Latin Square Design

Latin Square Design

Latin Square DesignAs a particular kind of configuration of integers, letters of the alphabet or other symbols, Latin square has received a lot of attention from the researchers for a long time. According to Fisher (1925), Latin square design is very useful in experimental researches to control the effects of extraneous variables. He emphasized that in order to properly use Latin square, the principle of randomly selecting a Latin square from the universe of possible Latin squares must be strictly adhered to, which is suitable for a particular research design. He also stressed that the Latin square design should be applied to the analysis of the results. In addition to agricultural experiments, many researchers realize that Latin square designs are also applicable to their work of other research fields. Especially in medical research, the Latin square designs can effectively control the effects of extraneous variables. And Latin square is excellent in controlling the effects of temporal order or sequence in repeated-measures designs.

The Latin square is a grid or matrix with the same number of rows and columns (e.g. k). A sequence of k symbols is built into the cell entries. These symbols may be integers from 1 to k or the first k letters of the alphabet. It should be noted that each symbol is displayed only once in each row and each column of the grid. The Sudoku puzzles are the most typical examples in nowadays. For example, as shown in Figure 1, this is a Latin square with four rows and four columns, containing the integers from 1 to 4, which is a standard form of Latin square and is also known as a reduced or normalized Latin square. In this kind of Latin square, the numbers in the first row and the first column are in their natural order. Researchers can obtain nonstandard Latin squares from standard forms if they exchange different rows or columns in a grid or exchange rows and columns at the same time.

A 4 × 4 Latin square.

Figure 1. A 4 × 4 Latin square.

Latin square design is a method that assigns treatments within a square block or field that allows these treatments to present in a balanced manner. Replicates are also included in this design. When trying to control two or more blocking factors, we may use Latin square design as the most popular alternative design of block design. In fact, we can consider a Latin square design as an extreme example of an incomplete block design, its all combinations of levels are assigned to only one treatment instead of to all. The rules for the Latin square design are as follows:

  • Randomly assigned treatments within rows and columns, and each treatment once per row and once per column;
  • Equal numbers of rows, columns, and treatments;
  • Suitable for controlling variation in two different directions.

The advantages of Latin square designs are as follows:

  • They are suitable for cases with several nuisance factors, and we cannot combine these factors into a single factor or we need to separate them.
  • Those experiments using Latin square design can have a relatively small amount of runs.

The disadvantages are:

  • The number of levels of each blocking variable and the number of levels of the treatment factor must be equal.
  • The Latin square design hypothesizes that there are no interactions between the blocking variables or between the treatment variable and the blocking variable.

At CD BioSciences, we can not only help you with the Latin square design but also help you make an appropriate choice of statistical strategy. If you have any questions, please feel free to contact us.

1. Richardson, J. T. E. (2018) ‘The use of latin-square designs in educational and psychological research’, Educational Research Review, 24, 84-97.
2. F., Z. H. (1925) ‘Statistical methods for research workers’, Protoplasma,23(1), 282-282.

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