Block structure

Imagine a set of 12 objects divided into four blocks of size three. Statisticians say that there is block factor with four levels; pure mathematicians say that the blocks form a partition of the whole set. They agree that the case when all blocks have the same size is important, but have various different names for it, such as balanced or uniform.

What happens when there are two or more factors (or partitions) on the same set?

Topics Some of my publications

What properties are needed to ensure a unique orthogonal decomposition of the corresponding vector space, or to ensure known eigenspaces of the corresponding covariance matrix?
R. A. Bailey: Distributive block structures and their automorphisms. In Combinatorial Mathematics VIII (ed. K. L. McAvaney), Lecture Notes in Mathematics, 884 Springer-Verlag, Berlin, 1981, pp. 115–124. doi: 10.1007/BFb0091813 [Maths Reviews 0641241 (83c: 62119)]
T. P. Speed and R. A. Bailey: On a class of association schemes derived from lattices of equivalence relations. In Algebraic Structures and Applications (eds. P. Schultz, C. E. Praeger and R. P. Sullivan), Marcel Dekker, New York, 1982, pp. 55–74. [Maths Reviews 0647166 (83f: 06023)]
R. A. Bailey: Block structures for designed experiments. In Applications of Combinatorics (ed. R. J. Wilson), Shiva Publishing, Nantwich, 1982, pp. 1–18. [Maths Reviews 0677504 (83j: 05001)]
R. A. Bailey: Contribution to the discussion of `Analysis of variance models in orthogonal designs' by Tue Tjur. International Statistical Review 52 (1984), pp. 65–77. [Maths Reviews 0967202 (89h: 62123)]
T. P. Speed and R. A. Bailey: Factorial dispersion models. International Statistical Review 55 (1987), 261–277. doi: 10.2307/1403405 [Maths Reviews 0963143 (89m: 62072)]
R. A. Bailey: Contribution to the discussion of `What is an analysis of variance?' by T. P. Speed. Annals of Statistics 15 (1987), pp. 913–916.
R. A. Bailey: Orthogonal partitions in designed experiments. Designs, Codes and Cryptography 8 (1996), 45–77. doi: 10.1023/A:1018072606346 [Maths Reviews 1393974 (97g: 62136a) and 1403872 (97g:62136b)]
R. A. Bailey: Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge University Press, Cambridge, 2004. 387pp. ISBN 0 521 82446 X. [Maths Reviews 2047311 (2005d: 05001)]
R. A. Bailey: Principles of designed experiments in J. A. Nelder's papers. In Methods and Models in Statistics: In Honour of Professor John Nelder, FRS (eds. N. M. Adams, M. J. Crowder, D. J. .Hand and D A. Stephens), Imperial College Press, London, 2004, pp. 171–194.
R. A. Bailey: Hasse diagrams in designed experiments: a pictorial aid to thinking about blocking, stratification, degrees of freedom, randomization, and analysis of variance, Região Brasileira da Sociedade Internacional de Biometria, Londrina, Brasil, 2005. 88pp.
R. A. Bailey: Design of Comparative Experiments, Cambridge University Press, Cambridge, 2008, 330pp. ISBN 978-0-521-86506-7 and 978-0-521-68357-9. [Maths Reviews 2422352]
R. A. Bailey: Structures defined by factors. Chapter 10 in Handbook of Design and Analysis of Experiments (eds. Angela Dean, Max Morris, John Stufken and Derek Bingham), Chapman and Hall/ CRC Handbooks of Modern Statistical Methods, Chapman and Hall/ CRC (2015), pp. 371–414.
R. A. Bailey, Sandra S. Ferreira, Dário Ferreia and Célia Nunes: Estimability of variance components when all model matrices commute. Linear Algebra and its Applications, 492 (2016), 144–160. doi: 10.1016/j.laa.2015.11.002
R. A. Bailey, Peter J. Cameron, Dário Ferreira, Sandra S. Ferreira and Célia Nunes: Designs for half-diallel experiments with commutative orthogonal block structure. Journal of Statistical Planning and Inference, 231 (2024), 106139. doi: 10.1016/j.jspi.2023.106139

