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Multi-part balanced incomplete-block designs R. A. Bailey University of St Andrews Combinatorics Seminar, Shanghai Jiao Tong University, 4 October 2018 Joint work with Peter Cameron (University of St Andrews) Bailey 2-part 2-designs 1/36

  1. Conditions on parameters Theorem In a 2 -part 2 -design with parameters v 1 , v 2 , b, k 1 , k 2 , λ 11 , λ 22 and λ 12 , the following hold. 1. Each cancer type occurs in r 1 blocks, where v 1 r 1 = bk 1 . 2. Each drug occurs in r 2 blocks, where v 2 r 2 = bk 2 . 3. λ 11 ( v 1 − 1 ) = r 1 ( k 1 − 1 ) . 4. λ 22 ( v 2 − 1 ) = r 2 ( k 2 − 1 ) . 5. bk 1 k 2 = v 1 v 2 λ 12 . Bailey 2-part 2-designs 9/36

  2. Conditions on parameters Theorem In a 2 -part 2 -design with parameters v 1 , v 2 , b, k 1 , k 2 , λ 11 , λ 22 and λ 12 , the following hold. 1. Each cancer type occurs in r 1 blocks, where v 1 r 1 = bk 1 . 2. Each drug occurs in r 2 blocks, where v 2 r 2 = bk 2 . 3. λ 11 ( v 1 − 1 ) = r 1 ( k 1 − 1 ) . 4. λ 22 ( v 2 − 1 ) = r 2 ( k 2 − 1 ) . 5. bk 1 k 2 = v 1 v 2 λ 12 . 6. b ≥ v 1 + v 2 − 1 . Bailey 2-part 2-designs 9/36

  3. Conditions on parameters Theorem In a 2 -part 2 -design with parameters v 1 , v 2 , b, k 1 , k 2 , λ 11 , λ 22 and λ 12 , the following hold. 1. Each cancer type occurs in r 1 blocks, where v 1 r 1 = bk 1 . 2. Each drug occurs in r 2 blocks, where v 2 r 2 = bk 2 . 3. λ 11 ( v 1 − 1 ) = r 1 ( k 1 − 1 ) . 4. λ 22 ( v 2 − 1 ) = r 2 ( k 2 − 1 ) . 5. bk 1 k 2 = v 1 v 2 λ 12 . 6. b ≥ v 1 + v 2 − 1 . Bailey 2-part 2-designs 9/36

  4. Conditions on parameters Theorem In a 2 -part 2 -design with parameters v 1 , v 2 , b, k 1 , k 2 , λ 11 , λ 22 and λ 12 , the following hold. 1. Each cancer type occurs in r 1 blocks, where v 1 r 1 = bk 1 . 2. Each drug occurs in r 2 blocks, where v 2 r 2 = bk 2 . 3. λ 11 ( v 1 − 1 ) = r 1 ( k 1 − 1 ) . 4. λ 22 ( v 2 − 1 ) = r 2 ( k 2 − 1 ) . 5. bk 1 k 2 = v 1 v 2 λ 12 . 6. b ≥ v 1 + v 2 − 1 . Items 1–5 are obtained by counting something in two different ways. Bailey 2-part 2-designs 9/36

  5. Conditions on parameters Theorem In a 2 -part 2 -design with parameters v 1 , v 2 , b, k 1 , k 2 , λ 11 , λ 22 and λ 12 , the following hold. 1. Each cancer type occurs in r 1 blocks, where v 1 r 1 = bk 1 . 2. Each drug occurs in r 2 blocks, where v 2 r 2 = bk 2 . 3. λ 11 ( v 1 − 1 ) = r 1 ( k 1 − 1 ) . 4. λ 22 ( v 2 − 1 ) = r 2 ( k 2 − 1 ) . 5. bk 1 k 2 = v 1 v 2 λ 12 . 6. b ≥ v 1 + v 2 − 1 . Items 1–5 are obtained by counting something in two different ways. Item 6 is like Fisher’s Inequality (in a 2-design, b ≥ v ). Bailey 2-part 2-designs 9/36

  6. A generalization of resolvability Definition A block design is resolvable if the set of blocks can be partitioned into r replicates of b / r blocks each, in such a way that each treatment occurs once in each replicate. Bailey 2-part 2-designs 10/36

  7. A generalization of resolvability Definition A block design is resolvable if the set of blocks can be partitioned into r replicates of b / r blocks each, in such a way that each treatment occurs once in each replicate. In general, r 1 � = r 2 , so we cannot use the usual definition of resolvable design here. Bailey 2-part 2-designs 10/36

