Relations among partitions

doi:10.1017/9781108332699.002

1 - Relations among partitions

Published online by Cambridge University Press: 21 July 2017

Edited by

David Manlove and

R. A. Bailey: Affiliation:
University of St Andrews
Anders Claesson: Affiliation:
University of Iceland, Reykjavik
Mark Dukes: Affiliation:
University College Dublin
Sergey Kitaev: Affiliation:
University of Strathclyde
David Manlove: Affiliation:
University of Glasgow
Kitty Meeks: Affiliation:
University of Glasgow

Book contents

Get access

Summary

Abstract

Combinatorialists often consider a balanced incomplete-block design to consist of a set of points, a set of blocks, and an incidence relation between them which satisfies certain conditions. To a statistician, such a design is a set of experimental units with two partitions, one into blocks and the other into treatments; it is the relation between these two partitions which gives the design its properties. The most common binary relations between partitions that occur in statistics are refinement, orthogonality and balance. When there are more than two partitions, the binary relations may not suffice to give all the properties of the system. I shall survey work in this area, including designs such as double Youden rectangles.

Introduction

Many combinatorialists think of a balanced incomplete-block design (BIBD) as a set P of points together with a collection B of subsets of P, called blocks, which satisfy various conditions. For example, see [52]. Some papers, such as [16, 65, 201], call a BIBD simply a design. Others think of it as the pair of sets P and B with a binary incidence relation between their elements. These views are both rather different from that of a statistician who is involved in designing experiments. The following examples introduce the statistical point of view, as well as serving as a basis for the combinatorial ideas in this paper.

Example 1.1 A horticultural enthusiast wants to compare three varieties of lettuce for people to grow in their own gardens. He enlists twelve people in his neighbourhood. Each of these prepares three patches in their vegetable garden, and grows one of the lettuce varieties on each patch, so that each gardener grows all three varieties.

Here the patches of land are experimental units. There may be some differences between the gardeners, so the three patches in a single garden form what is called a block. Each variety occurs just once in each block, and so the blocks are said to be complete. Complete-block designs were advocated by Fisher in [78], and are frequently used in practice.

Information

Type: Chapter
Information: Surveys in Combinatorics 2017 , pp. 1 - 86
DOI: https://doi.org/10.1017/9781108332699.002 [Opens in a new window]
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

[1] R., Julian R., Abel, Malcolm, Grieg & D. H., Rees, Existence of OBIBDs with k = 4 with and without nesting, Discrete Mathematics 266 (2003), 3–36.Google Scholar

[2] K., Afsarinejad & A., Hedayat, Some contributions to the theory of multistage Youden design, Annals of Statistics 3 (1975), 707–711.Google Scholar

[3] Hira Lal, Agrawal, Two way elimination of heterogeneity, Calcutta Statistical Association Bulletin 15 (1966), 32–38.Google Scholar

[4] Hira Lal, Agrawal, Some generalizations of distinct representatives with applications to statistical design, Annals of Mathematical Statistics 37 (1966), 525–528.Google Scholar

[5] Hira Lal, Agrawal, Some systematic methods of construction for twoway elimination of heterogeneity, Calcutta Statistical Association Bulletin 15 (1966), 93–108.Google Scholar

[6] Hira Lal, Agrawal, Some methods of construction of designs for twoway elimination of heterogeneity—1, Journal of the American Statistical Association 61 (1966), 1153–1171.Google Scholar

[7] H. L., Agrawal & R. I., Mishra, Some methods of constructing 4DIB designs, Calcutta Statistical Association Bulletin 20 (1971), 89–92.Google Scholar

[8] H. L., Agrawal & J., Prasad, Some methods of construction of balanced incomplete block designs with nested rows and columns, Biometrika 69 (1982), 481–483.Google Scholar

[9] H. L., Agrawal & J., Prasad, On construction of balanced incomplete block designs with nested rows and columns, Sankhyā, Series B 45 (1983), 345–350.Google Scholar

[10] H. L., Agrawal & K. L., Sharma, On construction of two-way designs, Journal of the Indian Statistical Association 13 (1975), 1–31.Google Scholar

[11] H. L., Agrawal & K. L., Sharma, On construction of four-dimensional incomplete block designs, Journal of the American Statistical Association 73 (1978), 844–849.Google Scholar

[12] D. A., Anderson & J. A., Eccleston, On the construction of a class of efficient row–column designs, Journal of Statistical Planning and Inference 11 (1985), 131–134.Google Scholar

[13] D. A., Anderson & W. T., Federer, Multidimensional balanced designs, Communications in Statistics – Theory and Methods 5 (1976), 1193–1204.Google Scholar

[14] I., Anderson, A hundred years of whist tournaments, Journal of Combinatorial Mathematics and Combinatorial Computation 19 (1995), 129–150.Google Scholar

[15] A., Ash, Generalized Youden designs: constructions and tables, Journal of Statistical Planning and Inference 5 (1981), 1–25.Google Scholar

[16] E. F., Assmus, Jr, On the theory of designs, in Surveys in Combinatorics, 1989 (ed. J., Siemons), London Mathematical Society Lecture Notes Series, 141, Cambridge University Press, Cambridge (1989), pp. 1–21.