What are desirable properties (such as orthogonality, or general balance) of a function between two such sets?
R. A. Bailey: Balance, orthogonality and efficiency factors in factorial design. Journal of the Royal Statistical Society, Series B 47 (1985), 453–458. doi: 10.1111/j.2517-6161.1985.tb01374.x [Maths Reviews 0844475 (87k: 62124)]
R. A. Bailey: Designs: mappings between structured sets. In Surveys in Combinatorics, 1989 (ed. J. Siemons), London Mathematical Society Lecture Note Series, 141, Cambridge University Press, Cambridge, 1989, pp. 22–51. [Maths Reviews 1036750 (90k: 05026)]
R. A. Bailey and C. A. Rowley: General balance and treatment permutations. Linear Algebra and its Applications 127 (1990), 183–225. doi: 10.1016/0024-3795(90)90343-B [Maths Reviews 1048802 (91d: 05014)]
R. A. Bailey: Contribution to the discussion of `The non-orthogonal design of experiments' by R. Mead. Journal of the Royal Statistical Society, Series A 153 (1990), p. 182.
R. A. Bailey: General balance: artificial theory or practical relevance? In Proceedings of the International Conference on Linear Statistical Inference LINSTAT '93 (ed. T. Caliński and R. Kala), 1994, Kluwer, Amsterdam, pp. 171–184. doi: 10.1007/978-94-011-1004-4_19 [Maths Reviews 1333664 (96d: 62144)]
J. P. Morgan and R. A. Bailey: Optimal design with many blocking factors. Annals of Statistics 28 (2000), 553–577. doi: 12.1214/aos/1016218230 [Maths Reviews 1790009 (2001i: 62086)]
R. A. Bailey: Principles of designed experiments in J. A. Nelder's papers. In Methods and Models in Statistics: In Honour of Professor John Nelder, FRS (eds. N. M. Adams, M. J. Crowder, D. J. .Hand and D A. Stephens), Imperial College Press, London, 2004, pp. 171–194.
R. A. Bailey and P. J. Cameron: What is a design? How should we classify them? Designs, Codes and Cryptography 44 (2007), 223–238. doi: 10.1007/s10623-007-9092-3 [Maths Reviews 2336407 (2008f: 05014)]
C. J. Brien and R. A. Bailey: Decomposition tables for experiments. I. A chain of randomizations. Annals of Statistics 37 (2009), 4184–4213. doi: 10.1214/09-AOS717 [Maths Reviews 2572457 (2010k: 62294)]
R. A. Bailey: Structures defined by factors. Chapter 10 in Handbook of Design and Analysis of Experiments (eds. Angela Dean, Max Morris, John Stufken and Derek Bingham), Chapman and Hall/ CRC Handbooks of Modern Statistical Methods, Chapman and Hall/ CRC (2015), pp. 371–414.
R. A. Bailey: Relations among partitions. In Surveys in Combinatorics 2017 (eds. Anders Claesson, Mark Dukes, Sergey Kitaev, David Manlove and Kitty Meeks), London Mathematical Society Lecture Notes, 440, Cambridge University Press, Cambridge, 2017, pp. 1–86.

