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) - PowerPoint PPT PresentationTRANSCRIPT
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 Day Chemical 1 2 3 4 5 6 7 Control A
19.8 G
16.8 F
16.7 E
15.8 D
17.3 C
18.1 B
18.0 Piperine B
13.0 A
5.3 G
14.0 F
7.2 E
14.1 D
10.8 C
14.7 Black Pepper C
13.0 B
11.0 A
12.3 G 8.6
F 14.5
E 15.8
D 12.7
Lemon Juice D 7.8
C 6.0
B 5.3
A 6.0
G 8.3
F 5.8
E 6.5
Hesperidin E 13.6
D 16.0
C 10.7
B 10.0
A 16.2
G 14.3
F 14.2
Ascorbic Acid F 15.0
E 12.2
D 11.7
C 12.2
B 13.2
A 16.0
G 11.8
Citric Acid G 14.5
F 14.7
E 11.0
D 11.2
C 9.5
B 17.2
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)
22
2
2
2
0:
0:
io
io
c
nLLH
nH
Power Analysis in Latin Squares
For replicated squares, the denominator df depends on the method of replication; see Montgomery
22
2
2
2
0:
0:
io
io
c
snLLH
snH
Suppose we have a Latin Square Design with a third blocking variable (indicated by font color):
A B C DB C D AC D A BD A B C
Graeco-Latin Square Design
Graeco-Latin Square Design
Suppose we have a Latin Square Design with a third blocking variable (indicated by font style):
A B C DB C D AC D A BD 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 DB A D CC D A BD C B A
Graeco-Latin Square Design
A B C D B A D CC D A BD C B A
Graeco-Latin Square Design
Column1 2 3 4
1 Aa Bb Cg DdRow 2 Bd Ag Db Ca
3 Cb Da Ad Bg4 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…