Optimal design with many blocking factors. (English) Zbl 1105.62361

Bailey, R. A.; Morgan, J. P.

Ann. Stat. 28, No. 2, 553-577 (2000).

Summary: Designs for sets of experimental units with many blocking factors are studied. It is shown that if the set of blocking factors satisfies a certain simple condition then the information matrix for the design has a simple form. In consequence, a design is optimal if it is optimal with respect to one particular blocking factor and regular with respect to all the rest, in a sense which is made precise in the paper. This encompasses several previous results for optimal designs with more than one blocking factor, and applications to many other situations are given.

MSC:

62K05

Optimal statistical designs

Keywords:

Block design; optimal design; orthogonality; nested factors; crossed factors

References:

[1]	Bagchi, S. (1988). On the optimality of nested multiway designs. In Probability, Statistics, and Design of Experiments (R. R. Bahadur, ed.) 23-31. Wiley Eastern, New Delhi.
[2]	Bagchi, S., Mukhopadhyay, A. C. and Sinha, B. K. (1990). A search for optimal nested rowcolumn designs. Sankhya Ser. B 52 93-104. · Zbl 0722.62048
[3]	Bailey, R. A. (1984). Discussion of ”Analysis of variance models in orthogonal designs,” by T. Tjur. Internat. Statist. Rev. 52 65-77.
[4]	Bailey, R. A. (1985). Factorial design and Abelian groups. Linear Algebra Appl. 70 349-368. · Zbl 0586.62123 · doi:10.1016/0024-3795(85)90064-3
[5]	Bailey, R. A. (1992). Efficient semi-Latin squares. Statist. Sinica 2 413-437. · Zbl 0822.62060
[6]	Bailey, R. A. (1993). Recent advances in experimental design in agriculture. Bull. Internat. Statist. Inst. 55 179-193.
[7]	Bailey, R. A. (1996). Orthogonal partitions in designed experiments. Designs, Codes and Cryptography 8 45-77. · Zbl 0877.05006 · doi:10.1007/BF00130568
[8]	Bailey, R. A., Kunert, J. and Martin, R. J. (1990). Some comments on gerechte designs I. Analysis for uncorrelated errors. J. Agron. Crop Sci. 165 121-130.
[9]	Bailey, R. A., Kunert, J. and Martin, R. J. (1991). Some comments on gerechte desings II. Randomization analysis, and other methods that allow for inter-plot dependence. J. Agron. Crop Sci. 166 101-111.
[10]	Behrens, W. U. (1956). Die Eignung verschiedener Feldversuchsanordnungen zum Ausgleich der Bodenunterschiede. Zeitschrift f ür Ackerund Pflanzenbau 101 243-278.
[11]	Beth, T., Jungnickel, D. and Lenz, H. (1986). Design Theory. Cambridge Univ. Press. · Zbl 0602.05001
[12]	Bose, R. C. (1939). On the construction of balanced incomplete block designs. Ann. Eugen. 9 353-399. · Zbl 0023.00102 · doi:10.1111/j.1469-1809.1939.tb02219.x
[13]	Bose, R. C., Shrikande, S. S. and Bhattacharya, K. N. (1953). On the construction of group divisible incomplete block designs. Ann. Math. Statist. 24 167-195. · Zbl 0050.14605 · doi:10.1214/aoms/1177729027
[14]	Chang, J. Y. and Notz, W. I. (1990). A method for constructing universally optimal block designs with nested rows and columns. Utilitas Math. 38 263-276. · Zbl 0723.62039
[15]	Chang, J. Y. and Notz, W. I. (1994). Some optimal nested row-column designs. Statist. Sinica 4 249-263. · Zbl 0824.62067
[16]	Cheng, C.-S. (1978). Optimal designs for the elimination of multi-way heterogeneity. Ann. Statist. 6 1262-1272. · Zbl 0401.62060 · doi:10.1214/aos/1176344372
[17]	Cheng, C.-S. (1979). Construction of Youden hyperrectangles. J. Statist. Plann. Inference 3 109-118. · Zbl 0413.62059 · doi:10.1016/0378-3758(79)90003-X
[18]	Cheng, C.-S. and Bailey, R. A. (1991). Optimality of some two-associate class partially balanced incomplete-block designs. Ann. Statist. 19 1667-1671. · Zbl 0741.62071 · doi:10.1214/aos/1176348270
[19]	Clatworthy, W. H. (1973). Tables of Two-Associate-Class Partially Balanced Designs. National Bureau of Standards, Washington, D.C. · Zbl 0289.05017
[20]	Darby, L. A. and Gilbert, N. (1958). The Trojan square. Euphytica 7 183-188.
[21]	Gupta, S. and Jones, B. (1983). Equireplicate balanced block deisgns with unequal block sizes. Biometrika 70 433-440. · Zbl 0521.62062 · doi:10.1093/biomet/70.2.433
[22]	Gupta, S. and Kageyama, S. (1992). Variance balanced designs with unequal block sizes and unequal replications. Utilitas Math. 42 15-24. · Zbl 0767.62062
[23]	Hartley, H. O., Shrikhande, S. S. and Taylor, W. B. (1953). A note on incomplete block designs with row balance. Ann. Math. Statist. 24 123-126. · Zbl 0050.14604 · doi:10.1214/aoms/1177729091
[24]	Houtman, A. M. and Speed, T. P. (1983). Balance in designed experiments with orthogonal block structure. Ann. Statist. 11 1069-1085. · Zbl 0566.62065
[25]	Jacroux, M. and Ray, R. S. (1991). On the determination and construction of optimal rowcolumn designs having unequal row and column sizes. Ann. Inst. Statist. Math. 43 377-390. · Zbl 0761.62104 · doi:10.1007/BF00118643