What about three or more sets?
C. J. Brien and R. A. Bailey: Multiple randomizations (with discussion). Journal of the Royal Statistical Society, Series B 68 (2006), 571–609. doi: 10.1111/j.1467-9868.2006.00557.x [Maths Reviews 2301010]
R. A. Bailey and P. J. Cameron: What is a design? How should we classify them? Designs, Codes and Cryptography 44 (2007), 223–238. doi: 10.1007/s10623-007-9092-3 [Maths Reviews 2336407 (2008f: 05014)]
C. J. Brien and R. A. Bailey: Decomposition tables for experiments. I. A chain of randomizations. Annals of Statistics 37 (2009), 4184–4213. doi: 10.1214/09-AOS717 [Maths Reviews 2572457 (2010k: 62294)]
C. J. Brien and R. A. Bailey: Decomposition tables for experiments. II. Two-one randomizations. Annals of Statistics 38 (2010), 3164–3190. doi: 10.1214/09-AOS785 [Maths Reviews 2722467 (2011j:62187)]
C. J. Brien, B. D. Harch, R. L. Correll and R. A. Bailey: Multiphase experiments with at least one later laboratory phase. I. Orthogonal designs. Journal of Agricultural, Biological and Environmental Statistics 16 (2011), 422–450. doi: 10.1007/s13253-011-0060-z [Maths Reviews 2843135]
R. A. Bailey: Structures defined by factors. Chapter 10 in Handbook of Design and Analysis of Experiments (eds. Angela Dean, Max Morris, John Stufken and Derek Bingham), Chapman and Hall/ CRC Handbooks of Modern Statistical Methods, Chapman and Hall/ CRC (2015), pp. 371–414.

What about possibly non-orthogonal partitions on the same set?
R. A. Bailey: Resolved designs viewed as sets of partitions. In Combinatorial Designs and their Applications (editors F. C. Holroyd, K. A. S. Quinn, C. Rowley and B. S. Webb), Chapman & Hall/CRC Press Research Notes in Mathematics 403, CRC Press LLC, Boca Raton, (1999), pp. 17–47. [Maths Reviews 1678589 (2000e: 05017)]
R. A. Bailey and P. J. Cameron: What is a design? How should we classify them? Designs, Codes and Cryptography 44 (2007), 223–238. doi: 10.1007/s10623-007-9092-3 [Maths Reviews 2336407 (2008f: 05014)]
R. A. Bailey: Relations among partitions. In Surveys in Combinatorics 2017 (eds. Anders Claesson, Mark Dukes, Sergey Kitaev, David Manlove and Kitty Meeks), London Mathematical Society Lecture Notes, 440, Cambridge University Press, Cambridge, 2017, pp. 1–86.

Topics	Some of my publications
What properties are needed to ensure a unique orthogonal decomposition of the corresponding vector space, or to ensure known eigenspaces of the corresponding covariance matrix?	R. A. Bailey: Distributive block structures and their automorphisms. In Combinatorial Mathematics VIII (ed. K. L. McAvaney), Lecture Notes in Mathematics, 884 Springer-Verlag, Berlin, 1981, pp. 115–124. doi: 10.1007/BFb0091813 [Maths Reviews 0641241 (83c: 62119)] T. P. Speed and R. A. Bailey: On a class of association schemes derived from lattices of equivalence relations. In Algebraic Structures and Applications (eds. P. Schultz, C. E. Praeger and R. P. Sullivan), Marcel Dekker, New York, 1982, pp. 55–74. [Maths Reviews 0647166 (83f: 06023)] R. A. Bailey: Block structures for designed experiments. In Applications of Combinatorics (ed. R. J. Wilson), Shiva Publishing, Nantwich, 1982, pp. 1–18. [Maths Reviews 0677504 (83j: 05001)] R. A. Bailey: Contribution to the discussion of `Analysis of variance models in orthogonal designs' by Tue Tjur. International Statistical Review 52 (1984), pp. 65–77. [Maths Reviews 0967202 (89h: 62123)] T. P. Speed and R. A. Bailey: Factorial dispersion models. International Statistical Review 55 (1987), 261–277. doi: 10.2307/1403405 [Maths Reviews 0963143 (89m: 62072)] R. A. Bailey: Contribution to the discussion of `What is an analysis of variance?' by T. P. Speed. Annals of Statistics 15 (1987), pp. 913–916. R. A. Bailey: Orthogonal partitions in designed experiments. Designs, Codes and Cryptography 8 (1996), 45–77. doi: 10.1023/A:1018072606346 [Maths Reviews 1393974 (97g: 62136a) and 1403872 (97g:62136b)] R. A. Bailey: Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge University Press, Cambridge, 2004. 387pp. ISBN 0 521 82446 X. [Maths Reviews 2047311 (2005d: 05001)] R. A. Bailey: Principles of designed experiments in J. A. Nelder's papers. In Methods and Models in Statistics: In Honour of Professor John Nelder, FRS (eds. N. M. Adams, M. J. Crowder, D. J. .Hand and D A. Stephens), Imperial College Press, London, 2004, pp. 171–194. R. A. Bailey: Hasse diagrams in designed experiments: a pictorial aid to thinking about blocking, stratification, degrees of freedom, randomization, and analysis of variance, Região Brasileira da Sociedade Internacional de Biometria, Londrina, Brasil, 2005. 88pp. R. A. Bailey: Design of Comparative Experiments, Cambridge University Press, Cambridge, 2008, 330pp. ISBN 978-0-521-86506-7 and 978-0-521-68357-9. [Maths Reviews 2422352] R. A. Bailey: Structures defined by factors. Chapter 10 in Handbook of Design and Analysis of Experiments (eds. Angela Dean, Max Morris, John Stufken and Derek Bingham), Chapman and Hall/ CRC Handbooks of Modern Statistical Methods, Chapman and Hall/ CRC (2015), pp. 371–414. R. A. Bailey, Sandra S. Ferreira, Dário Ferreia and Célia Nunes: Estimability of variance components when all model matrices commute. Linear Algebra and its Applications, 492 (2016), 144–160. doi: 10.1016/j.laa.2015.11.002 R. A. Bailey, Peter J. Cameron, Dário Ferreira, Sandra S. Ferreira and Célia Nunes: Designs for half-diallel experiments with commutative orthogonal block structure. Journal of Statistical Planning and Inference, 231 (2024), 106139. doi: 10.1016/j.jspi.2023.106139
What are desirable properties (such as orthogonality, or general balance) of a function between two such sets?	R. A. Bailey: Balance, orthogonality and efficiency factors in factorial design. Journal of the Royal Statistical Society, Series B 47 (1985), 453–458. doi: 10.1111/j.2517-6161.1985.tb01374.x [Maths Reviews 0844475 (87k: 62124)] R. A. Bailey: Designs: mappings between structured sets. In Surveys in Combinatorics, 1989 (ed. J. Siemons), London Mathematical Society Lecture Note Series, 141, Cambridge University Press, Cambridge, 1989, pp. 22–51. [Maths Reviews 1036750 (90k: 05026)] R. A. Bailey and C. A. Rowley: General balance and treatment permutations. Linear Algebra and its Applications 127 (1990), 183–225. doi: 10.1016/0024-3795(90)90343-B [Maths Reviews 1048802 (91d: 05014)] R. A. Bailey: Contribution to the discussion of `The non-orthogonal design of experiments' by R. Mead. Journal of the Royal Statistical Society, Series A 153 (1990), p. 182. R. A. Bailey: General balance: artificial theory or practical relevance? In Proceedings of the International Conference on Linear Statistical Inference LINSTAT '93 (ed. T. Caliński and R. Kala), 1994, Kluwer, Amsterdam, pp. 171–184. doi: 10.1007/978-94-011-1004-4_19 [Maths Reviews 1333664 (96d: 62144)] J. P. Morgan and R. A. Bailey: Optimal design with many blocking factors. Annals of Statistics 28 (2000), 553–577. doi: 12.1214/aos/1016218230 [Maths Reviews 1790009 (2001i: 62086)] R. A. Bailey: Principles of designed experiments in J. A. Nelder's papers. In Methods and Models in Statistics: In Honour of Professor John Nelder, FRS (eds. N. M. Adams, M. J. Crowder, D. J. .Hand and D A. Stephens), Imperial College Press, London, 2004, pp. 171–194. R. A. Bailey and P. J. Cameron: What is a design? How should we classify them? Designs, Codes and Cryptography 44 (2007), 223–238. doi: 10.1007/s10623-007-9092-3 [Maths Reviews 2336407 (2008f: 05014)] C. J. Brien and R. A. Bailey: Decomposition tables for experiments. I. A chain of randomizations. Annals of Statistics 37 (2009), 4184–4213. doi: 10.1214/09-AOS717 [Maths Reviews 2572457 (2010k: 62294)] R. A. Bailey: Structures defined by factors. Chapter 10 in Handbook of Design and Analysis of Experiments (eds. Angela Dean, Max Morris, John Stufken and Derek Bingham), Chapman and Hall/ CRC Handbooks of Modern Statistical Methods, Chapman and Hall/ CRC (2015), pp. 371–414. R. A. Bailey: Relations among partitions. In Surveys in Combinatorics 2017 (eds. Anders Claesson, Mark Dukes, Sergey Kitaev, David Manlove and Kitty Meeks), London Mathematical Society Lecture Notes, 440, Cambridge University Press, Cambridge, 2017, pp. 1–86.
What about three or more sets?	C. J. Brien and R. A. Bailey: Multiple randomizations (with discussion). Journal of the Royal Statistical Society, Series B 68 (2006), 571–609. doi: 10.1111/j.1467-9868.2006.00557.x [Maths Reviews 2301010] R. A. Bailey and P. J. Cameron: What is a design? How should we classify them? Designs, Codes and Cryptography 44 (2007), 223–238. doi: 10.1007/s10623-007-9092-3 [Maths Reviews 2336407 (2008f: 05014)] C. J. Brien and R. A. Bailey: Decomposition tables for experiments. I. A chain of randomizations. Annals of Statistics 37 (2009), 4184–4213. doi: 10.1214/09-AOS717 [Maths Reviews 2572457 (2010k: 62294)] C. J. Brien and R. A. Bailey: Decomposition tables for experiments. II. Two-one randomizations. Annals of Statistics 38 (2010), 3164–3190. doi: 10.1214/09-AOS785 [Maths Reviews 2722467 (2011j:62187)] C. J. Brien, B. D. Harch, R. L. Correll and R. A. Bailey: Multiphase experiments with at least one later laboratory phase. I. Orthogonal designs. Journal of Agricultural, Biological and Environmental Statistics 16 (2011), 422–450. doi: 10.1007/s13253-011-0060-z [Maths Reviews 2843135] R. A. Bailey: Structures defined by factors. Chapter 10 in Handbook of Design and Analysis of Experiments (eds. Angela Dean, Max Morris, John Stufken and Derek Bingham), Chapman and Hall/ CRC Handbooks of Modern Statistical Methods, Chapman and Hall/ CRC (2015), pp. 371–414.
What about possibly non-orthogonal partitions on the same set?	R. A. Bailey: Resolved designs viewed as sets of partitions. In Combinatorial Designs and their Applications (editors F. C. Holroyd, K. A. S. Quinn, C. Rowley and B. S. Webb), Chapman & Hall/CRC Press Research Notes in Mathematics 403, CRC Press LLC, Boca Raton, (1999), pp. 17–47. [Maths Reviews 1678589 (2000e: 05017)] R. A. Bailey and P. J. Cameron: What is a design? How should we classify them? Designs, Codes and Cryptography 44 (2007), 223–238. doi: 10.1007/s10623-007-9092-3 [Maths Reviews 2336407 (2008f: 05014)] R. A. Bailey: Relations among partitions. In Surveys in Combinatorics 2017 (eds. Anders Claesson, Mark Dukes, Sergey Kitaev, David Manlove and Kitty Meeks), London Mathematical Society Lecture Notes, 440, Cambridge University Press, Cambridge, 2017, pp. 1–86.

Page maintained by R. A. Bailey