[17] Sunanda, Bagchi, An infinite series of adjusted orthogonal designs with replication two, Statistica Sinica 6 (1996), 975–987.Google Scholar

[18] Sunanda, Bagchi, On two–way designs, Graphs and Combinatorics 14 (1998), 313–319.Google Scholar

[19] S., Bagchi & E. E. M. van, Berkum, On the optimality of a new class of adjusted orthogonal designs, Journal of Statistical Planning and Inference 28 (1991), 61–65.Google Scholar

[20] Sunanda, Bagchi, A. C., Mukhopadhyay & Bikas K., Sinha, A search for optimal nested row–column designs, Sankhyā, Series B 52 (1990), 93–104.Google Scholar

[21] S., Bagchi & K. R., Shah, On the optimality of a class of row–column designs, Journal of Statistical Planning and Inference 23 (1989), 397–402.Google Scholar

[22] R. A., Bailey, Designs: mappings between structured sets, in Surveys in Combinatorics, 1989 (ed. J., Siemons), London Mathematical Society Lecture Notes Series, 141, Cambridge University Press, Cambridge (1989), pp. 22–51.

[23] R. A., Bailey, Orthogonal partitions in designed experiments, Designs, Codes and Cryptography 8 (1996), 45–77.Google Scholar

[24] R. A., Bailey, Choosing designs for nested blocks, Listy Biometryczne 36 (1999), 85–126.Google Scholar

[25] R. A., Bailey, Resolved designs viewed as sets of partitions, in Combinatorial designs and their applications (eds. Fred C., Holroyd, Kathleen A. S., Quinn, Chris, Rowley & Bridget S., Webb), Research Notes in Mathematics, 403, Chapman & Hall, Boca Raton (1999), pp. 17– 47.

[26] R. A., Bailey, Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge Studies in Advanced Mathematics, 84, Cambridge University Press, Cambridge (2004).

[27] R. A., Bailey, Six families of efficient resolvable designs in three replicates, Metrika 62 (2005), 161–173.Google Scholar

[28] R. A., Bailey, Design of Comparative Experiments, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge (2008).

[29] R. A., Bailey, Donald Arthur Preece: A life in statistics, mathematics and music, arXiv 1402.2220.

[30] R. A., Bailey, Structures defined by factors, in Handbook of Design and Analysis of Experiments (eds. Angela, Dean, Max, Morris, John, Stufken & Derek, Bingham), Chapman and Hall/ CRC Handbooks of Modern Statistical Methods, Chapman and Hall/ CRC, Boca Raton (2015), pp. 371–414.

[31] R. A., Bailey & C. J., Brien, Randomization-based models for multitiered experiments: I. A chain of randomizations, Annals of Statistics 44 (2016), 1131–1164.Google Scholar

[32] R. A., Bailey & Peter J., Cameron, What is a design? How should we classify them?, Designs, Codes and Cryptography 44 (2007), 223– 238.Google Scholar

[33] R. A., Bailey & Peter J., Cameron, Combinatorics of optimal designs, in Surveys in Combinatorics 2009 (eds. Sophie, Huczynska, James D., Mitchell & Colva, Rone-Dougal), London Mathematical Society Lecture Note Series, 365, (2009), pp. 19–73.Google Scholar

[34] R. A., Bailey & Peter J., Cameron, Designs which allow each medical centre to treat only a limited number of cancer types with only a limited number of drugs, preprint, University of St Andrews, 2016.

[35] R. A., Bailey, Sandra S., Ferreira, Dário, Ferreia & Célia, Nunes, Estimability of variance components when all model matrices commute, Linear Algebra and its Applications 492 (2016), 144–160.Google Scholar

[36] R. A., Bailey, D. C., Goldrei & D. F., Holt, Block designs with block size two, Journal of Statistical Planning and Inference 10 (1984), 257–263.Google Scholar

[37] R. A., Bailey & P., Heidtmann, Extremal row–column designs with maximal balance and adjusted orthogonality, preprint, Goldsmiths' College, University of London, 1994.

[38] R. A., Bailey, H., Monod & J. P., Morgan, Construction and optimality of affine-resolvable designs, Biometrika 82 (1995), 187–200.Google Scholar

[39] R. A., Bailey & H. D., Patterson, A note on the construction of rowand- column designs with two replicates, Journal of the Royal Statistical Society, Series B 53 (1991), 645–648.Google Scholar

[40] R. A., Bailey, D. A., Preece & 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.Google Scholar

[41] R. A., Bailey & T. P., Speed, Rectangular lattice designs: efficiency factors and analysis, Annals of Statistics 14 (1986), 874–895.Google Scholar

[42] Simon T., Bate & Marion J., Chatfield, Identifying the structure of the experimental design, Journal of Quality Technology 48 (2016), 343–364.Google Scholar

[43] Simon T., Bate & Marion J., Chatfield, Using the structure of the experimental design and the randomization to construct a mixed model, Journal of Quality Technology 48 (2016), 365–387.Google Scholar

[44] Gerhard, Behrendt, Equivalence systems with finitely many relations, Monatshefte für Mathematik 103 (1987), 77–83.Google Scholar

[45] Thomas, Beth, Dieter, Jungnickel & Hanfried, Lenz, Design Theory, Volume I, second edition, Encyclopedia of Mathematics and its Applications, 69, Cambridge University Press, Cambridge (1999).

