Generates semi-Latin squares (W. van den Berg).

### Options

`PRINT` = string token |
Controls whether or not to print a plan of the design (`design` ); if unset in an interactive run `AGSEMILATIN` will ask whether the design is to be printed, in a batch run the default is not to print anything |
---|---|

`METHOD` = string token |
Method to use to construct the semi-Latin square (`Trojan` , `interleaving` , `inflated` ); if unset in an interactive run `AGSEMILATIN` will ask what type is required, in a batch run the default is `Trojan` |

`ANALYSE` = string token |
Controls whether or not to analyse the design, and produce a skeleton analysis-of-variance table using `ANOVA` (`no` , `yes` ); default is to ask if this is unset in an interactive run, and not to analyse if it is unset in a batch run |

### Parameters

`NROWS` = scalars |
Number of rows and columns of the semi-Latin square |
---|---|

`NUNITS` = scalars |
Number of units (i.e. treatments) within each block |

`SEED` = scalars |
Seed for randomization; a negative value implies no randomization |

`TREATMENTS` = factors |
Identifier for the treatment factor |

`ROWS` = factors |
Identifier for the row factor |

`COLUMNS` = factors |
Identifier for the column factor |

`UNITS` = factors |
Identifier for the unit factor |

`PSEUDOFACTOR` = factors |
Identifier for the pseudo-factor |

`STATEMENT` = texts |
Saves a command to recreate the design (useful if the design information has been specified in response to questions from `AGSEMILATIN` ) |

### Description

`AGSEMILATIN`

generates the factors and pseudo-factor required to define a semi-Latin square. It also sets the block and treatment formulae (using the `BLOCKSTRUCTURE`

and `TREATMENTSTRUCTURE`

directives) to allow the design, if balanced, to be analysed by `ANOVA`

.

An (*n* × *n*)/*k* semi-Latin square is like an *n* × *n* Latin square except that there are *k* letters in each cell. The combinations of the rows and columns of a semi-Latin square are called blocks. Each of the *n* × *k* letters occurs once in each row and once in each column. The design thus has *n* rows and columns, *k* (sub-) units within each row × column combination (or block), and *n* × *k* treatments. The analysis should contain strata for rows, columns, rows.columns and rows.columns.units, as well as treatment effects which may be estimated in either the rows.columns or the rows.columns.units strata. `AGSEMILATIN`

enables you to construct three types of semi-Latin square.

Trojan squares: a Trojan square consist of a set of *k* mutually orthogonal *n* × *n* Latin squares, on *k* disjoint sets of treatments. Each block of the semi-Latin square contains the treatments which occur in the corresponding cell of all the individual squares (Bailey 1988). `AGSEMILATIN`

can construct Trojan squares for any value of *n* for which a Graeco-Latin square exists. Thus, for example, no Trojan square exists for *n* = 6. In a Trojan square *k* must be greater than 1 and less than *n* (Edmondson 1998), and for some values of *n*, *k* must be less than that. The maximum values of *k* for *n* up to 15 for a Trojan square are

*n*: 3 4 5 7 8 9 11 12 13 14 15

*k*: 2 3 4 6 6 8 10 2 12 2 2

In a Trojan square, some treatment effects are estimated in both the rows.columns and the rows.columns.units strata, while others (which need to be represented by a pseudo-factor) are estimated only in the rows.columns.units stratum. Trojan squares are optimal semi-Latin squares (Bailey 1992).

Inflated Latin squares: an (*n* × *n*)/*k* inflated Latin square consists of an *n* × *n* Latin square with each letter replaced by *k* new symbols (Bailey 1988). `AGSEMILATIN`

can construct inflated Latin squares for any value of *n* greater than 2, and any value of *k* greater than 1. The analysis requires a pseudo-factor to distinguish the treatment contrasts that are estimated in the rows.columns stratum from those estimated in the rows.columns.units stratum.

Interleaving Latin squares: these are formed similarly to the Trojan square, except that there is no longer the requirement for the *k* Latin squares to be orthogonal (Bailey 1988). If the squares are orthogonal, the design is a Trojan square and can be analysed by `ANOVA`

with the help of a pseudo-factor as described above. For *n*=2 the design is an inflated Latin square and can be analysed by `ANOVA`

, again with the help of a pseudo-factor. Otherwise, the design is unbalanced. It is possible to generate a balanced analysis by omitting the row.column stratum, but this is not reasonable and Yates (1935) advises against such an analysis. `AGSEMILATIN`

can construct interleaving Latin squares for any value of *n* or *k* greater than 1.

