Single Factor Experiments

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).

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

MCQs General Knowledge

The article is about one way Analysis of Variance. In the analysis of variance, the total variation in the data of the sample is split up into meaningful components that measure different sources of variation. Each component yields an estimate of the population variance, and these estimates are tested for homogeneity by using the F-distribution.

One Way Classification (Single Factor Experiments)

The classification of observations based on a single criterion or factor is called a one-way classification.

In single factor experiments, independent samples are selected from $k$ populations, each with $n$ observations. For samples, the word treatment is used and each treatment has $n$ repetitions or replications. By treatment, we mean the fertilizers applied to the fields, the varieties of a crop sown, or the temperature and humidity to which an item is subjected in a production process. The collected data consisting of $kn$ observations ($k$ samples of $n$ observations each) can be presented as.

where

$X_{ij}$ is the $i$th observation receiving the $j$th treatment

$X_{\cdot j}=\sum\limits_{i=1}^n X_{ij}$ is the total observations receiving the $j$th treatment

$\overline{X}_{\cdot j}=\frac{X_{\cdot j}}{n}$ is the mean of the observations receiving the $j$th treatment

$X_{\cdot \cdot}=\sum\limits_{i=j}^n X_{\cdot j} = \sum\limits_{j=1}^k \sum\limits_{i=1}^n X_{ij}$ is the total of all observations

$\overline{\overline{X}} = \frac{X_{\cdot \cdot}}{kn}$ is the mean of all observations.

The $k$ treatments are assumed to be homogeneous, and the random samples taken from the same parent population are approximately normal with mean $\mu$ and variance $\sigma^2$.

One Way Analysis of Variance Model

The linear model on which the one way analysis of variance is based is

$$X_{ij} = \mu + \alpha_j + e_{ij}, \quad\quad i=1,2,\cdots, n; \quad j=1,2,\cdots, k$$

Where $X_{ij}$ is the $i$th observation in the $j$th treatment, $\mu$ is the overall mean for all treatments, $\alpha_j$ is the effect of the $j$th treatment, and $e_{ij}$ is the random error associated with the $i$th observation in the $j$th treatment.

The One Way Analysis of Variance model is based on the following assumptions:

• The model assumes that each observation $X_{ij}$ is the sum of three linear components
  • The true mean effect $\mu$
  • The true effect of the $j$th treatment $\alpha_j$
  • The random error associated with the $j$th observation $e_{ij}$
• The observations to which the $k$ treatments are applied are homogeneous.
• Each of the $k$ samples is selected randomly and independently from a normal population with mean $\mu$ and variance $\sigma^2_e$.
• The random error $e_{ij}$ is a normally distributed random variable with $E(e_{ij})=0$ and $Var(e_{ij})=\sigma^2_{ij}$.
• The sum of all $k$ treatments effects must be zero $(\sum\limits_{j=1}^k \alpha_j =0)$.

Suppose you are comparing crop yields that were fertilized with different mixtures. The yield (numerical) is the dependent variable, and fertilizer type (categorical with 3 levels) is the independent variable. ANOVA helps you determine if the fertilizer mixtures have a statistically significant effect on the average yield.

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