  8. A generalization of resolvability Definition A block design is resolvable if the set of blocks can be partitioned into r replicates of b / r blocks each, in such a way that each treatment occurs once in each replicate. In general, r 1 � = r 2 , so we cannot use the usual definition of resolvable design here. Definition A 2-part block design is c -partitionable if the set of blocks can be grouped into c classes of b / c blocks each, in such a way that every cancer type occurs the same number of times in each class and every drug occurs the same number of times in each class. Bailey 2-part 2-designs 10/36

  9. A generalization of resolvability Definition A block design is resolvable if the set of blocks can be partitioned into r replicates of b / r blocks each, in such a way that each treatment occurs once in each replicate. In general, r 1 � = r 2 , so we cannot use the usual definition of resolvable design here. Definition A 2-part block design is c -partitionable if the set of blocks can be grouped into c classes of b / c blocks each, in such a way that every cancer type occurs the same number of times in each class and every drug occurs the same number of times in each class. Theorem If a 2 -part 2 -design is c-partitionable then b ≥ v 1 + v 2 + c − 2 . Bailey 2-part 2-designs 10/36

  10. Notes on the theorem Theorem If a 2 -part 2 -design is c-partitionable then b ≥ v 1 + v 2 + c − 2 . Bailey 2-part 2-designs 11/36

  11. Notes on the theorem Theorem If a 2 -part 2 -design is c-partitionable then b ≥ v 1 + v 2 + c − 2 . (a) Every 2-part 2-design is 1-partitionable, so it is always true that b ≥ v 1 + v 2 − 1. Bailey 2-part 2-designs 11/36

  12. Notes on the theorem Theorem If a 2 -part 2 -design is c-partitionable then b ≥ v 1 + v 2 + c − 2 . (a) Every 2-part 2-design is 1-partitionable, so it is always true that b ≥ v 1 + v 2 − 1. (b) Bose’s Inequality states that, for a resolvable 2-design, b ≥ v + r − 1. Our new theorem generalizes that. Bailey 2-part 2-designs 11/36

  13. Easy construction I: Cartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Bailey 2-part 2-designs 12/36

  14. Easy construction I: Cartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Form all b 1 b 2 combinations of a block of each sort. Bailey 2-part 2-designs 12/36

  15. Easy construction I: Cartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Form all b 1 b 2 combinations of a block of each sort. For each block combination, form the Cartesian product of their sets of treatments. Bailey 2-part 2-designs 12/36

  16. Easy construction I: Cartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Form all b 1 b 2 combinations of a block of each sort. For each block combination, form the Cartesian product of their sets of treatments. The result is a 2-part 2-design, but it has b 1 b 2 blocks, which is often too large. Bailey 2-part 2-designs 12/36

  17. Easy construction II: Swap Given a 2-part 2-design, create another one, interchanging the values of k 1 and v 1 − k 1 , by replacing the set of cancer types in each block by the complementary set of cancer types. Bailey 2-part 2-designs 13/36

  18. Easy construction II: Swap Given a 2-part 2-design, create another one, interchanging the values of k 1 and v 1 − k 1 , by replacing the set of cancer types in each block by the complementary set of cancer types. The result is also a 2-part 2-design so long as v 1 − k 1 ≥ 2. Bailey 2-part 2-designs 13/36

  19. Easy construction II: Swap Given a 2-part 2-design, create another one, interchanging the values of k 1 and v 1 − k 1 , by replacing the set of cancer types in each block by the complementary set of cancer types. The result is also a 2-part 2-design so long as v 1 − k 1 ≥ 2. Similarly, swap drugs to interchange k 2 and v 2 − k 2 . Bailey 2-part 2-designs 13/36

  20. Easy construction III: Interchange Given a 2-part 2-design, create another one, interchanging the values of v 1 and v 2 , and the values of k 1 and k 2 , by interchanging the roles of cancer types and drugs. Bailey 2-part 2-designs 14/36

  21. Serious construction I: Subcartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Bailey 2-part 2-designs 15/36

  22. Serious construction I: Subcartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Suppose that ∆ 2 is resolvable with replication r , and that r divides b 1 . Bailey 2-part 2-designs 15/36

  23. Serious construction I: Subcartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Suppose that ∆ 2 is resolvable with replication r , and that r divides b 1 . Partition the set of blocks of ∆ 1 into r sets of b 1 / r blocks, in any way at all. Bailey 2-part 2-designs 15/36

