The different ways of using the semi-Latin square as a design for an experiment depend on which of these partitions correspond to nuisance factors and which to treatment factors.

All examples below use the same semi-Latin square.
Note that this square is deliberately chosen
to emphasise that a semi-Latin square is not the
same thing as a set of mutually orthogonal Latin
squares. This implies that the design given here is not best
possible if *B* has any effect.

If *B* has no effect then all semi-Latin
squares of the appropriate size are equally good.
Otherwise the efficiency of the design depends
on the relationship between *B* and *L*.

- Ten new vacuum cleaners are available for
comparison during a five-week period. Five
housewives volunteer to test them, each using two
of the vacuum cleaners in her house each week.
Week Housewife 1 2 3 4 5 a A J G E I C F D HB b B G A I E F H C J D c C F B D A H J E I G d I D H F B J A G C E e H E C J D G I B A F Before using such a design, randomly permute the rows (the way that the rows are assigned to actual weeks) and randomly permute the columns (the way that the columns are assigned to actual housewives). Where it makes sense (as in the following example), also randomly permute the order of the two letters in each cell.

- Ten treatments are to be applied to sugar-beet,
which is grown in a 5 × 10 rectangular array of plots.
Each plot is a single long North-South row of sugar
beet, so the 10 plots in a single row of the rectangle
are close to each other and these rows must be
regarded as a nuisance factor. The beet is sown
from five seed-drills on an arm which protrudes
from the right of the tractor. The tractor drives
Northwards up the left-hand side of the array,
sowing seed in the first five columns, then turns
round and drives Southwards down the right-hand side
of the array, sowing seed in the last five columns.
Thus the first and last column are sown by the same
drill, and drills form a second nuisance factor.
row drill 1 2 3 4 5 5 4 3 2 1 a A E C D B H F I G J b B A E C D J H F I G c F D H J I G E A B C d I H J G E C A B F D e H C G B A F I D J E

Now we simply have an orthogonal row-column
design and *B* is assumed to have no effect.

week | housewife | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | E | F | G | H | I | J | |

a | 1 | 5 | 3 | 4 | 2 | 4 | 2 | 5 | 3 | 1 |

b | 2 | 1 | 4 | 5 | 3 | 3 | 1 | 4 | 2 | 5 |

c | 3 | 2 | 1 | 2 | 4 | 1 | 5 | 3 | 5 | 4 |

d | 4 | 3 | 5 | 1 | 5 | 2 | 4 | 2 | 1 | 3 |

e | 5 | 4 | 2 | 3 | 1 | 5 | 3 | 1 | 4 | 2 |

Before using such a design, randomly permute the rows (the way that the rows are assigned to actual weeks) and randomly permute the columns (the way that the columns are assigned to actual housewives).

chamber | treatment combination | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | Aa | Cc | Fc | Ja | Bb | Id | He | Gb | Dd | Ee |

2 | Bc | Fd | Ga | Ab | Je | Ib | Ea | Dc | Ce | Hd |

3 | Hc | Ia | Jd | Ca | De | Ac | Bd | Eb | Ge | Fb |

4 | Hb | Gd | Be | Ie | Cb | Da | Jc | Ec | Fa | Ad |

5 | Ic | Db | Fe | Gc | Ha | Ed | Ae | Cd | Jb | Ba |

Before using such a design, randomly permute the way that the chambers are numbered and randomly permute the order of the treatment combinations within each chamber, doing this separately and independently for each chamber.

If there is no interaction between *R* and *C*
then all semi-Latin squares are equally good.
But we may assume that this interaction is non-zero,
in which case the efficiency of the design depends
upon the relationship between *B* and *L*.

chamber | treatment combination | ||||
---|---|---|---|---|---|

A | 1a | 2b | 3c | 4d | 5e |

B | 1b | 4e | 5a | 3d | 2c |

C | 2e | 3a | 5d | 1c | 4b |

D | 2c | 4a | 1d | 5b | 3e |

E | 1e | 3b | 5d | 2a | 4c |

F | 4a | 5e | 1c | 2d | 3b |

G | 5c | 2a | 4d | 3e | 1b |

H | 2d | 3c | 4b | 1e | 5a |

I | 5c | 3a | 1d | 2b | 4e |

J | 5b | 2e | 4c | 1a | 3d |

Before using such a design, randomly permute the way that the chambers are labelled and randomly permute the order of the treatment combinations within each chamber, doing this separately and independently for each chamber.

We have to assume that there is no interaction
between *R* and *L* or between *C* and *L*.
If there is no interaction between *R* and *C*
then all semi-Latin squares are equally good.
But we may assume that this interaction is non-zero,
in which case the efficiency of the design depends
upon the relationship between *B* and *L*.

The fifty treatment combinations are Aa1, Ja1, ..., Fe5.

Before such a design is used, the order of all the treatment combinations must be completely randomized.

Page maintained by R. A. Bailey

Modified 24/6/00