Lecture 26: Latin Square Design
Latin Square Design
Example: Greenhouse
We have seen how to use
blocking to account for
variability from a nuisance
factor.
What if we have two blocking
variables?
Less light
Heater
Warmer
Cooler
Doors
More light
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Windows
Nuisance Factors
Latin Square Design
There is variability in warmth
from side to side.
There is variability in light
from top to bottom.
Need a design that can account
for both of these nuisance
factors.
One set of treatments (these
could be levels of a single
factor or combinations
made by crossing several
factors of interest).
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Latin Square Design
Latin Square Design
Two blocking (nuisance)
factors.
The number of levels for each
blocking factor is determined
by the number of treatments.
Every combination of the
two blocking factors occurs
once. Treatments are
crossed with each blocking
factor.
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Lecture 26: Latin Square Design
Latin Square Design
3 Treatments
A Latin Square plan (or
layout) starts with a systematic
arrangement of the treatments
in rows (levels of one blocking
variable) and columns (levels
of the second blocking
variable).
A
B
C
B
C
A
C
A
B
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4 Treatments
A
B
C
D
B
C
D
A
C
D
A
B
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Comment
Note that in a Latin Square
plan that each treatment
appears once in each row
and once in each column.
D
A
B
C
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Comment
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Example: Movie Appeal
Response: How many people,
out of 50, would recommend the
movie to a friend.
Conditions: Four types of
movies; Action, Sci Fi, Comedy
or Drama.
Experimental Units: Audiences
of 50 people.
A Latin Square plan is a
systematic arrangement of
treatments. How does one
include randomization?
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Lecture 26: Latin Square Design
Blocking (Nuisance) Factors
First Blocking Factor
The people that attend
movies may be different
depending on the day of the
week or the time of day.
Day of the week. Because
there are four movies (levels
of the factor of interest) four
days of the week; M, T, W, Th
make up one blocking factor.
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Second Blocking Factor
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Randomization
Select a plan at random
from all possible plans (not
practical).
Randomize order of rows,
columns and treatments.
Time of day. Use four
times of day; Early
Matinee, Late Matinee,
Early Evening, Late
Evening.
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Start with systematic plan
1
2
3
4
1
A
B
C
D
2
B
C
D
A
3
C
D
A
B
4
D
A
B
C
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Randomize Columns
Use JMP to randomly
shuffle the numbers from 1
to 4.
Random order: 2, 3, 4, 1
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Lecture 26: Latin Square Design
Randomize Columns
1
2
3
4
Monday
2
Tuesday
3
B
C
D
A
C
D
A
B
Randomize Rows
Wednesday Thursday
4
1
D
A
B
C
Use JMP to randomly
shuffle the numbers from 1
to 4.
Random order: 4, 2, 1, 3
A
B
C
D
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Randomize Rows
EM
4
LM
2
EE
1
LE
3
Monday
Tuesday
C
A
B
D
D
B
C
A
Randomly Assign Treatments
Wednesday Thursday
A
C
D
B
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B
D
A
C
Use JMP to randomly
shuffle the numbers from 1
to 4.
Random order: 1, 3, 2, 4
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Comment
Randomly Assign Treatments
Monday
Tuesday
EM
C
Trmt 2
D
Trmt 4
A
Trmt 1
B
Trmt 3
LM
A
Trmt 1
B
Trmt 3
C
Trmt 2
D
Trmt 4
EE
B
Trmt 3
C
Trmt 2
D
Trmt 4
A
Trmt 1
LE
D
Trmt 4
A
Trmt 1
B
Trmt 3
C
Trmt 2
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Wednesday Thursday
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After randomizing rows,
columns and treatment
assignments the randomized
plan still has the characteristic
that each treatment appears
once in each row and once in
each column.
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