  24. Serious construction I: Subcartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Suppose that ∆ 2 is resolvable with replication r , and that r divides b 1 . Partition the set of blocks of ∆ 1 into r sets of b 1 / r blocks, in any way at all. Match these sets to the r resolution classes of ∆ 2 , in any way at all. Bailey 2-part 2-designs 15/36

  25. Serious construction I: Subcartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Suppose that ∆ 2 is resolvable with replication r , and that r divides b 1 . Partition the set of blocks of ∆ 1 into r sets of b 1 / r blocks, in any way at all. Match these sets to the r resolution classes of ∆ 2 , in any way at all. For each matched pair, construct the cartesian product design. Bailey 2-part 2-designs 15/36

  26. Serious construction I: Subcartesian product Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , and let ∆ 2 be a BIBD for v 2 treatments in b 2 blocks of size k 2 . Suppose that ∆ 2 is resolvable with replication r , and that r divides b 1 . Partition the set of blocks of ∆ 1 into r sets of b 1 / r blocks, in any way at all. Match these sets to the r resolution classes of ∆ 2 , in any way at all. For each matched pair, construct the cartesian product design. The result is a 2-part 2-design, and it has b 1 b 2 / r blocks. Bailey 2-part 2-designs 15/36

  27. An example of a subcartesian product: v 1 = 3 , v 2 = 4 ∆ 2 resolvable r = 3 ∆ 1 D1, D3 b = 3 D2, D4 C1, C2 D2, D3 C1, C3 D1, D4 C2, C3 D1, D2 D3, D4 Bailey 2-part 2-designs 16/36

  28. An example of a subcartesian product: v 1 = 3 , v 2 = 4 ∆ 2 resolvable r = 3 Block Cancer types Drugs ∆ 1 1 C1, C2 D1, D3 D1, D3 b = 3 D2, D4 2 C1, C2 D2, D4 C1, C2 D2, D3 C1, C3 D1, D4 C2, C3 D1, D2 D3, D4 Bailey 2-part 2-designs 16/36

  29. An example of a subcartesian product: v 1 = 3 , v 2 = 4 ∆ 2 resolvable r = 3 Block Cancer types Drugs ∆ 1 1 C1, C2 D1, D3 D1, D3 b = 3 D2, D4 2 C1, C2 D2, D4 C1, C2 3 C1, C3 D2, D3 D2, D3 C1, C3 4 C1, C3 D1, D4 D1, D4 C2, C3 5 C2, C3 D1, D2 D1, D2 6 C2, C3 D3, D4 D3, D4 Bailey 2-part 2-designs 16/36

  30. Serious construction II: Hadamard matrix If v 1 = v 2 = 2 k 1 = 2 k 2 = 2 n , write down a Hadamard matrix of order 4 n with all entries + 1 in the first row.   + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 − 1 − 1 − 1 − 1 − 1 − 1     + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 − 1 + 1 + 1 − 1     + 1 − 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 + 1     + 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1     + 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1     + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 + 1 − 1     + 1 − 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1     + 1 + 1 − 1 − 1 + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1      + 1 + 1 − 1 + 1 − 1 − 1 + 1 − 1 − 1 − 1 + 1 + 1     + 1 + 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 − 1 − 1    + 1 − 1 + 1 − 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1 + 1 Bailey 2-part 2-designs 17/36

  31. Serious construction II: Hadamard matrix If v 1 = v 2 = 2 k 1 = 2 k 2 = 2 n , write down a Hadamard matrix of order 4 n with all entries + 1 in the first row.   + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 − 1 − 1 − 1 − 1 − 1 − 1     + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 − 1 + 1 + 1 − 1     + 1 − 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 + 1     + 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1     + 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1     + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 + 1 − 1     + 1 − 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1     + 1 + 1 − 1 − 1 + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1      + 1 + 1 − 1 + 1 − 1 − 1 + 1 − 1 − 1 − 1 + 1 + 1     + 1 + 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 − 1 − 1    + 1 − 1 + 1 − 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1 + 1 Bailey 2-part 2-designs 17/36

  32. Serious construction II: Hadamard matrix If v 1 = v 2 = 2 k 1 = 2 k 2 = 2 n , write down a Hadamard matrix of order 4 n with all entries + 1 in the first row. Replace all ± entries in row 2 with levels of C/D.   + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 C1 C2 C3 C4 C5 C6 D1 D2 D3 D4 D5 D6     + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 − 1 + 1 + 1 − 1     + 1 − 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 + 1     + 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1     + 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1     + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 + 1 − 1     + 1 − 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1     + 1 + 1 − 1 − 1 + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1      + 1 + 1 − 1 + 1 − 1 − 1 + 1 − 1 − 1 − 1 + 1 + 1    + 1 + 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 − 1 − 1     + 1 − 1 + 1 − 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1 + 1 Bailey 2-part 2-designs 18/36

