Generates alpha designs (R.W. Payne).

### Option

`PRINT` = string token |
Whether to print the design (`design` ); default `*` i.e. no printing |
---|

### Parameters

`GENERATOR` = matrices |
generating array (of size number-of-plots-per-block by number-of-reps) |
---|---|

`LEVELS` = scalars or variates |
Defines the levels of each treatment factor; if this is omitted, the levels of the `TREATMENT` factor are used, if available, otherwise `LEVELS` is determined from the generating array on the assumption that the blocks are to be of equal size |

`SEED` = scalar |
Seed to be used to randomize the design, if required |

`TREATMENTS` = factors |
Specifies the treatment factor for each design |

`REPLICATES` = factors |
Specifies the replicate factor |

`BLOCKS` = factors |
Specifies the block factor |

`UNITS` = factors |
Specifies the factor to index the units within each block |

### Description

Alpha designs are a very flexible class of resolvable incomplete block designs. A resolvable design is one in which each block contains only a selection of the treatments, but the blocks can be grouped together into subsets in which each treatment is replicated once. The groupings of blocks thus form replicates, and the block structure of the design is

`Replicates / Blocks / Units`

Such designs are particularly useful when there are many treatments to examine and the variability of the units is such that the block size needs to be kept small. Alpha designs were thus devised originally for the analysis of plant breeding trials (Patterson & Williams 1976), where many varieties may need to be evaluated in a single trial, and have the advantage that they can provide effective designs for any number of treatments.

The construction of an alpha design requires a *k* × *r* array of integers between 0 and s-1, where *r* is the number of replicates, and *s* is the number of blocks per replicate. If the number of treatments, *v*, is a multiple of the number of blocks per replicate, *k* will be the number of units in each block, and *v* will be given by *s* × *k*. Otherwise, the design will have some blocks of size *k* and some of size *k*-1, and *v* will lie between *s* × (*k*-1) and *s* x *k*. Clearly, the properties of the design that is formed will be very dependent on the choice of array. Patterson, Williams and Hunter (1978) present 11 basic arrays to generate designs with up to 100 treatments and 2, 3 or 4 replicates when *k* is greater than 3 and *s* is greater than or equal to *k*; these arrays are reproduced in John (1987). Williams (1975) presents arrays for any sensible values of s and k with up to 100 treatments and 2 to 4 replicates.

Procedure `AFALPHA`

generates the treatment, replicate, block and unit factors for an alpha design. The design can be printed by setting option `PRINT=design`

, and the factors can be saved using the parameters `TREATMENTS`

, `REPLICATES`

, `BLOCKS`

and `UNITS`

. The generating array for the design must be specified as a *k* × *r* matrix using the `GENERATOR`

parameter, and the number of levels of the treatment factor can be defined by the `LEVELS`

parameter. If `LEVELS`

is omitted, `AFALPHA`

will see whether the `TREATMENTS`

parameter has been set to a factor whose levels have already been defined; if not, `AFALPHA`

will set `LEVELS`

to the scalar value *v* = *s* × *k*. By default the design is unrandomized, but randomization can be requested by setting the `SEED`

parameter.

Option: `PRINT`

.

Parameters: `GENERATOR`

, `LEVELS`

, `SEED`

, `TREATMENTS`

, `REPLICATES`

, `BLOCKS`

, `UNITS`

.

### Method

Each column of the generating array is used to form *s*-1 further columns by successively adding 1 modulo *s*. Next, *s* is added to row 2 of every column, 2*s* to row 3, and so on. Each resulting column then gives one of the blocks of the design, and the replicates are formed by the sets of columns that were all generated from the same initial column. If the design needs to have blocks of unequal sizes, procedure `SUBSET`

is used to omit the necessary plots to form the smaller blocks.

### References

Patterson, H.D. & Williams E.R. (1976). A new class of resolvable incomplete block designs. *Biometrika*, 63, 83-92.

Patterson, H.D., Williams E.R. & Hunter, E.A. (1978). Block designs for variety trials. *Journal of Agricultural Science, Cambridge*, 90, 395-400.

Williams, E.R. (1975). *A new class of resolvable block designs*. Ph.D. Thesis, University of Edinburgh.

### See also

Procedure: `AGALPHA`

.

Commands for: Design of experiments, REML analysis of linear mixed models.

### Example

CAPTION 'AFALPHA example',\ 'Data from Patterson & Williams (Biometrika 1976):',\ !t('1) Alpha design for 20 treatments (numbered 0-19)',\ 'with 3 replicates each containing 4 blocks of 5 plots;');\ STYLE=meta,plain,plain MATRIX [ROWS=5; COLUMNS=3; VALUES=0,0,0, 0,1,2, 0,2,3, 0,3,1, 0,3,2] Array FACTOR [LEVELS=!(0...19)] Treat AFALPHA [PRINT=design] Array; TREATMENTS=Treat; REPLICATES=Rep;\ BLOCKS=Block; UNITS=Plot CAPTION !t('2) Alpha design for 19 treatments (numbered 0-18)',\ 'with 3 replicates containing 4 blocks of 4 & 5 plots.') AFALPHA [PRINT=design] Array; LEVELS=!(0...18)