[46] R. C., Bose, On the construction of balanced incomplete block designs, Annals of Eugenics 9 (1939), 353–399.Google Scholar

[47] R. C., Bose, A note on the resolvability of balanced incomplete block designs, Sankhā 6 (1942), 105–110.Google Scholar

[48] Wayne, Broughton, Admissible parameters of symmetric designs satisfying v = 4(k - ƛ) + 2 and symmetric designs with inner balance, Designs, Codes and Cryptography 73 (2014), 77–83.Google Scholar

[49] A. R., Calderbank, P. J., Cameron, W. M., Kantor & J. J., Seidel, Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean linesets, Proceedings of the London Mathematical Society, Series 3 75 (1997), 436–480.Google Scholar

[50] Peter J., Cameron, On groups with several doubly-transitive permutation representations, Mathematische Zeitschrift 128 (1972), 1–14.Google Scholar

[51] P. J., Cameron, Multi-letter Youden rectangles from quadratic forms, Discrete Mathematics 266 (2003), 143–151.Google Scholar

[52] P. J., Cameron & J. H. van, Lint, Designs, Graphs, Codes and their Links, London Mathematical Society Student Texts, 22, Cambridge University Press, Cambridge (1991).

[53] P. J., Cameron & J. J., Seidel, Quadratic forms over GF(2), Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A 76 (1973), 1–8.Google Scholar

[54] B. D., Causey, Some examples of multi-dimensional incomplete block designs, Annals of Mathematical Statistics 39 (1968), 1577–1590.Google Scholar

[55] J. Y., Chang & W. I., Notz, A method for constructing universally optimal block designs with nested rows and columns, Utilitas Mathematica 38 (1990), 263–276.Google Scholar

[56] C.-S., Cheng, A method for constructing balanced incomplete-block designs with nested rows and columns, Biometrika 73 (1986), 695– 700.Google Scholar

[57] C., Christofi, Enumerating 4×5 and 5×6 double Youden rectangles, Discrete Mathematics 125 (1994), 129–135.Google Scholar

[58] C., Christofi, On the number of 6×7 double Youden rectangles, Ars Combinatoria 47 (1997), 223–241.Google Scholar

[59] G. M., Clarke, A second set of treatments in a Youden square design, Biometrics 19 (1963), 98–104.Google Scholar

[60] G. M., Clarke, Four-way balanced designs based on Youden squares with 5, 6, or 7 treatments, Biometrics 23 (1967), 803–812.Google Scholar

[61] Dominique, Collombier, Plans d'expérience factoriels, Mathématiques & Applications, 21, Springer-Verlag, Berlin (1996).

[62] L. C. A., Corsten, Rectangular lattices revisited, in Linear Statistical Inference (eds. T., Caliński & W., Klonecki), Lecture Notes in Statistics, 35, Springer, Berlin (1985), pp. 29–38.

[63] G., Dall'Aglio, Blocs incomplets éqilibrés orthogonaux, Colloques Internationaux du Centre National de la Recherche Scientifique 110 (1963), 105–214.Google Scholar

[64] A., Dey, U. S., Das & A. K., Banerjee, Construction of nested balanced incomplete block designs, Calcutta Statistical Association Bulletin 35 (1986), 161–167.Google Scholar

[65] Jean, Doyen, Designs and automorphism groups, in Surveys in Combinatorics, 1989 (ed. J., Siemons), London Mathematical Society Lecture Notes Series, 141, Cambridge University Press, Cambridge (1989), pp. 75–83.

[66] V., Duquenne, What can lattices do for experimental designs?, Mathematical Social Sciences 11 (1986), 243–281.Google Scholar

[67] Thomas, Durt, Berthold-Georg, Englert, Ingemar, Bengtsson & Karol, Życzkowski, On mutually unbiased bases, International Journal of Quantum Information 8 (2010), 535–640.Google Scholar

[68] J. A., Eccleston & J. A., John, Recovery of row and column information in row-column designs with adjusted orthogonality, Journal of the Royal Statistical Society, Series B 48 (1986), 238–243.Google Scholar

[69] J. A., Eccleston, J. A., John & D., Whitaker, Some row–column designs with adjusted orthogonality, Journal of Statistical Planning and Inference 36 (1993), 323–330.Google Scholar

[70] J. A., Eccleston & J., Kiefer, Relationships of optimality for individual factors of a design, Journal of Statistical Planning and Inference 5 (1981), 213–219.Google Scholar

[71] J. A., Eccleston & C. A., McGilchrist, Algebra of a row–column design, Journal of Statistical Planning and Inference 12 (1985), 305– 310.Google Scholar

[72] J., Eccleston & K., Russell, Connectedness and orthogonality in multifactor designs, Biometrika 62 (1975), 341–345.Google Scholar

[73] J. A., Eccleston & K. G., Russell, Adjusted orthogonality in nonorthogonal designs, Biometrika 64 (1977), 339–345.Google Scholar

[74] J. A., Eccleston & A., Street, Construction for adjusted orthogonal designs, Ars Combinatoria 28 (1990), 117–128.Google Scholar

[75] W. L., Edge, The geometry of the linear fractional group LF(4, 2), Proceedings of the London Mathematical Society, Series 3 4 (1954), 317–342.Google Scholar

[76] P., Eendebak & E., Schoen, Complete series of non-isomorphic orthogonal arrays, http://pietereendebak.nl/oapage/

[77] D. J., Finney, The fractional replication of factorial experiments, Annals of Eugenics 12 (1945), 291–301.Google Scholar

[78] R. A., Fisher, Design of Experiments, Oliver and Boyd, Edinburgh (1935).