  33. Serious construction II: Hadamard matrix If v 1 = v 2 = 2 k 1 = 2 k 2 = 2 n , write down a Hadamard matrix of order 4 n with all entries + 1 in the first row. Replace all ± entries in row 2 with levels of C/D.   + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 C1 C2 C3 C4 C5 C6 D1 D2 D3 D4 D5 D6     + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 − 1 + 1 + 1 − 1     + 1 − 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 + 1     + 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1     + 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1     + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 + 1 − 1     + 1 − 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1     + 1 + 1 − 1 − 1 + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1      + 1 + 1 − 1 + 1 − 1 − 1 + 1 − 1 − 1 − 1 + 1 + 1    + 1 + 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 − 1 − 1     + 1 − 1 + 1 − 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1 + 1 Row 3 → { C1,C3,C5 || D1,D4,D5 } Bailey 2-part 2-designs 18/36

  34. Serious construction II: Hadamard matrix If v 1 = v 2 = 2 k 1 = 2 k 2 = 2 n , write down a Hadamard matrix of order 4 n with all entries + 1 in the first row. Replace all ± entries in row 2 with levels of C/D.   + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 C1 C2 C3 C4 C5 C6 D1 D2 D3 D4 D5 D6     + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 − 1 + 1 + 1 − 1     + 1 − 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 + 1     + 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1     + 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1     + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 + 1 − 1     + 1 − 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1     + 1 + 1 − 1 − 1 + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1      + 1 + 1 − 1 + 1 − 1 − 1 + 1 − 1 − 1 − 1 + 1 + 1    + 1 + 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 − 1 − 1     + 1 − 1 + 1 − 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1 + 1 Row 3 → { C1,C3,C5 || D1,D4,D5 } and { C2,C4,C6 || D2,D3,D6 } . Bailey 2-part 2-designs 18/36

  35. Serious construction II: Hadamard matrix If v 1 = v 2 = 2 k 1 = 2 k 2 = 2 n , write down a Hadamard matrix of order 4 n with all entries + 1 in the first row. Replace all ± entries in row 2 with levels of C/D.   + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 C1 C2 C3 C4 C5 C6 D1 D2 D3 D4 D5 D6     + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 − 1 + 1 + 1 − 1     + 1 − 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 + 1     + 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 − 1     + 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1     + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 + 1 − 1     + 1 − 1 + 1 + 1 − 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1     + 1 + 1 − 1 − 1 + 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1      + 1 + 1 − 1 + 1 − 1 − 1 + 1 − 1 − 1 − 1 + 1 + 1    + 1 + 1 − 1 − 1 − 1 + 1 + 1 − 1 + 1 + 1 − 1 − 1     + 1 − 1 + 1 − 1 − 1 + 1 + 1 + 1 − 1 − 1 − 1 + 1 Row 3 → { C1,C3,C5 || D1,D4,D5 } and { C2,C4,C6 || D2,D3,D6 } . And so on, so b = 2 ( 4 n − 2 ) = 8 n − 4. Bailey 2-part 2-designs 18/36

  36. A good outcome of the Hadamard construction A Hadamard matrix of order 4 n leads to a 2-part 2-design with v 1 = v 2 = 2 n , k 1 = k 2 = n and b = 8 n − 4. Bailey 2-part 2-designs 19/36

  37. A good outcome of the Hadamard construction A Hadamard matrix of order 4 n leads to a 2-part 2-design with v 1 = v 2 = 2 n , k 1 = k 2 = n and b = 8 n − 4. It is c -partitionable for c = 4 n − 2. Bailey 2-part 2-designs 19/36

  38. A good outcome of the Hadamard construction A Hadamard matrix of order 4 n leads to a 2-part 2-design with v 1 = v 2 = 2 n , k 1 = k 2 = n and b = 8 n − 4. It is c -partitionable for c = 4 n − 2. Often, a subcartesian product can give a 2-part 2-design with the same parameters, but this is not usually ( 4 n − 2 ) -partitionable. Bailey 2-part 2-designs 19/36

  39. Serious construction III: Symmetric BIBD Start with a BIBD for v treatments in v blocks of size k , where each pair of blocks have λ treatments in common, and λ > 1 and 3 ≤ k ≤ v − k . Bailey 2-part 2-designs 20/36

