Isomorphism check for \(2^n\) factorial designs with randomization restrictions. (English) Zbl 1437.62315
Summary: Factorial designs with randomization restrictions are often used in industrial experiments when a complete randomization of trials is impractical. In the statistics literature, the analysis, construction, and isomorphism of factorial designs have been extensively investigated. Much of the work has been on a case-by-case basis – addressing completely randomized designs, randomized block designs, split-plot designs, etc. separately. In this paper, we take a more unified approach, developing theoretical results and an efficient relabeling strategy to both construct and check the isomorphism of multistage factorial designs with randomization restrictions. The examples presented in this paper particularly focus on split-lot designs.
MSC:
| 62K15 | Factorial statistical designs |
| 05B25 | Combinatorial aspects of finite geometries |
| 62K10 | Statistical block designs |
Keywords:
finite projective geometry; multistage factorial designs; split-lot designs; \((t-1)\)-spread; starsSoftware:
RReferences:
| [1] | Addelman S (1964) Some two-level factorial plans with split plot confounding’. Technometrics 6(3):253-258 |
| [2] | André J (1954) Über nicht-desarguessche ebenen mit transitiver translationsgruppe. Math Z 60(1):156-186 · Zbl 0056.38503 |
| [3] | Bailey RA (2004) Association schemes: designed experiments, algebra and combinatorics, vol 84. Cambridge University Press, Cambridge · Zbl 1051.05001 |
| [4] | Batten LM (1997) Combinatorics of finite geometries. Cambridge University Press, Cambridge · Zbl 0885.51012 |
| [5] | Bingham D, Sitter RR (1999) Minimum-aberration two-level fractional factorial split-plot designs. Technometrics 41(1):62-70 |
| [6] | Bingham D, Sitter R, Kelly E, Moore L, Olivas JD (2008) Factorial designs with multiple levels of randomization. Stat Sin 18(2):493-513 · Zbl 1135.62358 |
| [7] | Bose RC (1947) Mathematical theory of the symmetrical factorial design. Sankhyā: Indian J Stat 8:107-166 · Zbl 0038.09601 |
| [8] | Box GEP, Hunter WG, Hunter JS (1978) Statistics for experimenters. John Wiley & Sons, Inc., New York · Zbl 0394.62003 |
| [9] | Butler NA (2004) Construction of two-level split-lot fractional factorial designs for multistage processes. Technometrics 46:445-451 |
| [10] | Cheng CS (2016) Theory of factorial design: single-and multi-stratum experiments. Chapman and Hall/CRC, London |
| [11] | Cheng CS, Tang B (2005) A general theory of minimum aberration and its applications. Ann Stat 33(2):944-958. https://doi.org/10.1214/009053604000001228 · Zbl 1068.62086 · doi:10.1214/009053604000001228 |
| [12] | Cheng CS, Tsai PW (2011) Multistratum fractional factorial designs. Stat Sin 21:1001-1021 · Zbl 1534.62116 |
| [13] | Coxeter HSM (1969) Introduction to geometry, 2nd edn. Wiley, New York · Zbl 0181.48101 |
| [14] | Daniel C (1959) Use of half-normal plots in interpreting factorial two-level experiments. Technometrics 1(4):311-341 |
| [15] | Dean A, Voss D (1999) Design and analysis of experiments. Springer, New York · Zbl 0910.62066 |
| [16] | Eisfeld J, Storme L (2000) Partial t-spreads and minimal t-covers in finite projective spaces. Ghent University, Intensive Course on Finite Geometry and its Applications |
| [17] | Gordon NA, Shaw R, Soicher LH (2004) Classification of partial spreads in PG (4, 2). Available online as http://www.maths.qmul.ac.uk/ leonard/partialspreads/PG42new.pdf |
| [18] | Hedayat AS, Sloane NJA, Stufken J (2012) Orthogonal arrays: theory and applications. Springer, Berlin · Zbl 0935.05001 |
| [19] | Hirschfeld J (1998) Projective geometries over finite fields. Oxford mathematical monographs. Oxford University Press, New York · Zbl 0899.51002 |
| [20] | Honold T, Kiermaier M, Kurz S (2019) Classification of large partial plane spreads in PG(6, 2) and related combinatorial objects. J Geom 110(1):5 · Zbl 1407.05042 |
| [21] | Lidl R, Niederreiter H (1994) Introduction to finite fields and their applications. Cambridge University Press, Cambridge · Zbl 0820.11072 |
| [22] | Lin C, Sitter R (2008) An isomorphism check for two-level fractional factorial designs. J Stat Plan Inference 134:1085-1101 · Zbl 1130.62078 |
| [23] | Ma CX, Fang KT, Lin DK (2001) On the isomorphism of fractional factorial designs. J Complex 17(1):86-97 · Zbl 0979.62055 |
| [24] | Mateva ZT, Topalova ST (2009) Line spreads of PG (5, 2). J Comb Des 17(1):90-102 · Zbl 1177.51006 |
| [25] | McDonough T, Shaw R, Topalova S (2014) Classification of book spreads in PG (5, 2). Note di Mat 33(2):43-64 · Zbl 1293.51002 |
| [26] | Mee R (2009) A comprehensive guide to factorial two-level experimentation. Springer, Berlin |
| [27] | Mee RW, Bates RL (1998) Split-lot designs: experiments for multistage batch processes. Technometrics 40(2):127-140 |
| [28] | Miller A (1997) Strip-plot configurations of fractional factorials. Technometrics 39:153-161 · Zbl 0889.62069 |
| [29] | Mukerjee R, Wu C (2006) A modern theory of factorial design. Springer, New York · Zbl 1271.62179 |
| [30] | Nelder J (1965a) The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proc R Soc Lond A 283:147-162 · Zbl 0124.10703 |
| [31] | Nelder J (1965b) The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proc R Soc Lond A 283:163-178 · Zbl 0124.10703 |
| [32] | R Core Team (2014) R: A Language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/ |
| [33] | Ranjan P (2007) Factorial and fractional factorial designs with randomization restrictions-a projective geometric approach. Ph.D. thesis, Department of Statistics and Actuarial Science-Simon Fraser University |
| [34] | Ranjan P, Bingham DR, Dean AM (2009) Existence and construction of randomization defining contrast subspaces for regular factorial designs. Ann Stat 37(6A):3580-3599 · Zbl 1369.62187 |
| [35] | Ranjan P, Bingham DR, Mukerjee R (2010) Stars and regular fractional factorial designs with randomization restrictions. Stat Sin 20(4):1637-1653 · Zbl 1200.62087 |
| [36] | Ryan TP (2007) Modern experimental design. Wiley, London · Zbl 1119.62074 |
| [37] | Shaw R, Topalova ST (2014) Book spreads in PG (7, 2). Discrete Math 330:76-86 · Zbl 1295.51013 |
| [38] | Soicher L (2000) Computation of partial spreads, web preprint |
| [39] | Speed TP, Bailey RA (1982) On a class of association schemes derived from lattices of equivalence relations. Algebraic structures and applications. Marcel Dekker, New York, pp 55-74 · Zbl 0482.15008 |
| [40] | Tjur T (1984) Analysis of variance models in orthogonal designs. Int Stat Rev 52:33-81 · Zbl 0575.62068 |
| [41] | Topalova S, Zhelezova S (2010) 2-Spreads and transitive and orthogonal 2-parallelisms of PG (5, 2). Graphs Comb 26(5):727-735 · Zbl 1227.51003 |
| [42] | Wu CJ, Hamada M (2009) Experiments: planning, analysis and optimization, 2nd edn. Wiley, London · Zbl 1229.62100 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