[79] R. A., Fisher, An examination of the different possible solutions of a problem in incomplete blocks, Annals of Eugenics 10 (1940), 52–75.Google Scholar

[80] R. A., Fisher & F., Yates, Statistical Tables for Biological, Agricultural and Medical Research, Oliver and Boyd, Edinburgh (1938).

[81] Dmitri G., Fon-Der-Flaass, Arrays of distinct representatives — a very simple NP-complete problem, Discrete Mathematics 171 (1997), 295–298.Google Scholar

[82] G. H., Freeman, Some non-orthogonal partitions of 4 × 4, 5 × 5 and 6 × 6 Latin squares, Annals of Mathematical Statistics 37 (1966), 666–681.Google Scholar

[83] Dennis C., Gilliland, A note on orthogonal partitions and some wellknown structures in design of experiments, Annals of Statistics 5 (1977), 565–570.Google Scholar

[84] R. E., Greenwood & A. M., Gleason, Combinatorial relations and chromatic graphs, Canadian Journal of Mathematics 7 (1955), 1–7.Google Scholar

[85] M., Greig & D. H., Rees, Existence of balanced incomplete block designs for many sets of treatments, Discrete Mathematics 261 (2003), 299–324.Google Scholar

[86] Ulrike, Grömping & R. A., Bailey, Regular fractions of factorial arrays, in mODa 11—Advances in Model-Oriented Design and Analysis (eds. Joachim, Kunert, Christine H., Müller & Anthony C., Atkinson), Springer International Publishing, Switzerland (2016), pp. 143–151.

[87] Heiko, Grosmann, Automating the analysis of variance of orthogonal designs, Computational Statistics and Data Analysis 70 (2014), 1– 18.Google Scholar

[88] J. I., Hall, On identifying PG(3, 2) and the complete 3-design on seven points, Annals of Discrete Mathematics 7 (1980), 131–141.Google Scholar

[89] P., Hall, On representatives of subsets, Journal of the London Mathematical Society 10 (1935), 26–30.Google Scholar

[90] B., Harshbarger, Preliminary report on the rectangular lattices, Biometrics 2 (1946), 115–119.Google Scholar

[91] B., Harshbarger, Triple rectangular lattices, Biometrics 5 (1949), 1– 13.Google Scholar

[92] A., Hedayat & W. T., Federer, Pairwise and variance balanced incomplete block designs, Annals of the Institute of Statistical Mathematics 26 (1974), 331–338.Google Scholar

[93] A., Hedayat, E. T., Parker & W. T., Federer, The existence and construction of two families of designs for two successive experiments, Biometrika 57 (1970), 351–355.Google Scholar

[94] A., Hedayat, E., Seiden & W. T., Federer, Some families of designs for multistage experiments: mutually balanced Youden designs when the number of treatments is prime power or twin primes. I., Annals of Mathematical Statistics 43 (1972), 1517–1527.Google Scholar

[95] A. S., Hedayat, N. J. A., Sloan & J., Stufken, Orthogonal Arrays, Springer-Verlag, New York (1999).

[96] A., Hedayat & W. D., Wallis, Hadamard matrices and their applications, Annals of Statistics 6 (1978), 1184–1238.Google Scholar

[97] J. W. P., Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford Mathematical Monographs, Oxford University Press, Oxford (1985).

[98] T. N., Hoblyn, S. C., Pearce & G. H., Freeman, Some considerations in the design of successive experiments in fruit plantations, Biometrics 10 (1954), 503–515.Google Scholar

[99] A. T., James, The relationship algebra of an experimental design, Annals of Mathematical Statistics 28 (1957), 993–1002.Google Scholar

[100] A. T., James & G. N., Wilkinson, Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance, Biometrika 58 (1971), 279–294.Google Scholar

[101] M., Jimbo & S., Kuriki, Construction of nested designs, Ars Combinatoria 16 (1983), 275–285.Google Scholar

[102] J. A., John & J. A, Eccleston, Row–column α-designs, Biometrika 73 (1986), 301–306.Google Scholar

[103] Byron, Jones, Algorithms to search for optimal row-and-column designs, Journal of the Royal Statistical Society, Series B 41 (1979), 210–216.Google Scholar

[104] Byron, Jones, Algorithm AS156: Combining two component designs to form a row-and-column design, Applied Statistics 29 (1980), 334– 345.Google Scholar

[105] R. Morley, Jones, On a property of incomplete blocks, Journal of the Royal Statistical Society, Series B 21 (1959), 172–179.Google Scholar

[106] B., Kassanis & A., Kleczkowski, Inactivation of a strain of tobacco necrosis virus and of the RNA isolated from it, Photochemistry and Photobiology 4 (1965), 209–214.Google Scholar

[107] O., Kempthorne, G., Zyskind, S., Addelman, T. N., Throckmorton & R. N., White, Analysis of Variance Procedures, Aeronautical Research Laboratory, Ohio, Report No. 149, 1961.

