Designs for special layouts

Sometimes a single system of blocks is not enough, and a more complicated layout is needed. Then a design suitable for that layout is needed. Examples include Latin squares and semi-Latin squares.

Layouts Some of my publications

Observational units inside experimental units (m / n)
T. H. Sparks, R. A. Bailey and D. A. Elston: Pseudoreplication: common (mal)practice. SETAC (Society of Environmental Toxicology and Chemistry) News, 17(3) (1997), 12–13.
R. A. Bailey and J. J. D. Greenwood: Effects of neonicotinoids on bees: an invalid experiment. Ecotoxicology, 27 (2018), 1–7. doi: 10.1007/s10646-017-1877

Small blocks inside large blocks (n / b / k)
R. A. Bailey and T. P. Speed: Rectangular lattice designs: efficiency factors and analysis. Annals of Statistics 14 (1986), 874–895. doi: 12.1214/aos/117635009 [Maths Reviews 0856795 (88e: 62186)]
R. A. Bailey, H. Monod and J. P. Morgan: Construction and optimality of affine-resolvable designs. Biometrika 82 (1995), 187–200. doi: 10.1093/biomet/82.1.187 [Maths Reviews 1332849 (96k: 62212)]
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: Choosing designs for nested blocks. Listy Biometryczne 36 (1999), 85–126. url: https://sparrow.up.poznan.pl/biometrical.letters/full/BL%2036%202%201.pdf
R. A. Bailey: Six families of efficient resolvable designs in three replicates. Metrika 62 (2005), 161–173. doi: 10.1007/s00184-005-0405-0 [Maths Reviews 2274987]

Rows and columns (r × c)
R. A. Bailey, D. A. Preece and P. J. Zemroch: Totally symmetric Latin squares and cubes. Utilitas Mathematica 14 (1978), 161–170. [Maths Reviews 0511521 (80a: 05032)]
R. A. Bailey: Enumeration of totally symmetric Latin squares. Utilitas Mathematica 15 (1979), 193–216. [Maths Reviews 0531629 (80d: 05014a) and 0557001 (80d: 05014b)]
R. A. Bailey: Latin squares with highly transitive automorphism groups. Journal of the Australian Mathematical Society, Series A 33 (1982), 18–22. doi: 10.1017/S1446788700017560 [Maths Reviews 0662355 (83g: 05021)]
R. A. Bailey: Quasi-complete Latin squares: construction and randomization. Journal of the Royal Statistical Society, Series B 46 (1984), 323–334. doi: 10.1111/j.2517-6161.1984.tb01305.x [Maths Reviews 0781893 (86i: 62161)]
R. A. Bailey, D. A. Preece and C. A. Rowley: Randomization for a balanced superimposition of one Youden square on another. Journal of the Royal Statistical Society, Series B 57 (1995), 459–469. doi: 10.1111/j.2517-6161.1995.tb02040.x [Maths Reviews 1323350 (96i: 62083)]
Ian Anderson and R. A. Bailey: Completeness properties of conjugates of Latin squares based on groups, and an application to bipartite tournaments. Bulletin of the Institute of Combinatorics and its Applications 21 (1997), 95–99. [Maths Reviews 1470311 (98d: 05032)]
R. A. Bailey: Designs for two-colour microarray experiments. Applied Statistics 56 (2007), 365–394. doi: 10.1111/j.1467-9876.2007.00582.x [Maths Reviews 2409757]
C. J. Brien, R. A. Bailey, T. T. Tran and J. Bolund: Quasi-Latin designs. Electronic Journal of Statistics, 6 (2012), 1900–1925. doi: 10.1214/12-EJS732 [Maths Reviews 2988468]
R. A. Bailey, Peter J. Cameron and Tomas Nilson: Sesqui-arrays, a generalisation of triple arrays. Australasian Journal of Combinatorics, 71 (2018), 427–451. [Maths Reviews 3801275]