  40. Serious construction III: Symmetric BIBD Start with a BIBD for v treatments in v blocks of size k , where each pair of blocks have λ treatments in common, and λ > 1 and 3 ≤ k ≤ v − k . Choose one block, and identify its treatments with drugs (so v 2 = k ). Bailey 2-part 2-designs 20/36

  41. Serious construction III: Symmetric BIBD Start with a BIBD for v treatments in v blocks of size k , where each pair of blocks have λ treatments in common, and λ > 1 and 3 ≤ k ≤ v − k . Choose one block, and identify its treatments with drugs (so v 2 = k ). Identify the other treatments with cancer types (so v 1 = v − k ). Bailey 2-part 2-designs 20/36

  42. Serious construction III: Symmetric BIBD Start with a BIBD for v treatments in v blocks of size k , where each pair of blocks have λ treatments in common, and λ > 1 and 3 ≤ k ≤ v − k . Choose one block, and identify its treatments with drugs (so v 2 = k ). Identify the other treatments with cancer types (so v 1 = v − k ). Each remaining block gives a block of our 2-part 2-design, so b = v − 1 = k 2 λ k 1 = k − λ = λ 11 λ λ 12 = λ = λ − 1. λ 22 Bailey 2-part 2-designs 20/36

  43. An example from a symmetric BIBD: v 1 = 6 , v 2 = 5 rows are blocks 1 5 3 4 9 2 6 4 5 10 3 7 5 6 0 4 8 6 7 1 5 9 7 8 2 6 10 8 9 3 7 0 9 10 4 8 1 10 0 5 9 2 0 1 6 10 3 1 2 7 0 4 2 3 8 Bailey 2-part 2-designs 21/36

  44. An example from a symmetric BIBD: v 1 = 6 , v 2 = 5 rows are blocks 1 5 3 4 9 2 6 4 5 10 3 7 5 6 0 4 8 6 7 1 5 9 7 8 2 6 10 8 9 3 7 0 9 10 4 8 1 10 0 5 9 2 0 1 6 10 3 1 2 7 0 4 2 3 8 1 5 3 4 9 D1 D2 D3 D4 D5 Bailey 2-part 2-designs 21/36

  45. An example from a symmetric BIBD: v 1 = 6 , v 2 = 5 rows are blocks 1 5 3 4 9 2 6 4 5 10 3 7 5 6 0 4 8 6 7 1 5 9 7 8 2 6 10 8 9 3 7 0 9 10 4 8 1 10 0 5 9 2 0 1 6 10 3 1 2 7 0 4 2 3 8 1 5 3 4 9 0 2 6 7 8 10 D1 D2 D3 D4 D5 C1 C2 C3 C4 C6 C5 Bailey 2-part 2-designs 21/36

  46. An example from a symmetric BIBD: v 1 = 6 , v 2 = 5 rows are blocks 2-part 2-design 1 5 3 4 9 drugs cancer types 2 6 4 5 10 D2 D4 C2 C3 C5 3 7 5 6 0 D2 D3 C1 C3 C4 4 8 6 7 1 D1 D4 C3 C4 C6 5 9 7 8 2 D2 D5 C2 C4 C6 6 10 8 9 3 D3 D5 C3 C5 C6 7 0 9 10 4 D4 D5 C1 C4 C5 8 1 10 0 5 D1 D2 C1 C5 C6 9 2 0 1 6 D1 D5 C1 C2 C3 10 3 1 2 7 D1 D3 C2 C4 C5 0 4 2 3 8 D3 D4 C1 C2 C6 1 5 3 4 9 0 2 6 7 8 10 D1 D2 D3 D4 D5 C1 C2 C3 C4 C6 C5 Bailey 2-part 2-designs 21/36

  47. An example from a symmetric BIBD: v 1 = 6 , v 2 = 5 rows are blocks 2-part 2-design 1 5 3 4 9 drugs cancer types 2 6 4 5 10 D2 D4 C2 C3 C5 3 7 5 6 0 D2 D3 C1 C3 C4 4 8 6 7 1 D1 D4 C3 C4 C6 5 9 7 8 2 D2 D5 C2 C4 C6 6 10 8 9 3 D3 D5 C3 C5 C6 7 0 9 10 4 D4 D5 C1 C4 C5 8 1 10 0 5 D1 D2 C1 C5 C6 9 2 0 1 6 D1 D5 C1 C2 C3 10 3 1 2 7 D1 D3 C2 C4 C5 0 4 2 3 8 D3 D4 C1 C2 C6 1 5 3 4 9 0 2 6 7 8 10 D1 D2 D3 D4 D5 C1 C2 C3 C4 C6 C5 Bailey 2-part 2-designs 21/36