[26]	Kachlicka, D. and Mejza, S. (1995). Cox’s design in repeated row-column design with split plots and a control. Listy Biometryczne 32 81-92.
[27]	Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In A Survey of Statistical Design and Linear Models (J. N. Srivastava, ed.) 333-353. North-Holland, Amsterdam. · Zbl 0313.62057
[28]	Kunert, J. (1983). Optimal designs and refinement of the linear model with applications to repeated measures designs. Ann. Statist. 11 247-257. · Zbl 0522.62054 · doi:10.1214/aos/1176346075
[29]	Leeming, J. A. (1998). Comparing a resolvable 7/(2\times 4) nested row and column design with two non-resolvable designs.
[30]	Magda, C. G. (1980). Circular balanced repeated measurements designs. Comm. Statist. Theory Methods 9 1901-1918. · Zbl 0457.62061 · doi:10.1080/03610928008828004
[31]	Morgan, J. P. (1996). Nested designs. In Handbook of Statistics 13: Design and Analysis of Experiments (S. Ghosh and C. R. Rao, eds.) 939-976. North-Holland, Amsterdam. · Zbl 0900.62417 · doi:10.1080/03610929608831789
[32]	Morgan, J. P. and Uddin, N. (1993). Optimality and construction of nested row and column designs. J. Statist. Plann. Inference 37 81-93. · Zbl 0778.62065 · doi:10.1016/0378-3758(93)90081-G
[33]	Morgan, J. P. and Uddin, N. (1996). Optimal blocked main effects plans with nested rows and columns and related designs. Ann. Statist. 24 1185-1208. · Zbl 0862.62065 · doi:10.1214/aos/1032526963
[34]	Mukerjee, R. and Wu, C . F. J. (1995). On the existence of saturated and nearly saturated asymmetrical orthogonal arrays. Ann. Statist. 23 2102-2115. · Zbl 0897.62083 · doi:10.1214/aos/1034713649
[35]	Mukhopadhyay, A. C. (1981). Orthogonal main effects plans with variable numbers of levels for factors. In Combinatorics and Graph Theory. Lecture Notes in Statist. 885 347-357. Springer, New York. · Zbl 0479.05017
[36]	Mukhopadhyay, A. C. and Mukhopadhyay, S. (1984). Optimality in a balanced incomplete multiway heterogeneity set up. In Statistics: Applications and New Directions (J. K. Ghosh and J. Roy, eds.) 466-477. Indian Statistical Institute, Calcutta. Nelder, J. A. (1965a). The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proc. Roy. Soc. London Ser. A 283 147-162. Nelder, J. A. (1965b). The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proc. Roy. Soc. London Ser. A 283 163-178.
[37]	Pal, S. and Pal, S. (1988). Nonproper variance balanced designs and optimality. Comm. Statist. Theory Methods 17 1685-1695.
[38]	Patterson, H. D. and Robinson, D. L. (1989). Row-and-column designs with two replicates. J. Agric. Sci. 112 73-77.
[39]	Preece, D. A. (1967). Nested balanced incomplete block designs. Biometrika 43 479-486.
[40]	Preece, D. A. (1982). Balance and designs: another terminological tangle. Utilitas Mathematica 21C 85-186. · Zbl 0499.62064
[41]	Rao, C. R. (1947). Factorial experiments derivable from combinatorial arrangements of arrays. J. Roy. Statist. Soc. (Supp.) 9 128-139. · Zbl 0031.06201
[42]	Rao, C. R. (1973). Some combinatorial problems of arrays and applications to design of experiments. In A Survey of Combinatorial Theory (J. N. Srivastava, ed.) 349-359. NorthHolland, Amsterdam. · Zbl 0274.05017
[43]	Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. Wiley, New York. · Zbl 0850.62007
[44]	Seeger, P. (1986). Design and analysis of experiments with sugar beet seeds. J. Roy. Statist. Soc. Ser. C 35 262-268.
[45]	Shah, K. R. and Sinha, B. K. (1989). Theory of Optimal Designs. Springer, Berlin. · Zbl 0688.62043
[46]	Singh, M. and Dey, A. (1979). Block designs with nested rows and columns. Biometrika 66 321-326. · Zbl 0407.62051 · doi:10.1093/biomet/66.2.321
[47]	Stewart, F. P. and Bradley, R. A. (1991). Some universally optimum row-column designs with empty nodes. Biometrika 78 337-348. · Zbl 0735.62072 · doi:10.1093/biomet/78.2.337
[48]	Tjur, T. (1984). Analysis of variance models in orthogonal designs. Internat. Statist. Rev. 52 33-65. · Zbl 0575.62068 · doi:10.2307/1403242
[49]	Wang, J. C. and Wu, C . F. J. (1991). An approach to the construction of asymmetrical orthogonal arrays. J. Amer. Statist. Assoc. 86 450-456.
[50]	Williams, E. R. (1986). Row and column designs with contiguous replicates. Austral. J. Statist. 28 154-163.
[51]	Williams, E. R. and Matheson, A. C. (1994). Experimental Design and Analysis for Use in Tree Improvement. CSIRO, Melbourne.
[52]	Yates, F. (1935). Complex experiments. J. Roy. Statist. Soc. (Supp.) 2 181-247.
[53]	Yates, F. (1936). A new method of arranging variety trials involving a large number of varieties. J. Agric. Sci. 26 424-455.
[54]	Yates, F. (1937). The design and analysis of factorial experiments. Technical communication 35. Imperial Bureau of Soil Science, Harpenden. · JFM 63.1129.13
[55]	Yates, F. (1951). Bases logiques de la planification des expériences. Ann. Inst. H. Poincaré 12 97-112. · Zbl 0044.34003