Replicated Latin Squares Three types of
Description: 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
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slide1. Replicated Latin Squares Three types of replication in traditional (1 treatment, 2 blocks) latin squares
Case study (s=square, 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<br>
slide2. Replicated Latin Squares-Case 1 Column=Operator, Row=Batch
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<br>
slide3. Replicated Latin Squares-Case 2 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<br>
slide4. Replicated Latin Squares-Case 3a Case 3: Different Operator, Different Batch
Source df
Treatment n-1
Batch sn-1
Operator sn-1
Error By subtraction
Total sn2-1<br>
slide5. Replicated Latin Squares-Case 3b Case 3: Different Operator, Different Batch
Montgomery’s 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<br>
slide6. 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<br>
slide7. 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 Crossover Design as Replicated Latin Square<br>
slide8. Crossover Designs Yandell uses a different approach, in which
Sequence is a factor (basically the WP factor)
Subjects are nested in sequence
1 2 3 4 5 6
Period 1 A B C C A B
Period 2 B C A B C A
Period 3 C A B A B C<br>
slide9. Crossover Designs-Yandell Yandell uses a different approach, in which
Period is an effect (I’d call it a common SP)
Treatment (which depends on period and sequence) is the Latin letter effect (SP factor)
Carryover is eventually treated the same way we treat it<br>
slide10. Crossover Designs Notes The replicated Latin Square is an artifice, but helps to organize our thoughts
We will assume s Latin Squares with sn subjects
If you don’t have sn subjects, use as much of the last Latin Square as possible<br>
slide11. Example (n=4,s=2)
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 Crossover Designs-Example<br>
slide12. Crossover Designs-Case 2 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<br>
slide13. 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<br>
slide14. 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 Carry-over in Crossover Designs-Residual Treatment<br>
slide15. Carry-over in Crossover Designs-Balance 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<br>
slide16. Each pair is observed 1 time
A B C D
B C D A
D A B C
C D A B Residual Treatment in Crossover Designs-Ex 1<br>
slide17. 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 Residual Treatment in Crossover Designs-Ex 2<br>
slide18. These designs are not orthogonal since each treatment cannot follow itself. We analyze using Type III SS (i indexes period, j indexes treatment) Model with Residual Treatment<br>
slide19. Residual Treatment Example Example:
A B C D
B C D A
D A B C
C D A B<br>
slide20. Example First Two Rows<br>
slide21. Example Next Two Rows<br>
slide22. Residual Treatment Effects 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<br>
slide23. Residua Treatment ANOVA 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<br>
Case study (s=square, 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<br>
slide2. Replicated Latin Squares-Case 1 Column=Operator, Row=Batch
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<br>
slide3. Replicated Latin Squares-Case 2 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<br>
slide4. Replicated Latin Squares-Case 3a Case 3: Different Operator, Different Batch
Source df
Treatment n-1
Batch sn-1
Operator sn-1
Error By subtraction
Total sn2-1<br>
slide5. Replicated Latin Squares-Case 3b Case 3: Different Operator, Different Batch
Montgomery’s 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<br>
slide6. 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<br>
slide7. 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 Crossover Design as Replicated Latin Square<br>
slide8. Crossover Designs Yandell uses a different approach, in which
Sequence is a factor (basically the WP factor)
Subjects are nested in sequence
1 2 3 4 5 6
Period 1 A B C C A B
Period 2 B C A B C A
Period 3 C A B A B C<br>
slide9. Crossover Designs-Yandell Yandell uses a different approach, in which
Period is an effect (I’d call it a common SP)
Treatment (which depends on period and sequence) is the Latin letter effect (SP factor)
Carryover is eventually treated the same way we treat it<br>
slide10. Crossover Designs Notes The replicated Latin Square is an artifice, but helps to organize our thoughts
We will assume s Latin Squares with sn subjects
If you don’t have sn subjects, use as much of the last Latin Square as possible<br>
slide11. Example (n=4,s=2)
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 Crossover Designs-Example<br>
slide12. Crossover Designs-Case 2 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<br>
slide13. 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<br>
slide14. 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 Carry-over in Crossover Designs-Residual Treatment<br>
slide15. Carry-over in Crossover Designs-Balance 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<br>
slide16. Each pair is observed 1 time
A B C D
B C D A
D A B C
C D A B Residual Treatment in Crossover Designs-Ex 1<br>
slide17. 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 Residual Treatment in Crossover Designs-Ex 2<br>
slide18. These designs are not orthogonal since each treatment cannot follow itself. We analyze using Type III SS (i indexes period, j indexes treatment) Model with Residual Treatment<br>
slide19. Residual Treatment Example Example:
A B C D
B C D A
D A B C
C D A B<br>
slide20. Example First Two Rows<br>
slide21. Example Next Two Rows<br>
slide22. Residual Treatment Effects 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<br>
slide23. Residua Treatment ANOVA 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<br>