[108] J., Kiefer, On the nonrandomized optimality and randomized nonoptimality of symmetrical designs, Annals of Mathematical Statistics 29 (1958), 675–699.Google Scholar

[109] J., Kiefer, Balanced block designs and generalized Youden designs, I. Construction (patchwork), Annals of Statistics 3 (1975), 109–118.Google Scholar

[110] K., Kishen, Symmetrical unequal block arrangements, Sankhyā 5 (1940–1941), 329–344.Google Scholar

[111] A., Kleczkowski, Interpreting relationships between the concentrations of plant viruses and numbers of local lesions, Journal of General Microbiology 4 (1950), 53–69.Google Scholar

[112] A., Kobilinsky & H., Monod, Experimental designs generated by group morphisms, Scandinavian Journal of Statistics 18 (1991), 119–134.Google Scholar

[113] A. M., Kshirsagar, On balancing in designs in which heterogeneity is eliminated in two directions, Calcutta Statistical Assocation Bulletin 7 (1957), 161–166.Google Scholar

[114] Heinz, Lüneberg, Transitive Erweiterungen endlicher Permutationsgruppen, Lecture Notes in Mathematics, 4, Springer-Verlag, Berlin (1969).

[115] H. B., Mann, The algebra of a linear hypothesis, Annals of Mathematical Statistics 31 (1960), 1–15.Google Scholar

[116] John P., McSorley, Double arrays, triple arrays, and balanced grids with v = r + c - 1, Designs, Codes and Cryptography 37 (2005), 313–318.Google Scholar

[117] John P., McSorley & Nicholas C., Phillips, Complete enumeration and properties of binary pseudo-Youden designs PYD(9, 6, 6), Journal of Statistical Planning and Inference 137 (2007), 1464–1473.Google Scholar

[118] John P., McSorley, N. C. K., Phillips, W. D., Wallis & J. L., Yucas, Double arrays, triple arrays and balanced grids, Designs, Codes and Cryptography 35 (2005), 21–45.Google Scholar

[119] J. P., Morgan, Nested designs, in Design and Analysis of Experiments (eds. S., Ghosh & C. R., Rao), Handbook of Statistics, 13, Elsevier Science, Amsterdam (1996), pp. 939–976.

[120] J. P., Morgan, Properties of superimposed BIBDs, Journal of Statistical Planning and Inference 73 (1998), 135–148.Google Scholar

[121] J. P., Morgan, D. A., Preece & D. H., Rees, Nested balanced incomplete block designs, Discrete Mathematics 231 (2001), 351–389.Google Scholar

[122] John P., Morgan & Nizam, Uddin, Optimality and construction of nested row and column designs, Journal of Statistical Planning and Inference 37 (1993), 81–93.Google Scholar

[123] J. P., Morgan & Nizam, Uddin, Optimal blocked main effects plans with nested rows and columns and related designs, Annals of Statistics 24 (1996), 1185–1208.Google Scholar

[124] Rahul, Mukerjee & Sudhir, Gupta, Geometric construction of balanced block designs with nested rows and columns, Discrete Mathematics 91 (1991), 105–108.Google Scholar

[125] G., Nageswara Rao, Further contributions to balanced generalized two-way elimination of heterogeneity designs, Sankhyā 38 (1976), 72–79.Google Scholar

[126] J. A., Nelder, The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance, Proceedings of the Royal Society of London, Series A 283 (1965), 147–162.Google Scholar

[127] J. A., Nelder, The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance, Proceedings of the Royal Society of London, Series A 283 (1965), 163–178.Google Scholar

[128] J. A., Nelder, The combination of information in generally balanced designs, Journal of the Royal Statistical Society, Series B 30 (1968), 303–311.Google Scholar

[129] Tomas, Nilson, Row-column designs with adjusted orthogonality, Masters thesis, Mid Sweden University, Department of Engineering, Physics and Mathematics, 2007.

[130] Tomas, Nilson & Peter J., Cameron, Triple arrays from difference sets, arXiv 1609.00152.

[131] Tomas, Nilson & Pia, Heidtmann, Inner balance of symmetric designs, Designs, Codes and Cryptography 71 (2014), 247–260.Google Scholar

[132] Tomas, Nilson & Lars-Daniel, Öhman, Triple arrays and Youden squares, Designs, Codes and Cryptography 75 (2015), 429–451.Google Scholar

[133] H. D., Patterson & E. R., Williams, Some theoretical results on general block designs, Congressus Numerantium 15 (1976), 489–496.Google Scholar

[134] S. C., Pearce, The use and classification of non-orthogonal designs, Journal of the Royal Statistical Society, Series A 126 (1963), 353– 377.Google Scholar

[135] S. C., Pearce, The efficiency of block designs in general, Biometrika 57 (1970), 339–346.Google Scholar

[136] S. C., Pearce & J., Taylor, The changing of treatments in a long-term trial, Journal of Agricultural Science 38 (1948), 402–410.Google Scholar

[137] N. C. K., Phillips & D. A., Preece, Finding double Youden rectangles, in Designs 2002: Further Computational and Constructive Design Theory (ed. W. D., Wallis), Mathematics and its Applications, 563, Kluwer Academic Publishing, Boston MA (2003), pp. 301–315.

