latin square design

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1) What is a latin square?
2) What is a latin square design?
3) Orthogonal latin squares
4) Procedure to create a latin square design
5) The model for a latin experiment
6) Graeco-latin square designs
7) The model for a greaco-latin experiment
8) Advantages of latin square design
9) Disadvantages of latin square design

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WHAT IS A LATIN SQUARE?
A Latin square is an table filled with n × n different
symbols in such a way that each symbol occurs exactly
once in each row and exactly once in each column. Here
are few examples
a b
b a
a b c
b c a
c a b
a b c d
b c d a
c d a b
d a b c

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WHAT IS A LATIN SQUARE
DESIGN?
A Latin square design is a method of placing treatments so
that they appear in a balanced fashion within a square
block or field. Treatments appear once in each row and
column.
 Treatments are assigned at random within rows and
columns, with each treatment once per row and once
per column.
 There are equal numbers of rows, columns, and
treatments.

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ORTHOGONAL LATIN
SQUARES
Two n  n Latin squares L=[lij] and M =[mij] are orthogonal if the n2
pairs (lij, mij) are all different
a b c
b c a
c a b
a b c
c a b
b c a
aa bb cc
bc ca ab
cb ac ba

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PROCEDURE TO CREATE A
LATIN SQUARE DESIGN
An appropriate randomization strategy is as follows:
1) Write down any Latin square of the required size (it
could be a standard Latin square).
2) Randomize the order of the rows.
3) Randomize the order of the columns.
4)Randomize the allocation of treatments to the letters
of the square.

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EXAMPLE
In this Experiment the we are interested in how
weight gain (Y) in rats is affected by Source of
protein (Beef, Cereal, and Pork) and by Level of
Protein (High or Low).
There are a total of t = 3 X 2 = 6 treatment
combinations of the two factors.
• Beef -High Protein
• Cereal-High Protein
• Pork-High Protein
• Beef -Low Protein
• Cereal-Low Protein and
• Pork-Low Protein

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IN THIS EXAMPLE WE WILL CONSIDER USING
A LATIN SQUARE DESIGN
Six Initial Weight categories are identified for the
test animals in addition to Six Appetite categories.
• A test animal is then selected from each of the 6 X
6 = 36 combinations of Initial Weight and
Appetite categories.
• A Latin square is then used to assign the 6 diets to
the 36 test animals in the study.

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In the latin square the letter
• A represents the high protein-cereal diet
• B represents the high protein-pork diet
• C represents the low protein-beef Diet
• D represents the low protein-cereal diet
• E represents the low protein-pork diet and
• F represents the high protein-beef diet.

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The weight gain after a fixed period is measured for
each of the test animals and is tabulated below:
Appetite Category
1 2 3 4 5 6
1 62.1 84.3 61.5 66.3 73.0 104.7
A B C D E F
2 86.2 91.9 69.2 64.5 80.8 83.9
B F D C A E
Initial 3 63.9 71.1 69.6 90.4 100.7 93.2
Weight C D E F B A
Category 4 68.9 77.2 97.3 72.1 81.7 114.7
D A F E C B
5 73.8 73.3 78.6 101.9 111.5 95.3
E C A B F D
6 101.8 83.8 110.6 87.9 93.5 103.8
F E B A D C

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THE LATIN SQUARE DESIGN
Columns
Rows
1
2
2 3
t
3
⁞
t
3 1
1
2
All treats appear once in each row and
each column

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The Model for a Latin Experiment
ij k  k i j ij k  y       
i = 1,2,…, t j = 1,2,…, t
yij(k) = the observation in ith row and the jth
column receiving the kth treatment
 = overall mean
k = the effect of the ith treatment
i
= the effect of the ith row
ij(k) = random error
k = 1,2,…, t
j= the effect of the jth column
No interaction
between rows,
columns and
treatments

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GRAECO-LATIN SQUARE
DESIGNS
Mutually orthogonal Squares

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DEFINITION
A Greaco-Latin square consists of two latin squares (one
using the letters A, B, C, … the other using greek letters a,
b, c, …) such that when the two latin square are supper
imposed on each other the letters of one square appear
once and only once with the letters of the other square.
The two Latin squares are called mutually orthogonal.
Example: a 7 x 7 Greaco-Latin Square
Aa B Cb Df Ec F Gd
Bb Cf Dc E Fd Ga A
Cc D Ed Fa G Ab Bf
Dd Ea F Gb Af Bc C
E Fb Gf Ac B Cd Da
Ff Gc A Bd Ca D Eb
G Ad Ba C Db Ef Fc

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NOTE:
There exists at most (t –1) t x t Latin squares L1, L2,
…, Lt-1 such that any pair are mutually
orthogonal.
e.g. It is possible that there exists a set of six 7 x 7
mutually orthogonal Latin squares L1, L2, L3, L4, L5,
L6 .

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THE GREACO-LATIN SQUARE DESIGN - AN
EXAMPLE
A researcher is interested in determining the
effect of two factors
1) The percentage of lysine in the diet
and
2) Percentage of protein in the diet

18. THE MODEL FOR A GREACO-LATIN EXPERIMENT
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ij kl  k l i j ij kl  y        
i = 1,2,…, t
j = 1,2,…, t
yij(kl) = The observation in ith row and the jth column receiving the
kth latin treatment and the lth greek treatment
 = Overall mean
k = The effect of the kth latin treatment
l
= The effect of the lth greek treatment
i
= The effect of the ith row
j= The effect of the jth column
ij(k) = Random error
k = 1,2,…, t
l = 1,2,…, t

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• A Greaco-Latin Square experiment is assumed to
be a four-factor experiment.
• The factors are rows, columns, Latin treatments
and Greek treatments.
• It is assumed that there is no interaction between
rows, columns, Latin treatments and Greek
treatments.
• The degrees of freedom for the interactions is
used to estimate error.

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The Anova Table for a
Greaco-Latin Square Experiment
Source S.S. d.f. M.S. F p-value
Latin SSLa t-1 MSLa MSLa /MSE
Greek SSGr t-1 MSGr MSGr /MSE
Rows SSRow t-1 MSRow MSRow /MSE
Cols SSCol t-1 MSCol MSCol /MSE
Error SSE (t-1)(t-3) MSE
Total SST t2 - 1

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Controls more variation than CR or RCB designs
because of 2-way stratification. Results in a
smaller mean square for error.
Simple analysis of data
Analysis is simple even with missing plots.

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Number of treatments is limited to the number of
replicates which seldom exceeds 10.
If have less than 5 treatments, the df for
controlling random variation is relatively large and
the df for error is small.

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