Latin Square Design (LSD)

In Latin Square Design the treatments are grouped into replicates in two different ways, such that each row and each column is a complete block, and the grouping for balanced arrangement is performed by imposing the restriction that each of the treatments must appear once and only once in each of the rows and only once in each of the column. The experimental material should be arranged and the experiment conducted in such a way that the differences among the rows and columns represent a major source of variation.

Hence a Latin Square Design is an arrangement of k treatments in a $k\times k$ squares, where the treatments are grouped in blocks in two directions. It should be noted that in a Latin Square Design the number of rows, the number of columns, and the number of treatments must be equal.

In other words unlike Randomized Completely Block Design (RCBD) and Completely Randomized Design (CRD) a Latin Square Design is a two-restriction design, which provides the facility of two blocking factors that are used to control the effect of two variable that influences the response variable. Latin Square Design is called Latin Square because each Latin letter represents the treatment that occurs once in a row and once in a column in such a way that in respect of one criterion (restriction), rows are completely homogeneous blocks, and in respect of another criterion (second restriction) columns are completely homogeneous blocks.

The application of Latin Square Design is mostly in animal science, agriculture, industrial research, etc. A daily life example can be a simple game called Sudoku puzzle is also a special case of Latin square design. The main assumption is that there is no contact between treatments, rows, and columns effect.

The general model is defined as
\[Y_{ijk}=\mu+\alpha_i+\beta_j+\tau_k +\varepsilon_{ijk}\]

where $i=1,2,\cdots,t; j=1,2,\cdots,t$ and $k=1,2,\cdots,t$ with $t$ treatments, $t$ rows and $t$ columns,
$\mu$ is the overall mean (general mean) based on all of the observations,
$\alpha_i$ is the effect of the $i$th row,
$\beta_j$ is the effect of $j$th rows,
$\tau_k$ is the effect of the $k$th column.
$\varepsilon_{ijk}$ is the corresponding error term which is assumed to be independent and normally distributed with mean zero and constant variance i.e $\varepsilon_{ijk}\sim N(0, \sigma^2)$.

Latin Square Design Experimental Layout

Suppose we have 4 treatments (namely: $A, B, C$, and $D$), then it means that we have

Number of Treatments = Number of Rows = Number of Columns =4

The Latin Square Design’s Layout can be for example

A $Y_{111}$	B $Y_{122}$	C $Y_{133}$	D $Y_{144}$
B $Y_{212}$	C $Y_{223}$	D $Y_{234}$	A $Y_{241}$
C $Y_{313}$	D $Y_{324}$	A $Y_{331}$	B $Y_{342}$
D $Y_{414}$	A $Y_{421}$	B $Y_{432}$	C $Y_{443}$

The number in subscript represents a row, block, and treatment number respectively. For example, $Y_{421}$ means the first treatment in the 4th row, the second block (column).