¹Winona State University, Department of Mathematics and Statistics, Winona, United States. Assistant professor. Email: THooks@winona.edu
²University of Nebraska, Department of Statistics, Lincoln, United States. Professor. Email: DMarx1@unl.edu
³University of Nebraska, Department of Statistics, Lincoln, United States. Professor. Email: SKachman1@unl.edu
⁴University of Nebraska, USDA-ARS Research, Department of Agronomy and Horticulture, Lincoln, United States. Geneticist and professor. Email: JPedersen1@unl.edu

In the context of linear models, an optimality criterion is developed for models that include random effects. Traditional information-based criteria are premised on all model effects being regarded as fixed. When treatments and/or nuisance parameters are assumed to be random effects, an appropriate optimality criterion can be developed under the same conditions. This paper introduces such a criterion, and this criterion also allows for the inclusion of fixed and/or random nuisance parameters in the model and for the presence of a general covariance structure. Also, a general formula is presented for which all previously published optimality criteria are special cases.

Key words: Optimal design, Information matrix, Nuisance parameter, Covariance structure, Mixed model.

En el contexto de modelos lineales, los criterios de optimalidad se cons- truyen para los modelos que incluyen efectos aleatorios. Tradicionalmente los criterios basados en la información asumen que todos los efectos en el modelo se consideran fijos. Cuando los parámetros, tratamientos o molestias son considerados efectos aleatorios, un criterio adecuado de optimalidad se puede desarrollar en las mismas condiciones. En este trabajo se introduce ese criterio, que permite la inclusión en el modelo de parámetros que representan molestias fijas o al azar, además de una estructura general de covarianza. También, se presenta una fórmula general para la cual en todos los casos publicados anteriormente, los criterios de optimalidad son casos especiales.

Palabras clave: diseño óptimo, matrix informativa, parametros molestos, estructura de covarianza, modelo mixto.

Texto completo disponible en PDF

References

1. Atkinson, A. C. & Donev, A. N. (1992), Optimum Experimental Designs, Oxford University Press, New York, United States.         [ Links ]

2. Cook, R. D. & Nachtsheim, C. J. (1980), `A Comparison of Algorithms for Constructing Exact D-optimal Designs´, Technometrics 22(315-324).         [ Links ]

3. Cressie, N. A. C. (1993), Statistics for Spatial Data, John Wiley & Sons, Inc., New York, United States.         [ Links ]

4. Dette, H. (1995), `Designing of Experiments with Respect to ``Standardized'' Optimality Criteria´, Journal of the Royal Statistical Society 59, 97-110.         [ Links ]

5. Dette, H. & O'Brien, T. (1999), `Optimality Criteria for Regression Models Based on Predicted Variance´, Biometrika 86, 93-106.         [ Links ]

6. Draper, N. R. & Smith, H. (1998), Applied Regression Analysis, John Wiley & Sons, Inc., New York, United States.         [ Links ]

7. Dykstra, O. J. (1971), `The Augmentation of Experimental Data to Maximize |X' X|´, Technometrics 13, 682-688.         [ Links ]

8. Fedorov, V. V. (1972), Theory of Optimal Experiments, Academic Press, New York, United States.         [ Links ]

9. Harville, D. (1997), Matrix Algebra from a Statistician's Perspective, Springer-Verlag, New York, United States.         [ Links ]

10. Henderson, C. R. (1975), `Best Linear Unbiased Estimation and Prediction Under a Selection Model´, Biometrics 31, 423-447.         [ Links ]

11. Jacroux, M. (2001), `Determination and Construction of A-optimal Designs for Comparing two Sets of Treatments´, The Indian Journal of Statistics 63, 351-361.         [ Links ]

12. Johnson, M. E. & Nachtsheim, C. J. (1983), `Some Guidelines for Constructing Exact D-optimal Designs on Convex Design Spaces´, Technometrics 25, 271-277.         [ Links ]

13. Journel, A. G. & Huijbregts, C. J. (1978), Mining Geostatistics, Academic Press, New York, United States.         [ Links ]

14. Kiefer, J. (1958), `On the Non-randomized Optimality and Randomized Nonoptimality of Symmetrical Designs´, The Annals of Mathematical Statistics 29, 675-699.         [ Links ]

15. Kiefer, J. (1974), `General Equivalence Theory for Optimum Designs (Approximate Theory)´, Annals of Statistics 2, 849-879.         [ Links ]

16. Martin, R. J. (1986), `On the Design of Experiments Under Spatial Correlation´, Biometrika 73, 247-277.         [ Links ]

17. Marx, D. & Stroup, W. (1992), Designed Experiments in the Presence of Spatial Correlation, `Proceedings of the 1992 Kansas State University Conference of Applied Statistics in Agriculture´, p. 104-124.         [ Links ]

18. Mitchell, T. J. (1974), `An Algorithm for the Construction of D-optimal Experimental Designs´, Technometrics 16, 203-210.         [ Links ]

19. Mitchell, T. J. & Miller, F. L. Jr (1974), `An Algorithm for the Construction of D-optimal Experimental Designs´, Mathematlcs Division Annual Progress Report (ORNL-4661), 130-131.         [ Links ]

20. Müller, C. H. & Pazman, A. (1998), `Applications of Necessary and Sufficient Conditions for Maximin Efficient Designs´, Metrika 48, 1-19.         [ Links ]

21. Nguyen, N. K. & Miller, A. J. (1998), `A Review of Exchange Algorithms for Constructing Discrete D-optimal Designs´, Metrika 14, 489-498.         [ Links ]

22. Pazman, A. (1978), `Computation of the Optimum Designs under Singular Information Matrices´, Annals of Statistics 6, 465-467.         [ Links ]

23. SAS Institute Inc., (2007), SAS OnlineDoc 9.2, Cary, NC: SAS Institute Inc..         [ Links ]

24. Schmelter, T. (2007a), `The Optimality of Single-group Designs for Certain Mixed Models´, Metrika 65, 183-193.         [ Links ]

25. Schmelter, T. (2007b), `Considerations on Group-wise Identical Designs for Linear Mixed Models´, Journal of Statistical Planning and Inference 137, 4003-4010.         [ Links ]

26. Sebolai, B., Pedersen, J. F., Marx, D. B. & Boykin, D. L. (2005), `Effect of Grid Size, Control Plot Density, Control Plot Arrangement, and Assumption of Random or Fixed Effects on Non-replicated Experiments for Germplasm Screening´, Crop Science 45, 1978-1984.         [ Links ]

27. Silvey, S. D. (1978), `Optimal Design Measures with Singular Information Matrices´, Biometrika 65, 553-559.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX: