Latin Square Design

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Latin Square Design
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

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

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) not
avoiding the chemical
Latin Square Design
Chemical
Control
Piperine
Black Pepper
Lemon Juice
Hesperidin
Ascorbic Acid
Citric Acid
1
A
19.8
B
13.0
C
13.0
D
7.8
E
13.6
F
15.0
G
14.5
2
G
16.8
A
5.3
B
11.0
C
6.0
D
16.0
E
12.2
F
14.7
3
F
16.7
G
14.0
A
12.3
B
5.3
C
10.7
D
11.7
E
11.0
Day
4
E
15.8
F
7.2
G
8.6
A
6.0
B
10.0
C
12.2
D
11.2
5
D
17.3
E
14.1
F
14.5
G
8.3
A
16.2
B
13.2
C
9.5
6
C
18.1
D
10.8
E
15.8
F
5.8
G
14.3
A
16.0
B
17.2
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
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

Ho : L  0  
n
2
i
2
nL2

2
c
2
i
Power Analysis in Latin Squares

For replicated squares, the denominator df
depends on the method of replication; see
Montgomery
H o :  0

Ho : L  0  
sn 

2
i
2
2
snL
 2  ci2
Graeco-Latin Square Design

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

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
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
1
Row 2
3
4
Column
1
2
3
4
Aa Bb Cg Dd
Bd Ag Db Ca
Cb Da Ad Bg
Dg Cd Ba Ab
Graeco-Latin Square Design
Orthogonal designs do not exist for n=6
 Randomization

–
–
–
–
–
Standard square
Rows
Columns
Latin letters
Greek letters
Graeco-Latin Square Design
Total df is n2-1=(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

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…
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