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
SpringerVerlag, Berlin,
1981,
pp. 115–124.
[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.
[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.
[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 9780521865067 and 9780521683579.
[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

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.
[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.
[Maths Reviews 1048802 (91d: 05014)]
R. A. Bailey:
Contribution to the discussion of
`The nonorthogonal 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.
[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.
[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/s1062300790923
[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: 10.1214/09AOS717
[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.14679868.2006.00557.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/s1062300790923
[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: 10.1214/09AOS717
[Maths Reviews 2572457 (2010k: 62294)]
C. J. Brien and R. A. Bailey:
Decomposition tables for experiments. II. Twoone randomizations.
Annals of Statistics
38
(2010),
3164–3190.
doi: 10.1214/09AOS785
[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/s132530110060z
[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 nonorthogonal 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/s1062300790923
[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.
