Latin Square Design
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Traditionally, latin squares have two blocks, 1 treatment, all of size n
Yandell introduces latin squares as an incomplete factorial design instead
– Though his example seems to have at least one block
(batch)
Latin squares have recently shown up as parsimonious factorial designs for simulation studies

Latin Square Design
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Student project example
– 4 drivers, 4 times, 4 routes
– Y=elapsed time
Latin Square structure can be natural
(observer can only be in 1 place at 1 time)
Observer, place and time are natural blocks for a Latin Square

Latin Square Design
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Example
– Region II Science Fair years ago (7 by 7 design)
– Row factor—Chemical
– Column factor—Day (Block?)
– Treatment—Fly Group (Block?)
– Response—Number of flies (out of 20) avoiding the chemical not

Latin Square Design
Chemical
Control
Piperine
Black Pepper
Lemon Juice
Hesperidin
Ascorbic Acid
Citric Acid
2
G
16.8
A
5.3
B
11.0
C
6.0
D
16.0
E
12.2
F
14.7
1
A
19.8
B
13.0
C
13.0
D
7.8
E
13.6
F
15.0
G
14.5
Day
4
E
15.8
F
7.2
G
8.6
A
6.0
B
10.0
C
12.2
D
11.2
3
F
16.7
G
14.0
A
12.3
B
5.3
C
10.7
D
11.7
E
11.0
6
C
18.1
D
10.8
E
15.8
F
5.8
G
14.3
A
16.0
B
17.2
5
D
17.3
E
14.1
F
14.5
G
8.3
A
16.2
B
13.2
C
9.5
7
B
18.0
C
14.7
D
12.7
E
6.5
F
14.2
G
11.8
A
15.7

Power Analysis in Latin Squares
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For unreplicated squares, we increase power by increasing n (which may not be practical)
The denominator df is (n-2)(n-1)
H o
:
 
0
H o
: L

0




 n
2


2 nL
2

 i
2 c i
2

Power Analysis in Latin Squares
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For replicated squares, the denominator df depends on the method of replication; see
Montgomery
H o
:
 
0
H o
: L

0




 sn
2


2 snL
2

 c i
2 i
2

Graeco-Latin Square Design
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Suppose we have a Latin Square Design with a third blocking variable (indicated by font color):
A B C D
B C D A
C D A B
D A B C

Graeco-Latin Square Design
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Suppose we have a Latin Square Design with a third blocking variable (indicated by font style):
A B
C
D
B C D
A
C D A B
D A B C

Graeco-Latin Square Design
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Is the third blocking variable orthogonal to the treatment and blocks?
How do we account for the third blocking factor?
We will use Greek letters to denote a third blocking variable

Graeco-Latin Square Design
A B C D
B A D C
C D A B
D C B A

Graeco-Latin Square Design
A B C
D
B
A D C
C D A
B
D
C B A

Graeco-Latin Square Design
Column
1 2 3 4
1 A a B b C g D d
Row 2 B d A g D b C a
3 C b D a A d B g
4 D g C d B a A b

Graeco-Latin Square Design
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Orthogonal designs do not exist for n=6
Randomization
– Standard square
– Rows
– Columns
– Latin letters
– Greek letters

Graeco-Latin Square Design
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Total df is n 2 -1=(n-1)(n+1)
Maximum number of blocks is n-1
– n-1 df for Treatment
– n-1 df for each of n-1 blocks--(n-1) 2 df
– n-1 df for error
Hypersquares (# of blocks > 3) are used for screening designs

Conclusions
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We will explore some interesting extensions of Latin Squares in the text’s last chapter
– Replicated Latin Squares
– Crossover Designs
– Residual Effects in Crossover designs
But first we need to learn some more about blocking…