.

Once a good initial arrangement is known, plans can be quickly produced for a large number of trials with the same experimental structure.

Modern farming methods force agricultural experimenters to use long thin lines of plots for blocking, rather than more compact areas. Ordinary randomization of a long thin block can produce a plan with, say, all low levels of nitrogen at one end of the block. The first section of the article discusses why simple rerandomization is not a satisfactory method of avoiding poor layouts. Restricted randomization has been developed as a method of avoiding badly patterned plans while ensuring validity of the usual analysis.

One method of restricted randomization uses special groups of permutations of the plots. Catalogs of good restricted randomizations have been given for symmetric factorial designs.

The example described here is fairly typical of agricultural experiments in that the treatment structure is a full factorial plus control and the treatment factors do not have equal numbers of levels. In addition, there is some confounding of treatments with subblocks. The general method of restricted randomization can still be used, even though the catalogs contain nothing relevant.

The method uses group theory in two distinct ways. For subblocks of size 13 the first step is the choice of a suitable group G of permutations of 13 objects. Then the factorial structure of the treatments in a subblock is elucidated. Lattice diagrams prove particularly helpful and show how to match the noncontrol treatments to the elements of an abstract group C of order 12.

The experimenter specifies which layouts he would like to avoid. An initial arrangement of the 13 treatments is sought with the property that no permutation in G transforms it into any of the specified bad plans. In principle each possible initial arrangement would have to be tested on every permutation in G: the purpose of the previous matching of the treatments to C is to reduce the number of tests by a factor of 12.

Typically, the first specification is too stringent and there is no solution. Alternatively, an initial arrangement that satisfies the original specification may exhibit some other pattern that the experimenter then decides he wishes to avoid. In the light of the experience gained during the searching procedure, the specification is relaxed or strengthened until a satisfactory initial arrangement is found.

Once a good initial arrangement is known, plans can be quickly produced for a large number of trials with the same experimental structure.

Original language | English |
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Pages (from-to) | 712-719 |

Number of pages | 8 |

Journal | Journal of the American Statistical Association |

Volume | 82 |

Issue number | 399 |

Publication status | Published - 1987 |

## Keywords

- affine group
- blocking
- constrained randomization
- factorial design
- permutation group