  48. An example from a symmetric BIBD: v 1 = 6 , v 2 = 5 rows are blocks 2-part 2-design 1 5 3 4 9 drugs cancer types 2 6 4 5 10 D2 D4 C2 C3 C5 3 7 5 6 0 D2 D3 C1 C3 C4 4 8 6 7 1 D1 D4 C3 C4 C6 5 9 7 8 2 D2 D5 C2 C4 C6 6 10 8 9 3 D3 D5 C3 C5 C6 7 0 9 10 4 D4 D5 C1 C4 C5 8 1 10 0 5 D1 D2 C1 C5 C6 9 2 0 1 6 D1 D5 C1 C2 C3 10 3 1 2 7 D1 D3 C2 C4 C5 0 4 2 3 8 D3 D4 C1 C2 C6 1 5 3 4 9 0 2 6 7 8 10 D1 D2 D3 D4 D5 C1 C2 C3 C4 C6 C5 This is exactly the first 2-part 2-design that I showed you. Bailey 2-part 2-designs 21/36

  49. Serious construction IV: Augmentation Given a 2-part 2-design with v 2 = 2 k 2 + 1, add an extra drug, increasing v 2 to v 2 + 1, k 2 to k 2 + 1 and b to 2 b . Bailey 2-part 2-designs 22/36

  50. Serious construction IV: Augmentation Given a 2-part 2-design with v 2 = 2 k 2 + 1, add an extra drug, increasing v 2 to v 2 + 1, k 2 to k 2 + 1 and b to 2 b . Replace each previous block by two new blocks, both with the original subset of cancer types. Bailey 2-part 2-designs 22/36

  51. Serious construction IV: Augmentation Given a 2-part 2-design with v 2 = 2 k 2 + 1, add an extra drug, increasing v 2 to v 2 + 1, k 2 to k 2 + 1 and b to 2 b . Replace each previous block by two new blocks, both with the original subset of cancer types. One of these has the same drugs as before, plus the new drug. The other has all the remaining drugs. Bailey 2-part 2-designs 22/36

  52. Easy construction IV: Group-divisible designs If v 1 = v 2 and k 1 = k 2 then the concise form of a 2-part 2-design is a “semi-regular group-divisible incomplete-block design for two groups of treatments”. Look these up in Clatworthy’s Tables of Two-Associate Class Partially Balanced Designs . Bailey 2-part 2-designs 23/36

  53. Serious construction V: Permutation groups If there is a group G which acts doubly transitively on the set of cancer types and also acts doubly transitively on the set of drugs, then choose an initial block and then get the remaining blocks by applying the permutations in G to it. Interesting examples are too large to fit on a slide! Bailey 2-part 2-designs 24/36

  54. Extending the problem On 28 March 2016, Valerii sent me the png file of the first design in this talk. When I thanked him, he emailed back the next day with Dear Rosemary, It can be never ending story . . . . For instance, can we extend the table below and add another factor: oncogenes (biomarker)? . . . Bailey 2-part 2-designs 25/36

  55. 3 -part 2 -designs In a 3-part 2-design, we also have a set of v 3 biomarkers, such that (a) all medical centres involve k 1 cancer types, where k 1 < v 1 ; (b) all medical centres use k 2 drugs, where k 2 < v 2 ; (c) each pair of distinct cancer types occur together at λ 11 medical centres, where λ 11 > 0; (d) each pair of distinct drugs occur together at λ 22 medical centres, where λ 12 > 0; (e) each drug occurs with each type of cancer at λ 12 medical centres; Bailey 2-part 2-designs 26/36

  56. 3 -part 2 -designs In a 3-part 2-design, we also have a set of v 3 biomarkers, such that (a) all medical centres involve k 1 cancer types, where k 1 < v 1 ; (b) all medical centres use k 2 drugs, where k 2 < v 2 ; (c) each pair of distinct cancer types occur together at λ 11 medical centres, where λ 11 > 0; (d) each pair of distinct drugs occur together at λ 22 medical centres, where λ 12 > 0; (e) each drug occurs with each type of cancer at λ 12 medical centres; (f) all medical centres use k 3 biomarkers, where k 3 < v 3 ; Bailey 2-part 2-designs 26/36

  57. 3 -part 2 -designs In a 3-part 2-design, we also have a set of v 3 biomarkers, such that (a) all medical centres involve k 1 cancer types, where k 1 < v 1 ; (b) all medical centres use k 2 drugs, where k 2 < v 2 ; (c) each pair of distinct cancer types occur together at λ 11 medical centres, where λ 11 > 0; (d) each pair of distinct drugs occur together at λ 22 medical centres, where λ 12 > 0; (e) each drug occurs with each type of cancer at λ 12 medical centres; (f) all medical centres use k 3 biomarkers, where k 3 < v 3 ; (g) each pair of distinct biomarkers occur together at λ 33 medical centres, where λ 33 > 0; Bailey 2-part 2-designs 26/36

