Replicated Latin Squares

1
Replicated Latin Squares

• Three types of replication in traditional (1
  treatment, 2 blocks) latin squares
• Case study (ssquare, n of trt levels)
• Crossover designs
• Subject is one block, Period is another
• Yandell introduces crossovers as a special case
  of the split plot design

2
Replicated Latin Squares

• ColumnOperator, RowBatch
• Case 1 Same Operator, Same Batch Source df
• Treatment n-1
• Batch n-1
• Operator n-1
• Rep s-1
• Error By subtraction
• Total sn2-1

3
Replicated Latin Squares

• Case 2 Different Operator, Same Batch
• Source df
• Treatment n-1
• Batch n-1
• Operator sn-1
• O(S) s(n-1)
• Square s-1
• Error By subtraction
• Total sn2-1

4
Replicated Latin Squares

• Case 3 Different Operator, Different Batch
• Source df
• Treatment n-1
• Batch sn-1
• Operator sn-1
• Error By subtraction
• Total sn2-1

5
Replicated Latin Squares

• Case 3 Different Operator, Different Batch
• Montgomerys approach
• Source df
• Treatment n-1
• Batch(Square) s(n-1)
• Operator(Square) s(n-1)
• Square s-1
• Error By subtraction
• Total sn2-1

6
Crossover Design

• Two blocking factors subject and period
• Used in clinical trials
• Subject
• 1 2 3 4 5 6
• Period 1 A A B A B B
• Period 2 B B A B A A

7
Crossover Design

• Rearrange as a replicated Latin Square
• Subject
• 1 3 2 5 4 6
• Period 1 A B A B A B
• Period 2 B A B A B A

8
Crossover Designs

• Yandell uses a different approach, in which
• Sequence is a factor (basically the WP factor)
• Subjects are nested in sequence
• Period is an effect (Id call it a common SP)
• Treatment (which depends on period and sequence)
  is the Latin Square effect (SP factor)
• Carryover is eventually treated the same way we
  treat it

9
Crossover Designs

• The replicated Latin Square is an artifice, but
  helps to organize our thoughts
• We will assume s Latin Squares with sn subjects
• If you dont have sn subjects, use as much of the
  last Latin Square as possible

10
Crossover Designs

• Example (n4,s2)
• 1 2 3 4 5 6 7 8
• Period 1 A B C D A B C D
• Period 2 B C D A B C D A
• Period 3 C D A B C D A B
• Period 4 D A B C D A B C

11
Crossover Designs

• This is similar to Case 2
• The period x treatment interaction could be
  separated out as a separate test
• Block x treatment interaction
• Periods can differ from square to square--this is
  similar to Case 3

12
Carry-over in Crossover Designs

• Effects in Crossover Designs are confounded with
  the carry-over (residual effects) of previous
  treatments
• We will assume that the carry-over only persists
  for the treatment in the period immediately
  before the present period

13
Carry-over in Crossover Designs

• In this example, we observe the sequence AB, but
  never observe BA
• 1 2 3 4 5 6 7 8
• Period 1 A B C D A B C D
• Period 2 B C D A B C D A
• Period 3 C D A B C D A B
• Period 4 D A B C D A B C

14
Carry-over in Crossover Designs

• A crossover design is balanced with respect to
  carry-over if each treatment follows every other
  treatment the same number of times
• We can balance our example (in a single square)
  by permuting the third and fourth rows

15
Carry-over in Crossover Designs

• Each pair is observed 1 time
• A B C D
• B C D A
• D A B C
• C D A B

16
Carry-over in Crossover Designs

• For n odd, we will need a replicated design
• A B C A B C
• B C A C A B
• C A B B C A

17
Carry-over in Crossover Designs

• These designs are not orthogonal since each
  treatment cannot follow itself. We analyze using
  Type III SS

18
Carry-over in Crossover Designs

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

19
ExampleFirst Two Rows
20
ExampleNext Two Rows
21
Carry-over in Crossover Designs

• The parameter go is the effect of being in the
  first row--it is confounded with the period 1
  effect and will not be estimated
• Each of these factors loses a df as a result

22
Carry-over in Crossover Designs

• Source Usual df Type III df
• Treatment n-1 n-1
• Period n-1 n-2
• Subject sn-1 sn-1
• Res Trt n n-1
• Error By subtraction
• Total sn2-1 sn2-1