The type of semi-Latin square can be chosen using the `METHOD`

option with setting either `Trojan`

, `inflated`

, or `interleaving`

. In a batch run the default is `Trojan`

, while in an interactive run `AGSEMILATIN`

will ask what type you want. `AGSEMILATIN`

has two other options. The `PRINT`

option can be set to `design`

to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set `PRINT`

when running interactively, `AGSEMILATIN`

will ask whether or not you wish to print the design. Similarly the `ANALYSE`

option governs whether or not `AGSEMILATIN`

produces a skeleton analysis-of-variance table (containing just source of variation, degrees of freedom and efficiency factors). Again `AGSEMILATIN`

assumes that this is not required if `ANALYSE`

is unset in a batch run, and asks whether it is required if `ANALYSE`

is unset in an interactive run.

The information required to select the design and give identifiers to its factors can be defined using the parameters of `AGSEMILATIN`

. The number of rows and columns of the design (*n*) can be defined using the parameter `NROWS`

. Similarly, the number of units (*k*) for each row-column combination (that is, the number of treatments per block) can be defined by the parameter `NUNITS`

. Parameters `TREATMENTS`

, `ROWS`

, `COLUMNS`

, `UNITS`

and `PSEUDOFACTOR`

allow you to specify identifiers for the treatment, row, column and unit factors, and for the pseudo-factor. The `SEED`

parameter allows you to specify a seed to randomize the design. In a batch run, this has a default of -1, to suppress randomization. If `SEED`

is unset in an interactive run, you will be asked to provide a seed (and again a negative value will leave the design unrandomized). If one of the other parameters is unset in an interactive run, you will be asked to provide a name.

The `STATEMENT`

parameter allows you to save a Genstat text structure containing a command to recreate the design. This is particularly useful when you are running `AGSEMILATIN`

interactively, and specifying the information in response to questions.

Options: `PRINT`

, `METHOD`

, `ANALYSE`

.

Parameters: `NROWS`

, `NUNITS`

, `SEED`

, `TREATMENTS`

, `ROWS`

, `COLUMNS`

, `UNITS`

, `PSEUDOFACTOR`

, `STATEMENT`

.

### Method

The `QUESTION`

procedure is used to obtain the details of the required design.

Trojan squares are formed by constructing *k* orthogonal Latin squares with procedure `AGLATIN`

.

For constructing an inflated Latin square, first one of the possible orthogonal Latin squares is generated with procedure `AGLATIN`

. The generated treatment factor provides the “plot” factor. Each cell is split in *k* units with a corresponding “unit” treatment factor. Procedure `FACPRODUCT`

then forms the treatment factor from the *n* × *k* combinations of the plot and unit factors.

If an interleaving Latin square is chosen which fulfils the restrictions of a Trojan square, then a Trojan square is generated because Trojan squares are optimal semi-Latin squares. When *n* and *k* do not fulfil the restrictions of a Trojan square, interleaving Latin squares are generated by first generating a Trojan square with *k* as large as possible. After that the generated Latin squares are duplicated (inflated) until the required interleaving Latin square is obtained. For an interleaving Latin square with *n* equal to 2, the *n* × *k* treatment levels are laid out from 1 to *n* × *k* in the first row, and from *n* × *k* to 1 in the second row.

The randomization is performed with `BLOCKSTRUCTURE=(rows*columns)/units`

and, in addition, the treatment levels are permuted at random.

### References

Bailey, R.A. (1988). Semi-Latin squares. *Journal of Statistical Planning and Inference*, **8**, 299-312.

Bailey, R.A. (1992). Efficient semi-Latin squares. *Statistica Sinica*, **2**, 413-437.

Edmondson, R.N. (1998). Trojan square and incomplete Trojan square designs for crop research. *Journal of Agricultural Science, Cambridge*, **131**, 135-142.

Yates, F. (1935). Complex experiments (with Discussion). *Supplement to the Journal of the Royal Statistical Society*, **2**, 181-247. [Reprinted (without Discussion) in Yates, F. (1970). *Experimental Design: Selected Papers*, 69-117. Griffin, London.

### See also

Procedures: `AGCROSSOVERLATIN`

, `AGLATIN`

, `AGQLATIN`

, `AGYOUDENSQUARE`

.

Commands for: Design of experiments, Analysis of variance.

### Example

CAPTION 'AGSEMILATIN example'; STYLE=meta AGSEMILATIN [PRINT=design; METHOD=trojan; ANALYSE=yes]\ NROWS=5; NUNITS=4; SEED=135143; TREATMENTS=Treat;\ COLUMNS=Column; ROWS=Row; UNITS=Plot; PSEUDOFACTOR=Pseudo AGSEMILATIN [PRINT=design; METHOD=inflated; ANALYSE=yes]\ NROWS=5; NUNITS=4; SEED=314612 AGSEMILATIN [PRINT=design; METHOD=interleaving; ANALYSE=yes]\ NROWS=5; NUNITS=6; SEED=235978