[138] N. C. K., Phillips, D. A., Preece & D. H., Rees, Double Youden rectangles for the four biplanes with k = 9, Journal of Combinatorial Mathematics and Combinatorial Computing 44 (2003), 169–176.Google Scholar

[139] N. C. K., Phillips, D. A., Preece & W. D., Wallis, The seven classes of 5 × 6 triple arrays, Discrete Mathematics 293 (2005), 213–218.Google Scholar

[140] N. C. K., Phillips & W. D., Wallis, An elusive array, Bulletin of the Institute of Combinatorics and its Applications 39 (2003), 39–40.Google Scholar

[141] Giovanni, Pistone & Maria-Piera, Rogantin, Indicator function and complex coding for mixed fractional factorial designs, Journal of Statistical Planning and Inference 138 (2008), 787–802.Google Scholar

[142] R. L., Plackett & J. P., Burman, The design of optimum multifactorial experiments, Biometrika 33 (1946), 305–325.Google Scholar

[143] Richard F., Potthoff, Three-factor additive designs more general than the Latin square, Technometrics 4 (1962), 187–208.Google Scholar

[144] Richard F., Potthoff, Some illustrations of four-dimensional incomplete block constructions, Calcutta Statistical Association Bulletin 12 (1963), 19–30.Google Scholar

[145] D. A., Preece, Some row and column designs for two sets of treatments, Biometrics 22 (1966), 1–25.Google Scholar

[146] D. A., Preece, Classifying Youden rectangles, Journal of the Royal Statistical Society, Series B 28 (1966), 118–130.Google Scholar

[147] D. A., Preece, Some balanced incomplete block designs for two sets of treatments, Biometrika 53 (1966), 497–506.Google Scholar

[148] D. A., Preece, Nested balanced incomplete block designs, Biometrika 54 (1967), 479–486.Google Scholar

[149] D. A., Preece, Balanced 6 × 6 designs for 9 treatments, Sankhyā, Series B 30 (1968), 443–446.Google Scholar

[150] D. A., Preece, Some new balanced row-and-column designs for two non-interacting sets of treatments, Biometrics 27 (1971), 426–430.Google Scholar

[151] D. A., Preece, Non-orthogonal Graeco-Latin designs, in Combinatorial Mathematics IV (eds. Louis R. A., Casse & Walter D., Wallis), Lecture Notes in Mathematics, 560, Springer-Verlag, Berlin (1976), pp. 7–26.

[152] D. A., Preece, Some designs based on 11×5 Youden ‘squares’, Utilitas Mathematica 9 (1976), 139–146.Google Scholar

[153] D. A., Preece, A second domain of balanced 6 × 6 designs for nine equally-replicated treatments, Sankhyā, Series B 38 (1976), 192– 194.Google Scholar

[154] D. A., Preece, Orthogonality and designs: a terminological muddle, Utilitas Mathematica 12 (1977), 201–223.Google Scholar

[155] D. A., Preece, Balance and designs: another terminological tangle, Utilitas Mathematica 21C (1982), 85–186.Google Scholar

[156] D. A., Preece, Some partly cyclic 13 × 4 Youden ‘squares’ and a balanced arrangement for a pack of cards, Utilitas Mathematica 22 (1982), 255–263.Google Scholar

[157] D. A., Preece, Genstat analyses for complex balanced designs with non-interacting factors, Genstat Newsletter 21 (March 1988), 33–45.Google Scholar

[158] D. A., Preece, Fifty years of Youden squares: a review, Bulletin of the Institute of Mathematics and its Applications 26 (1990), 65–75.Google Scholar

[159] D. A., Preece, Double Youden rectangles of size 6×11, Mathematical Scientist 16 (1991), 41–45.Google Scholar

[160] D. A., Preece, Enumeration of some 7×15 Youden squares and construction of some 7 × 15 double Youden rectangles, Utilitas Mathematica 41 (1992), 51–62.Google Scholar

[161] D. A., Preece, A set of double Youden rectangles of size 8 × 15, Ars Combinatoria 36 (1993), 215–219.Google Scholar

[162] D. A., Preece, Double Youden rectangles—an update with examples of size 5 × 11, Discrete Mathematics 125 (1994), 309–317.Google Scholar

[163] D. A., Preece, Triple Youden rectangles: A new class of fully balanced combinatorial arrangements, Ars Combinatoria 37 (1994), 175–182.Google Scholar

[164] D. A., Preece, Multi-factor balanced block designs with complete adjusted orthogonality for all pairs of treatment factors, Australian Journal of Statistics 38 (1996), 223–230.Google Scholar

[165] D. A., Preece, Some 6×11 Youden squares and double Youden rectangles, Discrete Mathematics 167/168 (1997), 527–541.Google Scholar

[166] D. A., Preece, P. W., Brading, C. W. H., Lam & M., Côté, Balanced 6 × 6 designs for 4 equally replicated treatments, Discrete Mathematics 125 (1994), 319–327.Google Scholar

[167] D. A., Preece & P. J., Cameron, Some new fully-balanced Graeco- Latin Youden ‘squares’, Utilitas Mathematica 8 (1975), 193–204.Google Scholar

[168] D. A., Preece & J. P., Morgan, Multi-layered Youden rectangles, Journal of Combinatorial Designs, 25 (2017), 75–84. doi:10.1002/jcd.21252Google Scholar

[169] D. A., Preece & N. C. K., Phillips, A new type of Freeman-Youden rectangle, Journal of Combinatorial Mathematics and Combinatorial Computing 25 (1997), 65–78.Google Scholar

[170] D. A., Preece, D. H., Rees & J. P., Morgan, Doubly nested balanced incomplete block designs, Congressus Numerantium 137 (1999), 5– 18.Google Scholar

[171] D. A., Preece & B. J., Vowden, Graeco-Latin squares with embedded balanced superpositions of Youden squares, Discrete Mathematics 138 (1995), 353–363.Google Scholar

[172] D. A., Preece, B. J., Vowden & N. C. K., Phillips, Double Youden rectangles of sizes p×(2p+1) and (p+1)×(2p+1), Ars Combnatoria 51 (1999), 161–171.Google Scholar

[173] D. A., Preece, B. J., Vowden & N. C. K., Phillips, Double Youden rectangles of sizes (p + 1) × (p2 + p + 1), Utilitas Mathematica 59 (2001), 139–154.Google Scholar

[174] D. A., Preece, W. D., Wallis & J. L., Yucas, Paley triple arrays, Australasian Journal of Combinatorics 33 (2005), 237–246.Google Scholar

[175] The R Project for Statistical Computing, https://www.r-project.org.

