a **Latin square** is an *n* × *n* array filled with *n* different symbols, each occurring exactly once in each row and exactly once in each column. Here is an example:

A | B | C |

C | A | B |

B | C | A |

For the past three decades, Latin Squares techniques have been widely used in many **statistical applications**. Much effort has been devoted to Latin Square Design. In this paper, I introduce the **mathematical properties of Latin square**s and the **application of Latin squares in experimental design**. Some examples and SAS codes are provided that illustrates these methods.

——Lei gao 2005

## The Latin Square design

The Latin square design is used where the researcher desires to **control the variation in an experiment** that is related to rows and columns in the field.

**Field marks:**

- Treatments are assigned
**at random**within**rows**and**columns**, with each treatment**once per row**and**once per column**. - There are
**equal numbers of rows, columns**, and treatments. - Useful where the experimenter desires to control
**variation in two different directions**

This is just one of many 4×4 squares that you could create. In fact, you can **make any size square you want**, for any number of treatments – it just needs to have the following property associated with it – that **each treatment occurs only once in each row and once in each column**.

Note that **a Latin Square is an incomplete design**, which means that it does not include observations for all possible combinations of *i*, *j* and *k*. This is why we use notation *k* = *d*(*i, j*). **Once we know the row and column of the design, then the treatment is specified**. In other words, if we know *i* and *j*, then *k* is specified by the Latin Square design.

This property has an impact on how we calculate **means** and **sums of squares**, and for this reason **we can not use the balanced ANOVA** command in Minitab even though it looks perfectly balanced. We will see later that although it has the property of orthogonality, you still cannot use the balanced ANOVA command in Minitab because it is not complete.

The randomization procedure for assigning treatments that you would like to use when you actually apply a Latin Square, is somewhat restricted to **preserve the structure of the Latin Square.** The ideal randomization would be to select a square from the set of all possible Latin squares of the specified size. However, a more practical randomization scheme would be to** select a standardized Latin square at random** (these are tabulated) and then:

- randomly permute the columns,
- randomly permute the rows, and then
- assign the treatments to the Latin letters in a random fashion.

via https://onlinecourses.science.psu.edu/stat503/node/21