The Latin Square Designs is an effective tool that can simultaneously handle two sources of variation among the treatments, which are treated as two independent blocking criteria. These blocks are known as row-block and column-block, also called double-block. Both sources of variations (blocks) are perpendicular to each other. Latin Square Designs are used to simultaneously eliminate (or control) the two sources of nuisance variability (Rows and Columns).
Table of Contents
Introduction
In a Latin square, treatments are arranged in a square matrix such that each treatment appears exactly once in each row and once in each column. This structure helps mitigate the influence of extraneous variables, allowing researchers to focus on the effects of the treatments themselves.
Latin square designs are widely used in agriculture (field experiments), psychology, and many fields where controlled experiments are necessary. The Latin Square Designs are applied in field trials, where
- the experimental area has two fertility gradients running perpendicular to each other
- in the greenhouse experiments in which the experimental pots are arranged in straight lines perpendicular to the sheets or walls of the greenhouse such that the difference between rows and the distance from the wall is expected to be two major extraneous sources of variation,
- in laboratory experiments where the trials are replicated over time such that the difference between the experimental units conducted at the same time and those conducted over different time period constitute the two known sources of variations
| Rows of Tree | |||
| Water Channel | A | B | C |
| B | C | A | |
| C | A | B | |
Key Features of Latin Square Designs
The Latin square designs have the following key features:
- Control for Two Variables: The design simultaneously accounts for variability in two factors (e.g., time and location).
- Efficient Use of Resources: These designs allow for the evaluation of multiple treatments without requiring a full factorial design, which can be resource-intensive.
- Simple Analysis: The data collected can be analyzed using standard statistical techniques such as ANOVA.
Randomization and Layout Plan for Latin Square Designs
Suppose, there are five treatments (A, B, C, D, E) for this we need $5 \times 5$ LS-Designs, which means we should layout the experiment with five rows and five columns:
| A | B | C | D | E |
| B | C | D | E | A |
| C | D | E | A | B |
| D | E | A | B | C |
| E | A | B | C | D |
First of all, randomize the row arrangement by using random numbers then randomize the column arrangement by using random numbers. One can generate five random numbers on your calculator or computer. For example,
| Random Numbers | Sequence | Rank |
|---|---|---|
| 628 | 1 | 3 |
| 846 | 2 | 4 |
| 475 | 3 | 2 |
| 902 | 4 | 5 |
| 452 | 5 | 1 |
The first rank is 3, treatment c is allocated to cell-1 in column-1, then treatment D is allocated to cell-2 of column-1, and so on.
| C | D | A | E | B |
| D | E | B | A | C |
| B | A | E | C | D |
| E | C | D | B | A |
| A | B | C | D | E |
Now, generate random numbers for the columns
| Random Numbers | Sequence | Rank |
|---|---|---|
| 792 | 1 | 4 |
| 032 | 2 | 1 |
| 947 | 3 | 5 |
| 293 | 4 | 3 |
| 196 | 5 | 2 |
For the layout of LS-Designs, the 4th column from the first random generation is used as the 1st column of LS-Designs, then the 1st column as the 2nd of LS-Design, and so on. The complete Design is:
ANOVA Table for Latin Square Designs
For a statistical analysis, the ANOVA table for LS-Designs is used given as follows:
| SOV | df | SS | MS | Fcal | F tab/P-value |
|---|---|---|---|---|---|
| Rows | $r-1 = 4$ | ||||
| Columns | $c-1 = 4$ | ||||
| Treatments | $t-1 = 4$ | ||||
| Error | $12$ | ||||
| Total | $rc-1 = 24$ |
Example: An experiment was conducted with three maize varieties and a check variety, the experiment was laid out under Latin Square Designs, Analyse the data given below
| $C$-1 | $C$-2 | $C$-3 | $C$-4 | $Total$ | |
| $R$-1 | 1640(B) | 1210(D) | 1425(C) | 1345(A) | |
| $R$-2 | 1475(C) | 1185(A) | 1400(D) | 1290(B) | |
| $R$-3 | 1670(A) | 710(C) | 1665(B) | 1180(D) | |
| $R$-4 | 1565(D) | 1290(B) | 1655(A) | 660(C) | |
| $Total$ |
Solution:
| A | B | C | D |
| 1670 | 1640 | 1475 | 1565 |
| 1185 | 1290 | 710 | 1210 |
| 1655 | 1665 | 1425 | 1400 |
| 1345 | 1290 | 660 | 1180 |
The following formulas may be used for the computation of Latin Square Design’s ANOVA Table.
\begin{align*}
CF &= \frac{GT^2}{N}\\
SS_{Total} &= \sum\limits_{j=1}^t \sum\limits_{i=1}^r y_{ij}^2 -CF\\
SS_{Treat} &= \frac{\sum\limits_{j=1}}{r} r_j^2 – CF\\
SS_{Rows} &= \frac{\sum\limits_{r=1}^r R_i^2}{t} – CF\\
SS_{Col} &= \frac{\sum\limits_{r=1}^b c_j^2}{t} – CF\\
SS_{Error} &=SS_{Total} – SS_{Treat} – SS_{Rows} – SS_{Col}
\end{align*}
| SOV | df | SS | MS | Fcal | F tab (5%) | F tab (1%) |
|---|---|---|---|---|---|---|
| Rows | 3 | 30154.69 | 10051.56 | 0.465NS | 4.7571 | 9.7795 |
| Columns | 3 | 827342.19 | 275780.73 | 12.769** | 4.7571 | 9.7795 |
| Treatments | 3 | 426842.19 | 142280.73 | 6.588* | 4.7571 | 9.7795 |
| Error | 6 | 129584.38 | 21597.40 | |||
| Total | 15 | 1413923.44 |
In summary, the Latin square design is an effective tool for researchers looking to control for variability and conduct efficient, straightforward analyses in their experiments.
Learn about the Introduction of Design of Experiments