[176] D., Raghavarao, A note on some balanced generalized two-way elimination of heterogeneity designs, Journal of the Indian Society of Agricultural Statistics 22 (1970), 49–52.Google Scholar

[177] D., Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, John Wiley and Sons, New York (1971).

[178] D., Raghavarao & W. T., Federer, On connectedness in two-way elimination of heterogeneity designs, Annals of Statistics 3 (1975), 730– 735.Google Scholar

[179] D., Raghavarao & G., Nageswararao, A note on a method of construction of designs for two-way elimination of heterogeneity, Communications in Statistics 3 (1974), 197–199.Google Scholar

[180] D., Raghavarao & K. R., Shah, A class of D0 designs for two-way elimination of heterogeneity, Communications in Statistics—Theory and Methods 9 (1980), 75–80.Google Scholar

[181] C. R., Rao, Factorial arrangements derivable from combinatorial arrangements of arrays, Journal of the Royal Statistical Society, Supplement 9 (1947), 128–139.Google Scholar

[182] C. R., Rao, On a class of arrangements, Proceedings of the Edinburgh Mathematical Society 8 (1949), 119–125.Google Scholar

[183] D. H., Rees & D. A., Preece, Perfect Graeco-Latin balanced incomplete block designs (pergolas), Discrete Mathematics 197/198 (1999), 691–712.Google Scholar

[184] K. B., Reid & E., Brown, Doubly regular tournaments are equivalent to skew Hadamard matrices, Journal of Combinatorial Theory, Series A 12 (1972), 332–338.Google Scholar

[185] Kenneth Graham, Russell, On the theory of row-column designs, PhD thesis, University of New South Wales, 1977.

[186] K. G., Russell, A comparison of six methods of analysing row-column designs with inter-block information, Statistics and Computing 9 (1999), 239–246.Google Scholar

[187] K. G., Russell, J. A., Eccleston & G. J., Knudsen, Algorithms for the construction of (M,S) - optimal block designs and row–column designs, Journal of Statistical Computation and Simulation 12 (1981), 93–105.Google Scholar

[188] G. M., Saha & A. D., Das, A note on construction of mutually balanced Youden designs, Utilitas Mathematica 33 (1988), 5–8.Google Scholar

[189] Jennifer, Seberry, A note on orthogonal Graeco-Latin designs, Ars Combinatoria 8 (1979), 85–94.Google Scholar

[190] E., Seiden & C. Y., Wu, A geometric construction of generalized Youden designs for v a power of a prime, Annals of Statistics 6 (1978), 452–460.Google Scholar

[191] K. R., Shah, On uniformly better combined estimates in row-column designs with adjusted orthogonality, Communications in Statistics— Theory and Methods 17 (1988), 3121–3124.Google Scholar

[192] K. R., Shah & J. A., Eccleston, On some aspects of row–column designs, Journal of Statistical Planning and Inference 15 (1986), 87–95.Google Scholar

[193] Kirti R., Shah & Bikas K., Sinha, Theory of Optimal Designs, Lecture Notes in Statistics, 54, Springer-Verlag, New York (1989).

[194] S. S., Shrikande, Designs for two-way elimination of heterogeneity, Annals of Mathematical Statistics 22 (1951), 235–247.Google Scholar

[195] M., Singh & A., Dey, Two-way elimination of heterogeneity, Journal of the Royal Statistical Society, Series B 40 (1978), 58–63.Google Scholar

[196] M., Singh & A., Dey, Block designs with nested rows and columns, Biometrika 66 (1979), 321–326.Google Scholar

[197] N. P., Singh & Gular, Singh, Analysis of row-column experiments involving several non-interacting-sets of treatments and multistage Youden square designs, Biometrical Journal 26 (1984), 893–899.Google Scholar

[198] C. A. B., Smith & H. O., Hartley, The construction of Youden squares, Journal of the Royal Statistical Society, Series B 10 (1948), 262–263.Google Scholar

[199] T. P., Speed & R. A., Bailey, On a class of association schemes derived from lattices of equivalence relations, in Algebraic Structures and Applications (eds. P., Schulz, C. E., Praeger & R. P., Sullivan), Marcel Dekker, New York (1982), pp. 55–74.

[200] T. P., Speed & R. A., Bailey, Factorial dispersion models, International Statistical Review 55 (1987), 261–277.Google Scholar

[201] Edward, Spence, Construction and classification of combinatorial designs, in Surveys in Combinatorics, 1995 (ed. Peter, Rowlinson), London Mathematical Society Lecture Notes Series, 218, Cambridge University Press, Cambridge (1995), pp. 191–213.