Small blocks inside rows and columns ((r × c) / n)
R. A. Bailey: Semi-Latin squares. Journal of Statistical Planning and Inference 18 (1988), 299–312. doi: 10.1016/0378-3758(88)90107-3 [Maths Reviews 0926634 (89b: 62160)]
R. A. Bailey: An efficient semi-Latin square for twelve treatments in blocks of size two. Journal of Statistical Planning and Inference 26 (1990), 263–266. doi: 10.1016/0378-3758(90)90130-M [Maths Reviews 1086100]
R. A. Bailey: Efficient semi-Latin squares. Statistica Sinica 2 (1992), 413–437. url: http://www.jstor.org/stable/24304868 [Maths Reviews 1187951 (93j: 62199)]
R. A. Bailey: Recent advances in experimental design in agriculture. Bulletin of the International Statistical Institute 55 (1) (1993), 179–193.
R. A. Bailey: A Howell design admitting A₅. Discrete Mathematics 167-168 (1997), 65–71. doi: 10.1016/S0012-365X(96)00216-6 [Maths Reviews 1446733 (98h: 05039)]
R. A. Bailey and P. E. Chigbu: Enumeration of semi-Latin squares. Discrete Mathematics 167-168 (1997), 73–84. doi: 10.1016/S0012-365X(96)00217-8 [Maths Reviews 1446734 (98g: 05026)]
R. A. Bailey and G. Royle: Optimal semi-Latin squares with side six and block size two. Proceedings of the Royal Society, Series A 453 (1997), 1903–1914. doi: 10.1098/rspa.1997.0102 [Maths Reviews 1478138 (98k: 62130)]
R. A. Bailey and H. Monod: Efficient semi-Latin rectangles: designs for plant disease experiments. Scandinavian Journal of Statistics 28 (2001), 257–270. doi: 10.1111/1467-9469.00235 [Maths Reviews 1842248 (2002c: 05038)]
R. A. Bailey: Semi-Latin squares. Wiley StatsRef: Statistics Reference Online, 2017, pp. 1–8. doi: 10.1002/9781118445112.stat02644.pub2
N. P. Uto and R. A. Bailey: Balanced semi-Latin rectangles: Properties, existence and constructions for block size two. Journal of Statistical Theory and Practice, 14:51 (2020). doi: 10.1007/s42519-020-00118-3
R. A. Bailey and Leonard H. Soicher: Uniform semi-Latin squares and their pairwise variance aberrations. Journal of Statistical Planning and Inference, 213 (2021), 282‐291. doi: 10.1016/j.jspi.2020.12.003
N. P. Uto and R. A. Bailey: Constructions for regular-graph semi-Latin rectangles with block size two. Journal of Statistical Planning and Inference, 221 (2022), 81–89. doi: 10.1016/j.jspi.2022.02.007

Rows and columns inside large blocks b / (r × c)
R. A. Bailey and H. D. Patterson: A note on the construction of row-and-column designs with two replicates. Journal of the Royal Statistical Society, Series B 53 (1991), 645–648. doi: 10.1111/j.2517-6161.1991.tb01853.x [Maths Reviews 1125721]
H. Monod and R. A. Bailey: Pseudofactors: normal use to improve design and facilitate analysis. Applied Statistics 41 (1992), 317–336. doi: 10.2307/2347564
R. A. Bailey: Recent advances in experimental design in agriculture. Bulletin of the International Statistical Institute 55 (1) (1993), 179–193.
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 E. R. Williams: Optimal nested row-column designs with specified components. Biometrika 94 (2007), 459–468. doi: 10.1093/biomet/asm039 [Maths Reviews 2331490]
C. J. Brien, R. A. Bailey, T. T. Tran and J. Bolund: Quasi-Latin designs. Electronic Journal of Statistics, 6 (2012), 1900–1925. doi: 10.1214/12-EJS732 [Maths Reviews 2988468]
R. A. Bailey and A. Łacka: Nested row-column designs for near-factorial experiments with two treatment factors and one control treatment. Journal of Statistical Planning and Inference, 165 (2015), 63–77. doi: 10.1016/j.jspi.2015.04.003

Small blocks inside large blocks, with rows running across all small blocks (r × (b / k))
R. A. Bailey and T. P. Speed: Rectangular lattice designs: efficiency factors and analysis. Annals of Statistics 14 (1986), 874–895. doi: 12.1214/aos/117635009 [Maths Reviews 0856795 (88e: 62186)]
R. A. Bailey: Recent advances in experimental design in agriculture. Bulletin of the International Statistical Institute 55 (1) (1993), 179–193.
C. J. Brien, R. A. Bailey, T. T. Tran and J. Bolund: Quasi-Latin designs. Electronic Journal of Statistics, 6 (2012), 1900–1925. doi: 10.1214/12-EJS732 [Maths Reviews 2988468]

Smalls rows inside large rows at the same time as small columns inside large columns ((r / n) × (c / m))
R. A. Bailey, J. Kunert and R. J. Martin: Some comments on gerechte designs. I. Analysis for uncorrelated errors. Journal of Agronomy & Crop Science 165 (1990), 121–130. doi: 10.1111/j.1439-037X.1990.tb00842.x
R. A. Bailey, J. Kunert and R. J. Martin: Some comments on gerechte designs. II. Randomization analysis, and other methods that allow for inter-plot dependence. Journal of Agronomy & Crop Science 166 (1991), 101–111. doi: 10.1111/j.1439-037X.1991.tb00891.x
R. A. Bailey, Peter J. Cameron and Robert Connolly: Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes. The American Mathematical Monthly 115 (2008), 383–404. url: https://www.jstor.org/stable/27642500
(Translated into Chinese in Mathematical Advance in Translation 28 (2009), 21–39.) [Maths Reviews 2408485]

