8 Latin Square Design (LSD)

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Design and Analysis of Agricultural Experiments - Dr. Awadallah Belal Dafaallah
Design and Analysis of
Single Factor
Experiments

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Design and Analysis of Single Factor
Experiments:
A single factor experiment is a simple
experiment that consists of limited number of
treatment belonging to a certain factor.

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Examples of single factors:
 Different varieties of a crop:
 Different levels of a pesticide.
 Different genotypes of animals.
 Different strains of bacteria or fungi.
 Different types of harvesters or tractors.

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Latin Square Design
(LSD)

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Latin square design is one of most
experimental design used in agricultural research,
particularly in field experiments.

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Experimental units are heterogeneous and the
heterogeneity is uniform and in two directions.
Experimental units are grouped or randomized
into blocks called rows and again vertically in
blocks called columns.
Rows are randomized separately and
independently.
columns are randomized separately and
independently.

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Each treatment is found once in each row and
once in each column.
The number of treatment equal number of rows
equal number of columns equal number of
replicates. T=R=C=r=T
The total number of experimental units=
T*T or R*R or C*C or r*r = r 2.
Treatments are represented by Latin letters (A,
B, …Z).
So, this design is called LSD.

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Total variation
Treatment
variation
Row
variation
Column
variation
Experimental
error variation

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Advantages of LSD:
 Simple to use.
 Easy statistical analysis.
 More efficient than RCBD.

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Disadvantages of LSD:
 It has to be used when the number of
treatments ranged between 4 and 8.

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Uses of LSD:
 Greenhouse experiments.
 Field experiments.

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Mathematical model of LSD:
Where:
Character measured
Grand mean
Row effect
Column effect
Treatment effect
Experimental error effect

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An example:
Research problem:
 An agricultural engineer wanted to study the
efficiency of four combine harvesters of wheat
on header losses in the field in LSD.

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 At harvest, the engineer thought that the speed
of the harvester (km/h) and the speed of the reel
(r/min) might affect the header losses (fed).
 So, the experimental units were divided into
row to correct the speed of the harvester and
into column to correct the speed of the reel.

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Objective of the experiment:
Efficiency of four combine harvesters of wheat on
header losses.

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Hypothesis:
Null hypothesis: There are no significant differences
between the four combine harvesters of wheat on header
losses.
Alternative hypothesis: There are significant differences
between the four combine harvesters of wheat on header
losses.

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Randomization and layout of experiment:

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Data collection and organization
Table of rows and column
Column (harvester speed) Total of
rows
5 km/h 6 km/h 7 km/h 8 km/h
Row
(Reel
speed)
5 r/min 6
A= 2
C= 3
D= 4
B= 15
10
r/min
2
D= 3
B= 4
C= 6
A= 15
15
r/min
3
B= 4
D= 7
A= 6
C= 20
20
r/min
4
C= 7
A= 6
B= 7
D= 24
Total of columns 15 16 20 23 74

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Table of treatments
Column (harvester speed)
Treatment
total
Treatment
mean
5
km/h
6
km/h
7
km/h
8
km/h
Harvester 1 6 7 7 6 26 6.5
Harvester 2 3 3 6 4 16 4
Harvester 3 4 2 4 6 16 4
Harvester 4 2 4 3 7 16 4
Grand total 74
4.625

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Testing hypothesis: Steps for analysis of
variance:
1) Sum of squares
2) Degree of freedom
3) Mean squares
4) F values
5) Analysis of variance Table (ANOVA table)
6) Testing significance
7) General conclusion
8) Summary of results

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1) Sum squares:
Before calculation of sum of squares, the
correction factor is, firstly, calculated as follow:

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Sum of squares for total:

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Sum of squares for rows:

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Sum of squares for columns:

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Sum of squares for treatments:

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Sum of squares for error:

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2) Degree of freedom:
Degree of freedom for total:
Degree of freedom for rows:
Degree of freedom for columns:

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Degree of freedom for treatments:
Degree of freedom for error:

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3) Mean squares:
Mean squares for rows:
Mean squares for columns:

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Mean squares for treatments:
Mean squares for error:

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4) F values
Calculated F value for rows
Calculated F value for columns
Calculated F value for treatments

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Tabulated F value
Tabulated F value for rows, columns and
treatments

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5) ANOVA table:
SOV SS DF MS F-cal
F-tab
0.05 0.01
Rows 14.25 3 4.75 6.333 4.76 9.78
Columns 10.25 3 3.417 4.556Ns 4.76 9.78
Treatments 18.75 3 6.25 8.333 4.76 9.78
Experimental
error
4.5 6 0.75
Total 47.75 15

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 Since the calculated f value for rows (6.333) is
greater than tabulated f value for rows at 0.05
(4.76) level of significance, null hypothesis is
rejected.

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 Since the calculated f value for columns
(4.556) is smaller than tabulated f value for
columns at 0.05 (4.76) level of significance,
null hypothesis is accepted.

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 Since the calculated f value for treatments
(8.333) is greater than tabulated f value for
treatments at 0.05 (4.76) level of significance,
null hypothesis is rejected.

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7) General conclusion:
 Rows is efficient in removing the variation in
the experimental units due to harvester speed.
 Column is not efficient in removing the
variation in the experimental units due to reel
speed.

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 There are significant differences between the
four combine harvesters of wheat on header
losses.

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8) Summary of results:
Standard error (SE±) =
Coefficient of variation (CV%) =

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Table. 1. Effect of four combine harvesters of
wheat on header losses
Treatments Harvester 1 Harvester 2 Harvester 3 Harvester 4
Mean of
treatment
6.5 4.0 4.0 4.0
SE± 0.433
CV % 18.725

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Fig. 1. Effect of four combine harvesters of wheat
on header losses
0
1
2
3
4
5
6
7
Harvester
1
Harvester
2
Harvester
3
Harvester
4
Loss
(kg/fed)
Type of combine harvester

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Reference:
Dafaallah, A. B. (2017).Fundamentals of Design and
Analysis of Agricultural. Experiments
(Observation – Experimentation –Discussion), Part
One. First Edition. University of Gezira House for
Printing and Publishing, Wad Medani, Sudan. Pp
246.
Dafaallah, A. B. (2017). Fundamentals of Design and
Analysis of Agricultural Experiments (Observation
– Experimentation –Discussion), Part Two. First
Edition. University of Gezira House for Printing
and Publishing , Wad Medani, Sudan. Pp 204.

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Thanks
Dr. Awadallah Belal Dafaallah
E-mail: awadna@hotmail.com;
awadna@uofg.edu.sd
Tel: +249902295166

8 Latin Square Design (LSD)

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