  58. 3 -part 2 -designs In a 3-part 2-design, we also have a set of v 3 biomarkers, such that (a) all medical centres involve k 1 cancer types, where k 1 < v 1 ; (b) all medical centres use k 2 drugs, where k 2 < v 2 ; (c) each pair of distinct cancer types occur together at λ 11 medical centres, where λ 11 > 0; (d) each pair of distinct drugs occur together at λ 22 medical centres, where λ 12 > 0; (e) each drug occurs with each type of cancer at λ 12 medical centres; (f) all medical centres use k 3 biomarkers, where k 3 < v 3 ; (g) each pair of distinct biomarkers occur together at λ 33 medical centres, where λ 33 > 0; (h) each biomarker occurs with each type of cancer at λ 13 medical centres; Bailey 2-part 2-designs 26/36

  59. 3 -part 2 -designs In a 3-part 2-design, we also have a set of v 3 biomarkers, such that (a) all medical centres involve k 1 cancer types, where k 1 < v 1 ; (b) all medical centres use k 2 drugs, where k 2 < v 2 ; (c) each pair of distinct cancer types occur together at λ 11 medical centres, where λ 11 > 0; (d) each pair of distinct drugs occur together at λ 22 medical centres, where λ 12 > 0; (e) each drug occurs with each type of cancer at λ 12 medical centres; (f) all medical centres use k 3 biomarkers, where k 3 < v 3 ; (g) each pair of distinct biomarkers occur together at λ 33 medical centres, where λ 33 > 0; (h) each biomarker occurs with each type of cancer at λ 13 medical centres; (i) each biomarker occurs with each drug at λ 23 medical centres. Bailey 2-part 2-designs 26/36

  60. Serious new construction: Orthogonal array Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , ∆ 2 a BIBD for v 2 treatments in b 2 blocks of size k 2 , and ∆ 3 a BIBD for v 3 treatments in b 3 blocks of size k 3 . Bailey 2-part 2-designs 27/36

  61. Serious new construction: Orthogonal array Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , ∆ 2 a BIBD for v 2 treatments in b 2 blocks of size k 2 , and ∆ 3 a BIBD for v 3 treatments in b 3 blocks of size k 3 . Use an orthogonal array of strength 2, with three columns, where column i has b i symbols. Bailey 2-part 2-designs 27/36

  62. Serious new construction: Orthogonal array Let ∆ 1 be a BIBD for v 1 treatments in b 1 blocks of size k 1 , ∆ 2 a BIBD for v 2 treatments in b 2 blocks of size k 2 , and ∆ 3 a BIBD for v 3 treatments in b 3 blocks of size k 3 . Use an orthogonal array of strength 2, with three columns, where column i has b i symbols. For each row of the orthogonal array, construct the cartesian product of the three blocks, one in each of ∆ 1 , ∆ 2 and ∆ 3 . Bailey 2-part 2-designs 27/36

  63. An example using an orthogonal array: v 1 = v 2 = v 3 = 3 design ∆ 1 design ∆ 2 design ∆ 1 Block 1 C1, C2 Block 1 D1, D2 Block 1 B1, B2 Block 2 C1, C3 Block 2 D1, D3 Block 2 B1, B3 Block 3 C2, C3 Block 3 D2, D3 Block 3 B2, B3 Bailey 2-part 2-designs 28/36

  64. An example using an orthogonal array: v 1 = v 2 = v 3 = 3 design ∆ 1 design ∆ 2 design ∆ 1 Block 1 C1, C2 Block 1 D1, D2 Block 1 B1, B2 Block 2 C1, C3 Block 2 D1, D3 Block 2 B1, B3 Block 3 C2, C3 Block 3 D2, D3 Block 3 B2, B3 Orthogonal array 1 1 1 2 2 2 3 3 3 1 3 2 2 1 3 3 2 1 1 2 3 2 3 1 3 1 2 Bailey 2-part 2-designs 28/36