Layouts	Some of my publications
Observational units inside experimental units (m / n)	T. H. Sparks, R. A. Bailey and D. A. Elston: Pseudoreplication: common (mal)practice. SETAC (Society of Environmental Toxicology and Chemistry) News, 17(3) (1997), 12–13. R. A. Bailey and J. J. D. Greenwood: Effects of neonicotinoids on bees: an invalid experiment. Ecotoxicology, 27 (2018), 1–7. doi: 10.1007/s10646-017-1877
Small blocks inside large blocks (n / b / k)	R. A. Bailey and T. P. Speed: Rectangular lattice designs: efficiency factors and analysis. Annals of Statistics 14 (1986), 874–895. doi: 12.1214/aos/117635009 [Maths Reviews 0856795 (88e: 62186)] R. A. Bailey, H. Monod and J. P. Morgan: Construction and optimality of affine-resolvable designs. Biometrika 82 (1995), 187–200. doi: 10.1093/biomet/82.1.187 [Maths Reviews 1332849 (96k: 62212)] 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: Choosing designs for nested blocks. Listy Biometryczne 36 (1999), 85–126. url: https://sparrow.up.poznan.pl/biometrical.letters/full/BL%2036%202%201.pdf R. A. Bailey: Six families of efficient resolvable designs in three replicates. Metrika 62 (2005), 161–173. doi: 10.1007/s00184-005-0405-0 [Maths Reviews 2274987]
Rows and columns (r × c)	R. A. Bailey, D. A. Preece and P. J. Zemroch: Totally symmetric Latin squares and cubes. Utilitas Mathematica 14 (1978), 161–170. [Maths Reviews 0511521 (80a: 05032)] R. A. Bailey: Enumeration of totally symmetric Latin squares. Utilitas Mathematica 15 (1979), 193–216. [Maths Reviews 0531629 (80d: 05014a) and 0557001 (80d: 05014b)] R. A. Bailey: Latin squares with highly transitive automorphism groups. Journal of the Australian Mathematical Society, Series A 33 (1982), 18–22. doi: 10.1017/S1446788700017560 [Maths Reviews 0662355 (83g: 05021)] R. A. Bailey: Quasi-complete Latin squares: construction and randomization. Journal of the Royal Statistical Society, Series B 46 (1984), 323–334. doi: 10.1111/j.2517-6161.1984.tb01305.x [Maths Reviews 0781893 (86i: 62161)] R. A. Bailey, D. A. Preece and C. A. Rowley: Randomization for a balanced superimposition of one Youden square on another. Journal of the Royal Statistical Society, Series B 57 (1995), 459–469. doi: 10.1111/j.2517-6161.1995.tb02040.x [Maths Reviews 1323350 (96i: 62083)] Ian Anderson and R. A. Bailey: Completeness properties of conjugates of Latin squares based on groups, and an application to bipartite tournaments. Bulletin of the Institute of Combinatorics and its Applications 21 (1997), 95–99. [Maths Reviews 1470311 (98d: 05032)] R. A. Bailey: Designs for two-colour microarray experiments. Applied Statistics 56 (2007), 365–394. doi: 10.1111/j.1467-9876.2007.00582.x [Maths Reviews 2409757] C. J. Brien, R. A. Bailey, T. T. Tran and J. Bolund: Quasi-Latin designs. Electronic Journal of Statistics, 6 (2012), 1900–1925. doi: 10.1214/12-EJS732 [Maths Reviews 2988468] R. A. Bailey, Peter J. Cameron and Tomas Nilson: Sesqui-arrays, a generalisation of triple arrays. Australasian Journal of Combinatorics, 71 (2018), 427–451. [Maths Reviews 3801275]
Small blocks inside rows and columns ((r × c) / n)	R. A. Bailey: Semi-Latin squares. Journal of Statistical Planning and Inference 18 (1988), 299–312. doi: 10.1016/0378-3758(88)90107-3 [Maths Reviews 0926634 (89b: 62160)] R. A. Bailey: An efficient semi-Latin square for twelve treatments in blocks of size two. Journal of Statistical Planning and Inference 26 (1990), 263–266. doi: 10.1016/0378-3758(90)90130-M [Maths Reviews 1086100] R. A. Bailey: Efficient semi-Latin squares. Statistica Sinica 2 (1992), 413–437. url: http://www.jstor.org/stable/24304868 [Maths Reviews 1187951 (93j: 62199)] R. A. Bailey: Recent advances in experimental design in agriculture. Bulletin of the International Statistical Institute 55 (1) (1993), 179–193. R. A. Bailey: A Howell design admitting A₅. Discrete Mathematics 167-168 (1997), 65–71. doi: 10.1016/S0012-365X(96)00216-6 [Maths Reviews 1446733 (98h: 05039)] R. A. Bailey and P. E. Chigbu: Enumeration of semi-Latin squares. Discrete