Latin Square Design

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Latin Square Design
Row Col Trt Row Col Trt 1 1 A 2 1
C 3 1 D 4 1 B 1 2 D 2
2 B 3 2 C 4 2 A 1 3 B 2
3 D 3 3 A 4 3 C 1 4 C
2 4 A 3 4 B 4 4 D
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The Latin Squares model
Yijk m ai tj gk eijk m overall
mean ai ith treatment effect i 1,, L tj
jth row block effect j 1,, L gk kth column
block effect k 1,, L eijk random error term
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ANOVA Source df Model 3(L-1) Error
L(L-3)2 Total L2-1
L gt 2 (Firm mathematical necessity) L lt 11
(Approximate practicality)
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Graeco-Latin Square (extension of Latin Square)
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A university, while developing a retraining
program to teach general computer repair skills
to persons displaced from their previous
occupations, conducted an experiment to evaluate
three different incentive methods on achievement
during the program. The factors age and IQ were
thought to be important blocking factors.
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The Latin Squares model
Yijk m ai tj gk eijk m overall
mean ai ith method effect i 1,, 3 tj jth
IQ block effect j 1,, 3 gk kth age block
effect k 1,, 3 eijk random error term
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If age and IQ were ignored, then approximately 10
times as many subjects (90 people) would be
required for the study to obtain the equivalent
precision in estimating the differences between
the 3 methods.
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• Advantages
• The use of 2 blocking variables often permits
  greater reductions in variability of experimental
  errors than can be obtained by either blocking
  variable alone
• Treatment effects can be studied in a small scale
  experiment (pilot study).

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• Disadvantages
• The number of levels of each factor must be
  equal. This restriction is often difficult to
  meet in practice.
• The assumptions of the model are restrictive
  (e.g., that there are no interactions between
  either blocking variable and the treatment, and
  also none between the two blocking variables.
• The randomization required is somewhat more
  complex than for earlier designs considered.

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An experiment was conducted to study the effects
of different types of background music on the
productivity of bank tellers. The treatments were
various combinations of tempo music (slow,
medium, fast) and style of music (instrumental
and vocal, instrumental only). The experiment
took place over 5 weeks during each weekday.
Mon Tue Wed Thu Fri Week 1
18(D) 17(C) 14(A) 21(B) 17(E) Week 2 13(C)
34(B) 21(E) 16(A) 15(D) Week 3 7(A) 29(D)
32(B) 27(E) 13(C) Week 4 17(E) 13(A) 24(C)
31(D) 25(B) Week 5 21(B) 26(E) 26(D) 31(C)
7(A)
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Treatments explanation A Slow, Instrumental
vocal B Medium, Instrumental vocal C Fast,
Instrumental vocal D Medium, Instrumental
only E Fast, Instrumental only
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• Homework (due 3/6)
• Write out the model.
• Perform the appropriate post hoc analysis with
  90 confidence.
• Test for a difference in productivity between
  fast and medium music with 90 confidence.
• Test for a difference in productivity between
  fast and slow music with 90 confidence.
• Test for a difference in productivity between
  medium and slow music with 90 confidence.
• Test for a difference in productivity between
  instrumental only and instrumental with vocal
  with 90 confidence.
• Recommend one of the treatments for the bank to
  use. Defend your choice.
• Was taking day of the week and week into account
  a worthwhile idea? Defend your choice.