Abstract
The interference model has been widely used and studied in block designs where the treatment in a particular plot effects on ones in its neighbor plots. There are many experiments specially in agricultre that block size (k) is greater than the number of treatment (t). If \(k>t\), the universally optimal designs under the interference models are usually difficult to obtain. In this paper, we consider an interference model with equal left- and right-neighbor effects for uncorrelated errors when \(k>t\). Based on Kushner’s method, we obtain the universally optimal designs on the class of circular block designs. We also present some methods to construct these designs. Then, we generalize our results for one-sided interference models.
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References
Ahmed R, Akhtar M (2008) Construction of neighbor balanced block designs. Journal Statistical Theory Practice 2(4):551–558
Ai MY, Ge GN, Chan LY (2007) Circular neighbor-balanced designs universally optimal for total effects. Sci China, Ser A Math 50(6):821–828
Azais JM, Bailey RA, Monod H (1993) A catalogue of efficient neighbour-designs with border plots. Biometrics 93:1252–1261
Bailey RA (2003) Designs for one-sided neighbour effects. Journal Indian Society Agricultural Statistics 56(3):302–314
Chai FS, Majumdar D (2000) Optimal designs for nearest-neighbor analysis. Journal statistical Planning Inference 86(1):265–275
Das A, Dey A (1989) A note on balanced block designs. Journal statistical Planning Inference 22(2):265–268
Druilhet P (1999) Optimality of neighbour balanced designs. Journal Statistical Planning Inference 81(1):141–152
Druilhet P, Tinsson W (2014) Optimal cross-over designs for total effects under a model with self and mixed carryover effects. Journal Statistical Planning Inference 154:54–61
Filipiak K (2012) Universally optimal designs under an interference model with equal left-and right-neighbor effects. Statist Probab Lett 82(3):592–598
Filipiak K, Markiewicz A (2003) Optimality of neighbor balanced designs under mixed effects model. Statist Probab Lett 61(3):225–234
Filipiak K, Markiewicz A (2004) Optimality of type I orthogonal arrays for general interference model with correlated observations. Statist Probab Lett 68(3):259–265
Filipiak K, Markiewicz A (2005) Optimality and efficiency of circular neighbor balanced designs for correlated observations. Metrika 61(1):17–27
Filipiak K, Markiewicz A (2007) Optimal designs for a mixed interference model. Metrika 65(3):369–386
Filipiak K, Markiewicz A (2012) On universal optimality of circular weakly neighbor balanced designs under an interference model. Communications Statistics-Theory Methods 41(13–14):2356–2366
Filipiak K, Markiewicz A (2014) On the optimality of circular block designs under a mixed interference model. Communications Statistics-Theory Methods 43(21):4534–4545
Filipiak K, Markiewicz A (2017) Universally optimal designs under interference models with and without block effects. Communications in Statistics-Theory and Methods 46(3):1127–1143
Hwang FK (1973) Constructions for some classes of neighbor designs. Ann Stat 1(4):786–790
Ipinyomi RA (1986) Equineighboured xperimental designs. Australian Journal of Statistics 28(1):79–88
Iqbal I, Amanullah M, Nasir JA (2006) The construction of second order neighbor designs. Journal Research (Science) 17(3):191–199
Iqbal I, Tahir MH, Ghazali SSA (2009) Circular neighbor-balanced designs using cyclic shifts. Sci China, Ser A Math 52(10):2243–2256
Jenkyn JF, Dyke GV (1985) Interference between plots in experiments with plant pathogens. Asp Appl Biol 10:75–85
Kempton RA, Lockwood G (1984) Inter-plot competition in variety trials of field beans (Vicia faba L.). J Agric Sci 103(2):293–302
Khodsiani R, Pooladsaz S (2017) Universal optimal block designs under hub correlation structure. Statist Probab Lett 129:387–392
Kiefer J (1975) Construction and optimality of generalized youden designs. In: Sri-vastava JN (ed) A Survey of Statistical Designs and Linear Models. North Holland, Amsterdam, pp 333–353
Kiefer J (1958) On the nonrandomized optimality and randomized nonoptimality of symmetrical designs. Ann Math Stat 29(3):675–699
Kunert J, Martin RJ (2000) On the determination of optimal designs for an interference model. Ann Stat 28(6):1728–1742
Kunert J, Martin RJ, Pooladsaz S (2003) Optimal designs under two related models for interference. Metrika 57(2):137–143
Kunert J, Mersmann S (2011) Optimal designs for an interference model. Journal statistical Planning Inference 141(4):1623–1632
Kushner HB (1997) Optimal repeated measurements designs: the linear optimality equations. Ann Stat 25(6):2328–2344
Lawless JF (1971) A note on certain types of BIBD’s balanced for residual effects. Ann Math Stat 42(4):1439–1441
Li K, Zheng W, Ai M (2015) Optimal designs for the proportional interference model. Ann Stat 43(4):1596–1616
Majumdar D, Martin RJ (2004) Efficient designs based on orthogonal arrays of type I and type II for experiments using units ordered over time or space. Statistical Methodology 1(1–2):19–35
Pooladsaz S, Martin RJ (2005) Optimal extended complete block designs for dependent observations. Metrika 61(2):185–197
Rees DH (1967) Some designs of use in serology. Biometrics 23:779–791
Acknowledgements
This paper is a part of the first author’s Ph.D. thesis. The authors would like to thank the referees for their valuable comments and suggestions that improved the presentation of this paper.
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Fakhari-Esferizi, A., Pooladsaz, S. Universally optimal balanced block designs for interference model. Metrika 85, 1049–1061 (2022). https://doi.org/10.1007/s00184-022-00870-5
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DOI: https://doi.org/10.1007/s00184-022-00870-5