Mathematics 167-168 (1997), 73–84. doi: 10.1016/S0012-365X(96)00217-8 [Maths Reviews 1446734 (98g: 05026)] R. A. Bailey and G. Royle: Optimal semi-Latin squares with side six and block size two. Proceedings of the Royal Society, Series A 453 (1997), 1903–1914. doi: 10.1098/rspa.1997.0102 [Maths Reviews 1478138 (98k: 62130)] R. A. Bailey and H. Monod: Efficient semi-Latin rectangles: designs for plant disease experiments. Scandinavian Journal of Statistics 28 (2001), 257–270. doi: 10.1111/1467-9469.00235 [Maths Reviews 1842248 (2002c: 05038)] R. A. Bailey: Semi-Latin squares. Wiley StatsRef: Statistics Reference Online, 2017, pp. 1–8. doi: 10.1002/9781118445112.stat02644.pub2 N. P. Uto and R. A. Bailey: Balanced semi-Latin rectangles: Properties, existence and constructions for block size two. Journal of Statistical Theory and Practice, 14:51 (2020). doi: 10.1007/s42519-020-00118-3 R. A. Bailey and Leonard H. Soicher: Uniform semi-Latin squares and their pairwise variance aberrations. Journal of Statistical Planning and Inference, 213 (2021), 282‐291. doi: 10.1016/j.jspi.2020.12.003 N. P. Uto and R. A. Bailey: Constructions for regular-graph semi-Latin rectangles with block size two. Journal of Statistical Planning and Inference, 221 (2022), 81–89. doi: 10.1016/j.jspi.2022.02.007
Rows and columns inside large blocks b / (r × c)	R. A. Bailey and H. D. Patterson: A note on the construction of row-and-column designs with two replicates. Journal of the Royal Statistical Society, Series B 53 (1991), 645–648. doi: 10.1111/j.2517-6161.1991.tb01853.x [Maths Reviews 1125721] H. Monod and R. A. Bailey: Pseudofactors: normal use to improve design and facilitate analysis. Applied Statistics 41 (1992), 317–336. doi: 10.2307/2347564 R. A. Bailey: Recent advances in experimental design in agriculture. Bulletin of the International Statistical Institute 55 (1) (1993), 179–193. 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 E. R. Williams: Optimal nested row-column designs with specified components. Biometrika 94 (2007), 459–468. doi: 10.1093/biomet/asm039 [Maths Reviews 2331490] C. J. Brien, R. A. Bailey, T. T. Tran and J. Bolund: Quasi-Latin designs. Electronic Journal of Statistics, 6 (2012), 1900–1925. doi: 10.1214/12-EJS732 [Maths Reviews 2988468] R. A. Bailey and A. Łacka: Nested row-column designs for near-factorial experiments with two treatment factors and one control treatment. Journal of Statistical Planning and Inference, 165 (2015), 63–77. doi: 10.1016/j.jspi.2015.04.003
Small blocks inside large blocks, with rows running across all small blocks (r × (b / k))	R. A. Bailey and T. P. Speed: Rectangular lattice designs: efficiency factors and analysis. Annals of Statistics 14 (1986), 874–895. doi: 12.1214/aos/117635009 [Maths Reviews 0856795 (88e: 62186)] R. A. Bailey: Recent advances in experimental design in agriculture. Bulletin of the International Statistical Institute 55 (1) (1993), 179–193. C. J. Brien, R. A. Bailey, T. T. Tran and J. Bolund: Quasi-Latin designs. Electronic Journal of Statistics, 6 (2012), 1900–1925. doi: 10.1214/12-EJS732 [Maths Reviews 2988468]
Smalls rows inside large rows at the same time as small columns inside large columns ((r / n) × (c / m))	R. A. Bailey, J. Kunert and R. J. Martin: Some comments on gerechte designs. I. Analysis for uncorrelated errors. Journal of Agronomy & Crop Science 165 (1990), 121–130. doi: 10.1111/j.1439-037X.1990.tb00842.x R. A. Bailey, J. Kunert and R. J. Martin: Some comments on gerechte designs. II. Randomization analysis, and other methods that allow for inter-plot dependence. Journal of Agronomy & Crop Science 166 (1991), 101–111. doi: 10.1111/j.1439-037X.1991.tb00891.x R. A. Bailey, Peter J. Cameron and Robert Connolly: Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes. The American Mathematical Monthly 115 (2008), 383–404. url: https://www.jstor.org/stable/27642500 (Translated into Chinese in Mathematical Advance in Translation 28 (2009), 21–39.) [Maths Reviews 2408485]

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