Latin Square Design

1
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

2
Latin Square Design

• Student project example
• 4 drivers, 4 times, 4 routes
• Yelapsed 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

3
Latin Square Design

• Example
• Region II Science Fair years ago (7 by 7 design)
• Row factorChemical
• Column factorDay (Block?)
• TreatmentFly Group (Block?)
• ResponseNumber of flies (out of 20) not avoiding
  the chemical

4
Latin Square Design
5
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)

6
Power Analysis in Latin Squares

• For replicated squares, the denominator df
  depends on the method of replication see
  Montgomery

7
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

8
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

9
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

10
Graeco-Latin Square Design

• A B C D
• B A D C
• C D A B
• D C B A

11
Graeco-Latin Square Design

• A B C D
• B A D C
• C D A B
• D C B A

12
Graeco-Latin Square Design

• Column
• 1 2 3 4
• 1 Aa Bb Cg Dd
• Row 2 Bd Ag Db Ca
• 3 Cb Da Ad Bg
• 4 Dg Cd Ba Ab

13
Graeco-Latin Square Design

• Orthogonal designs do not exist for n6
• Randomization
• Standard square
• Rows
• Columns
• Latin letters
• Greek letters

14
Graeco-Latin Square Design

• Total df is n2-1(n-1)(n1)
• 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 gt 3) are used for
  screening designs

15
Conclusions

• We will explore some interesting extensions of
  Latin Squares in the texts last chapter
• Replicated Latin Squares
• Crossover Designs
• Residual Effects in Crossover designs
• But first we need to learn some more about
  blocking