Abstract
The subject of optimality of block designs under the mixed model has been undertaken in the eighties. Up to today there are not many papers considering the problem, contrary to the case of the fixed model. The papers of Bagchi (1987a,b), Mukhopdhyay (1981), Khatri and Shah (1984), Bhattacharya and Shah (1984) or Jacroux (1989) deal with the optimality of block designs under mixed model of a special simple kind.
Different ways of double randomization (depending on structure of experimental material) give more complicated mixed models. Many authors considered randomization models from the estimation and testing point of view, and they underlined the adequacy (by the nature of the problem) and applicability of the model to practical experiments (Fisher, 1926; Neyman, Iwaszkiewicz and Kolodziejczyk, 1935; Nelder, 1965; Ogawa and Ikeda, 1973; Caliński and Kageyama, 1991; Kala, 1991). We consider a class of block designs under such models having the property of general balance which makes considerations of optimality easier. We present conditions for a design to be optimal design in a general sense, i.e. optimal with respect to a wide class of criteria (considered formerly under the fixed model).
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References
Bagchi, S. (1987a). On the E-optimality of certain asymetrical designs under mixed effects model. Metrika 37 95–105.
Bagchi, S. (1987b). On the optimalities of the MBGDD’s under mixed effects model. Report, Computer Science Unit, Indian Statistical Institute, Calcuta, India.
Bailey, R.A. (1981). A unified approach to design of experiments. Journal of the Royal Statistical Society A 144 214–223.
Bailey, R.A. (1991). Strata for randomized experiments (with discussion). Journal of the Royal Statistical Society B 53 27–78.
Bhattacharya, C. G. and Shah, K. R. (1984). On the optimality of block designs under a mixed effects model. Utilitas Mathematica 26 339–345.
Bogacka, B. (1992). Optimality of experimental block designs under mixed models. Ph.D. thesis. Agricultural University of Poznań, Poland, (in Polish).
Bondar, J. V. (1983). Universal optimality of experimental designs: definitions and a criterion. The Canadian Journal of Statistics 11 325–331.
Calinski, T. (1977). On the notion of balance in block designs. In: J. R. Barra, F. Brodeau, G. Romier and B. van Cutsem, Eds., Recent Developments in Statistics. North-Holland, Amsterdam, 365–374.
Calinski, T. (1993). The basic contrasts of a block experimental design with special references to the notion of general balance. Listy Biometryczne - Biometrical Letters 30 13–38.
Caliński, T. and Kageyama, S. (1991). On the randomization theory of intra-block and inter-block analysis. Listy Biometryczne - Biometrical Letters 28 97–122.
Cheng, C. S. (1978). Optimality of certain asymetrical experimental designs. The Annals of Statistics 6 1239–1261.
Fisher, R.A. (1926). The arrangement of field experiments. Journal of Ministry of Agriculture. 23 503–513.
Houtman, A. M. and Speed, T. P. (1983). Balance in designed experiments with orthogonal block structure. The Annals of Statistics 11 1069–1085.
Jacroux, M. (1989). On the E-optimality of block designs under the assumption of random block effects. Sankhya B 51 1–12.
Kala, R. (1991). Elements of the randomization theory. III. Randomization in block experiments. Listy Biometryczne - Biometrical Letters 28 3–23, (in Polish).
Khatri, C.G. and Shah, K.R. (1984). Optimality of block designs. In: Proceedings of Indian Statistical Institute Golden Jubilee International Conference on Statistics: Applications and New Directions. Calcutta, 326–332.
Kiefer, J. (1974). General equivalence theory for optimum designs (Approximate theory). The Annals of Statistics 2 849–879.
Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In: J. N. Srivastava, Ed., A Survey of Statistical Design and Linear Models. North-Holland, Amsterdam, 333–353.
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
Mejza, S. (1992). On some aspects of general balance in designed experiments. Statistica 2 263–278.
Mukhopadhyay, S. (1981). On the optimality of block designs under mixed effects model. Calcutta Statistical Association Bulletin 30 171–185.
Nelder, J.A. (1965). 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 A 283 147–178.
Neyman, J., Iwaszkiewicz, K. and Kolodziejczyk, S. (1935). Statistical problems in agricultural experimentation (with discussion). Journal of the Royal Statistical Society, Supplement 2 107–180.
Ogawa, J. and Ikeda, S. (1973). On the randomization of block designs. In: J. N. Srivastava, Ed., A Survey of Combinatorial Theory. North-Holland, Amsterdam, 335–347.
Payne, R.W. and Tobias, R.D. (1992). General balance, combination of information and the analysis of covariance. Scandinavian Journal of Statistics
Pukelsheim, F. (1980). On linear regression designs which maximize information. Journal of Statistical Planning and Inference 4, 339–364.
Pukelsheim, F. (1983). On optimality properties of simple block designs in the approximate design theory. Journal of Statistical Planning and Inference 8, 193–208.
Shah, K. R. and Sinha, B. K. (1989). Theory of Optimal Designs. Lecture Notes in Statistics 54. Springer-Verlag, New York.
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Bogacka, B., Mejza, S. (1994). Optimality of Generally Balanced Experimental Block Designs. In: Caliński, T., Kala, R. (eds) Proceedings of the International Conference on Linear Statistical Inference LINSTAT ’93. Mathematics and Its Applications, vol 306. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1004-4_20
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DOI: https://doi.org/10.1007/978-94-011-1004-4_20
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