  65. An example using an orthogonal array: v 1 = v 2 = v 3 = 3 design ∆ 1 design ∆ 2 design ∆ 1 Block 1 C1, C2 Block 1 D1, D2 Block 1 B1, B2 Block 2 C1, C3 Block 2 D1, D3 Block 2 B1, B3 Block 3 C2, C3 Block 3 D2, D3 Block 3 B2, B3 Orthogonal Cancer Bio- array Block types Drugs markers 1 1 1 1 C1, C2 D1, D2 B1, B2 2 2 2 2 C1, C3 D1, D3 B1, B3 3 3 3 3 C2, C3 D2, D3 B2, B3 1 3 2 4 C1, C2 D2, D3 B1, B3 2 1 3 5 C1, C3 D1, D2 B2, B3 3 2 1 6 C2, C3 D1, D3 B1, B2 1 2 3 7 C1, C2 D1, D3 B2, B3 2 3 1 8 C1, C3 D2, D3 B1, B2 3 1 2 9 C2, C3 D1, D2 B1, B3 Bailey 2-part 2-designs 28/36

  66. General multi-part BIBDs The foregoing definition extends to m different types of thing. Bailey 2-part 2-designs 29/36

  67. General multi-part BIBDs The foregoing definition extends to m different types of thing. Theorem Let ∆ be an m-part 2 -design with v i things of type i, for i = 1 , . . . , m. If the parameters are b, v i , k i , λ ii and λ ij for 1 ≤ i < j ≤ m, then the following hold. 1. For i = 1 , . . . , m, each thing of type i occurs in r i blocks, where v i r i = bk i . 2. For i = 1 , . . . , m, λ ii ( v i − 1 ) = r i ( k i − 1 ) . 3. For 1 ≤ i < j ≤ m, bk i k j = v i v j λ ij . 4. If ∆ is c-partitionable then b ≥ v 1 + · · · + v m + c − m. 5. In particular, b ≥ v 1 + · · · + v m − m + 1 . Bailey 2-part 2-designs 29/36

  68. Nothing new under the sun In May 2018 we learnt that designs like these had already been proposed by Randy Sitter ( Biometrika , 1993) and Rahul Mukerjee ( Journal of Statistical Planning and Inference , 1998). Bailey 2-part 2-designs 30/36

  69. Nothing new under the sun In May 2018 we learnt that designs like these had already been proposed by Randy Sitter ( Biometrika , 1993) and Rahul Mukerjee ( Journal of Statistical Planning and Inference , 1998). Mukerjee’s main construction is the general orthogonal array (possibly trivial, i.e. all possible rows) applied to parts of m component BIBDs which are all c -partitionable (possibly with c = 1). Bailey 2-part 2-designs 30/36

  70. Nothing new under the sun In May 2018 we learnt that designs like these had already been proposed by Randy Sitter ( Biometrika , 1993) and Rahul Mukerjee ( Journal of Statistical Planning and Inference , 1998). Mukerjee’s main construction is the general orthogonal array (possibly trivial, i.e. all possible rows) applied to parts of m component BIBDs which are all c -partitionable (possibly with c = 1). There are three main differences between their approach and ours. Bailey 2-part 2-designs 30/36

  71. Different applications: I Sitter is concerned with sampling. The population consists of m distinct strata of sizes v 1 , . . . , v m . He wants to draw a random sample of size k i from stratum i , for i = 1, . . . , m , measure something on each element sampled, and hence estimate something about the population. Bailey 2-part 2-designs 31/36

  72. Different applications: II Mukerjee’s proposed application is to designed experiments, but it is different from ours. Bailey 2-part 2-designs 32/36

  73. Different applications: II Mukerjee’s proposed application is to designed experiments, but it is different from ours. For him, each of our “blocks” is a single experimental unit, to which subsets of different types of treatment are applied. Block Cancer types Drugs 1 C1, C2, C3 D1, D5 Experimental Chemical mixture Variety unit applied to the soil of wheat 1 C1, C2, C3 D1, D5 one measurement a mixture of variety obtained by three chemicals cross-breeding varieties D1 and D5 Bailey 2-part 2-designs 32/36

  74. Different constraints Sitter’s and Mukerjee’s applications do not need the constraint that λ ii > 0. So their designs allow k i = 1, which ours do not. Bailey 2-part 2-designs 33/36

  75. Different terminology Sitter and Mukerjee called their designs balanced orthogonal multi-arrays, following Brickell ( Congressus Numerantium , 1984). Bailey 2-part 2-designs 34/36

  76. Different terminology Sitter and Mukerjee called their designs balanced orthogonal multi-arrays, following Brickell ( Congressus Numerantium , 1984). Brickell does not require the separate designs to be balanced, but he does require λ ij = 1 if i � = j . Sitter and Mukerjee explicitly dropped this last condition. Bailey 2-part 2-designs 34/36

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