[202] S. K., Srivastav & J. P., Morgan, On the class of 2×2 balanced incomplete block designs with nested rows and columns, Communications in Statistics—Theory and Methods 25 (1996), 1859–1870.Google Scholar

[203] Leon S., Sterling & Nicholas, Wormald, A remark on the construction of designs for two-way elimination of heterogeneity, Bulletin of the Australian Mathematical Society 14 (1976), 383–388.Google Scholar

[204] Anne Penfold, Street & Deborah J., Street, Combinatorics of Experimental Design, Oxford University Press, Oxford (1987).

[205] Deborah J., Street, Graeco-Latin and nested row and column designs, in Combinatorial Mathematics VIII (ed. Kevin L., McAvaney), Lecture Notes in Mathematics, 884, Springer-Verlag, Berlin (1981), pp. 304–313.

[206] W. B., Taylor, Incomplete block designs with row balance and recovery of inter-block information, Biometrics 13 (1957), 1–12.Google Scholar

[207] T. N., Throckmorton, Structures of classification data, PhD thesis, Ames, Iowa, 1961.

[208] Tue, Tjur, Analysis of variance models in orthogonal designs, International Statistical Review 52 (1984), 33–65.Google Scholar

[209] Tue, Tjur, Analysis of variance and design of experiments, Scandinavian Journal of Statistics 18 (1991), 273–322.Google Scholar

[210] K. D., Tocher, The design and analysis of block experiments, Journal of the Royal Statistical Society, Series B 14 (1952), 45–91.Google Scholar

[211] Nizam, Uddin, Constructions for some balanced incomplete block designs with nested rows and columns, Journal of Statistical Planning and Inference 31 (1992), 253–261.Google Scholar

[212] Nizam, Uddin, On recursive construction for balanced incomplete block designs with nested rows and columns, Metrika 42 (1995), 341–345.Google Scholar

[213] Nizam, Uddin & John P., Morgan, Some constructions for balanced incomplete block designs with nested rows and columns, Biometrika 77 (1990), 193–202.Google Scholar

[214] Nizam, Uddin & John P., Morgan, Two constructions for balanced incomplete block designs with nested rows and columns, Statistica Sinica 1 (1991), 229–232.Google Scholar

[215] P. M. van de, Ven & A. Di, Bucchianico, On the equivalence of definitions for regular fractions of mixed-level factorial designs, Journal of Statistical Planning and Inference 139 (2009), 2351–2361.Google Scholar

[216] Barry, Vowden, Infinite series of double Youden rectangles, Discrete Mathematics 125 (1994), 385–391.Google Scholar

[217] B. J., Vowden, A new infinite series of double Youden rectangles, Ars Combinatoria 56 (2000), 133–145.Google Scholar

[218] B. J., Vowden & D. A., Preece, Some new infinite series of Freeman- Youden rectangles, Ars Combinatoria 51 (1999), 49–63.Google Scholar

[219] W. D., Wallis, Triple arrays and related designs, Discrete Applied Mathematics 163 (2014), 220–236.Google Scholar

[220] W. D., Wallis & J. L., Yucas, Note on Agrawal's “Designs for Two-way Elimination of Heterogeneity”, Journal of Combinatorial Mathematics and Combinatorial Computation 46 (2003), 155–160.Google Scholar

[221] E. R., Williams, Efficiency-balanced designs, Biometrika 62 (1975), 686–689.Google Scholar

[222] E. R., Williams, A note on rectangular lattice designs, Biometrics 33 (1977), 410–414.Google Scholar

[223] E. R., Williams, H. D., Patterson & J. A., John, Resolvable designs with two replications, Journal of the Royal Statistical Society, Series B 38 (1976), 296–301.Google Scholar

[224] E. R., Williams, H. D., Patterson & J. A., John, Efficient two-replicate resolvable designs, Biometrics 33 (1977), 713–717.Google Scholar

[225] R. M., Wilson, An existence theorem for pairwise balanced designs. III. Proof of the existence conjecture, Journal of Combinatorial Theory, Series A 18 (1975), 71–79.Google Scholar

[226] F., Yates, Complex experiments, Journal of the Royal Statistical Society, Supplement 2 (1935), 181–247.Google Scholar

[227] F., Yates, Incomplete randomized blocks, Annals of Eugenics 7 (1936), 121–140.Google Scholar

[228] F., Yates, Incomplete Latin squares, Journal of Agricultural Science 26 (1936), 301–315.Google Scholar

[229] F., Yates, A new method of arranging variety trials involving a large number of varieties, Journal of Agricultural Science 26 (1936), 424– 455.Google Scholar

[230] F., Yates, A further note on the arrangement of variety trials: quasi- Latin squares, Annals of Eugenics 7 (1937), 319–332.Google Scholar

[231] F., Yates, Lattice squares, Journal of Agricultural Science 30 (1940), 672–787.Google Scholar

[232] W. J., Youden, Use of incomplete block replications in estimating tobacco-mosaic virus, Contributions from Boyce Thompson Institute 9 (1937), 41–48.Google Scholar

[233] Joseph L., Yucas, The structure of a 7 × 15 triple array, Congressus Numerantium 154 (2002), 43–47.Google Scholar

[234] G., Zyskind, On structure, relation, sigma, and expectation of mean squares, Sankhyā, Series A 24 (1962), 115–148.Google Scholar

Accessibility standard: Unknown

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

Accessibility compliance for the PDF of this chapter is currently unknown and may be updated in the future.