Latin squares are a special form of fractional factorial design. They can be used as a form of blocking when (a) there are two blocking factors to be used; (b) each blocking factor is to be examined at exactly k-levels; (c) the single treatment effect is to be evaluated at k-levels, i.e. the treatment effect levels and blocking factor levels must match; (d) each row and column of the kxk Latin square design receives each treatment exactly once. In agricultural trials it is common for the rows and columns to be plot positions in a field, matching the number of treatments to be applied. In this way spatial factors can be minimized and/or understood (e.g. fertility variations across the trial area), whilst attention is focused on the treatment effects.

A simple example helps to clarify the application of such designs. Menzler, cited in Cox (1958, [COX1]) gives the example of an experiment in which the fuel consumption of buses is to be examined. The treatment to be considered is the effect of tyre pressure on fuel consumption. Four different tyre pressures (levels) are to be tested, A,B,C,D. The design uses four buses and is carried out over four days. In order to eliminate variations between buses, and between days, these factors provide the row and column blocking elements of the design. The basic design applied is then of the form shown in the table below:

4x4 Latin square |
Bus 1 |
Bus 2 |
Bus 3 |
Bus 4 |

Day 1 |
A |
B |
C |
D |

Day 2 |
B |
C |
D |
A |

Day 3 |
C |
D |
A |
B |

Day 4 |
D |
A |
B |
C |

The pattern shown above is systematic, not randomized, and prior to conducting a trial of this type the design should be randomized. A simple random permutation of rows and columns would suffice. Thus a random permutation for columns might yield the sequence 3,4,2,1, and for rows 1,3,2,4. The table utilized would then be:

4x4 Latin square |
Bus 1 |
Bus 2 |
Bus 3 |
Bus 4 |

Day 1 |
C |
D |
B |
A |

Day 2 |
A |
B |
D |
C |

Day 3 |
D |
A |
C |
B |

Day 4 |
B |
C |
A |
D |

This 4x4 arrangement requires a total of 16 trials, as opposed to the full factorial experiment which would require 4x4x4=64 trials to obtain every possible combination of day, bus and treatment. It is thus a fractional experiment (and an unbalanced design) and as such loses some information that could be obtained through a complete experiment (the loss is in the interaction effects between the pairs of factors, e.g. days and buses) but it does retain all the main effects at a considerable saving in time and cost.

As with other forms of randomized block designs, a simple model can be used to describe the general form of Latin square designs. Let yijk represent the data obtained from the experiment (the measured outcome or result) conducted on the jth replicate that receives the ith treatment; let Tk be the effect attributable to the kth treatment, Ri be the effect attributable to the ith row-wise block, Cj be the effect attributable to the jth column-wise block and let e denote residual error, unexplained by other factors. Then the statistical model for this kind of experiment is of the form:

This model states that the measured response is a simple linear function of the overall mean value, μ, for all the data, plus a treatment effect (tyre pressure in this example), plus a row block and a column block effect (days and buses in this example), plus some residual error. The overall mean is estimated from the mean of all the sample data, whilst the treatment mean values and block (row and column) mean values are estimated from the means for each treatment or block. The treatment effects are estimated as the difference between the overall mean and the individual treatments means.

The Latin square arrangement is a so-called complete design. If, in the example above, only 3 buses are available for the trial on any one day, the design would be incomplete. An example of a design (not randomized at this stage) which seeks to address this problem is shown below, where x marks the unavailable entries:

4x4 Latin square |
Bus 1 |
Bus 2 |
Bus 3 |
Bus 4 |

Day 1 |
A |
B |
C |
x |

Day 2 |
A |
B |
x |
D |

Day 3 |
A |
x |
C |
D |

Day 4 |
x |
B |
C |
D |

A feature of this design is that each treatment occurs with every other treatment on two of the test days, i.e. the treatments A and B occur together on days 1 and 2, A and D occur together on days 2 and 3. This applies to each treatment, and is a characteristic of so-called balanced incomplete block designs. These designs generally have the following properties:

•each block (Day in the example above) contains the same number of units (3 buses in the example above)

•each treatment occurs the same number of times in total (3 in the example above, 3 occurrences of A, B, C and D)

•taking any two treatments (e.g. A and B in the example above) the number of times they occur in each block will be the same for every such pair (i.e. not necessarily twice, but possibly more times if more days were available)

Balanced incomplete designs are only available for certain combinations of the number of units per treatment, treatments, blocks and replicates. Cox (1958, p222, [COX1]) gives a summary table of available designs with fuller details being provided in Cochrane and Cox (1957, section 9.6, [COC1]). A special type of balanced incomplete block design, known as a Youden square, can be used in certain special cases. Such designs are available in situations where the number of treatments equals (or can be matched by) the number of blocks, and the number of available units is less than the number of treatments. With a Youden square the columns of the design matrix form a balanced incomplete block design whilst the rows contain every treatment (or treatment symbol). They can be produced from Latin square designs by omitting either a row or column. An example 7x3 Youden square is shown below:

A |
B |
C |
D |
E |
F |
G |

B |
C |
D |
E |
F |
G |
A |

D |
E |
F |
G |
A |
B |
C |

References

[COX1] Cox D R (1958) Planning of experiments. John Wiley & Sons, New York

[COC1] Cochrane W G, Cox G M (1957) Experimental designs. John Wiley & Sons, New York