STAM101 :: Lecture 17 :: Latin square design – description – layout – analysis – advantages and disadvantages
Latin Square Design
When the experimental material is divided into rows and columns and the treatments are allocated such that each treatment occurs only once in each row and each column, the design is known as L S D.
In LSD the treatments are usually denoted by A B C D etc.
For a 5 x 5 LSD the arrangements may be
A 
B 
C 
D 
E 
B 
A 
E 
C 
D 
C 
D 
A 
E 
B 
D 
E 
B 
A 
C 
E 
C 
D 
B 
A 
Square 1 
B 
C 
D 
E 

B 
A 
D 
E 
C 
C 
E 
A 
B 
D 
D 
C 
E 
A 
B 
E 
D 
B 
C 
A 
Square 2 
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 
Square 3 
Analysis
The ANOVA model for LSD is
Yijk = µ + ri + cj + tk + eijk
ri is the ith row effect
cj is the jth column effect
tk is the kth treatment effect and
eijk is the error term
The analysis of variance table for LSD is as follows:
Sources of Variation 
d.f. 
S S 
M S 
F 
Rows 
t1 
RSS 
RMS 
RMS/EMS 
Columns 
t1 
CSS 
CMS 
CMS/EMS 
Treatments 
t1 
TrSS 
TrMS 
TrMS/EMS 
Error 
(t1)(t2) 
ESS 
EMS 

Total 
t21 
TSS 


F table value
F [t1),(t1)(t2)] degrees of freedom at 5% or 1% level of significance
Steps to calculate the above Sum of Squares are as follows:
Correction Factor
Total Sum of Squares
Row sum of squares
Column sum of squares
Treatment sum of squares
Error Sum of Squares = TSSRSSCSSTrSS
These results can be summarized in the form of analysis of variance table.
Calculation of SE, SE (d) and CD values
where r is the number of rows
.
CD= SE (d). t
where t = table value of t for a specified level of significance and error degrees of freedom
Using CD value the bar chart can be drawn and the conclusion may be written.
Advantages
 LSD is more efficient than RBD or CRD. This is because of double grouping that will result in small experimental error.
 When missing values are present, missing plot technique can be used and analysed.
 This design is not as flexible as RBD or CRD as the number of treatments is limited to the number of rows and columns. LSD is seldom used when the number of treatments is more than 12. LSD is not suitable for treatments less than five.
Because of the limitations on the number of treatments, LSD is not widely used in agricultural experiments.
Note: The number of sources of variation is two for CRD, three for RBD and four for LSD.
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