Replicated Latin Squares

1. 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

2. Replicated Latin Squares • Column=Operator, Row=Batch • Case 1: Same Operator, Same Batch Sourcedf 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 Sourcedf 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 Sourcedf 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 • Montgomery’s approach Sourcedf 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 (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

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 don’t have sn subjects, use as much of the last Latin Square as possible

10. Crossover Designs • 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

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 (i indexes period, j indexes treatment)

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 SourceUsual dfType 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