Published

2009-01-01

CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS

OPTIMALITY CRITERIA FOR MODELS WITH RANDOM EFFECTS

Keywords:

diseño óptimo, matrix informativa, parámetros molestos, estructura de covarianza, modelo mixto (es)
Optimal design, Information matrix, Nuisance parameter, Covariance structure, Mixed model (en)

Downloads

Authors

  • Tisha Hooks Winona State University
  • David Marx University of Nebraska
  • Stephen Kachman University of Nebraska
  • Jeffrey Pedersen University of Nebraska
En el contexto de modelos lineales, los criterios de optimalidad se construyen 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.
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.
Untitled Document
Optimality Criteria for Models with Random Effects

Criterios de optimalidad para los modelos con efectos aleatorios
TISHA HOOKS1, DAVID MARX2, STEPHEN KACHMAN3, JEFFREY PEDERSEN4

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 

Abstract

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.

Resumen

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.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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.

18. Mitchell, T. J. (1974), `An Algorithm for the Construction of D-optimal Experimental Designs´, Technometrics16, 203-210.

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.

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

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

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

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

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

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

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.

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

[Recibido en agosto de 2008. Aceptado en diciembre de 2008]

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

@ARTICLE{RCEv32n1a02, 
    AUTHOR  = {Hooks, Tisha and Marx, David and Kachman, Stephen and Pedersen, Jeffrey}, 
    TITLE   = {{Optimality Criteria for Models with Random Effects}}, 
    JOURNAL = {Revista Colombiana de Estadística}, 
    YEAR    = {2009}, 
    volume  = {32}, 
    number  = {1}, 
    pages   = {17-31} 
}

How to Cite

APA

Hooks, T., Marx, D., Kachman, S. and Pedersen, J. (2009). CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS. Revista Colombiana de Estadística, 32(1), 17–31. https://revistas.unal.edu.co/index.php/estad/article/view/29751

ACM

[1]
Hooks, T., Marx, D., Kachman, S. and Pedersen, J. 2009. CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS. Revista Colombiana de Estadística. 32, 1 (Jan. 2009), 17–31.

ACS

(1)
Hooks, T.; Marx, D.; Kachman, S.; Pedersen, J. CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS. Rev. colomb. estad. 2009, 32, 17-31.

ABNT

HOOKS, T.; MARX, D.; KACHMAN, S.; PEDERSEN, J. CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS. Revista Colombiana de Estadística, [S. l.], v. 32, n. 1, p. 17–31, 2009. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/29751. Acesso em: 18 apr. 2024.

Chicago

Hooks, Tisha, David Marx, Stephen Kachman, and Jeffrey Pedersen. 2009. “CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS”. Revista Colombiana De Estadística 32 (1):17-31. https://revistas.unal.edu.co/index.php/estad/article/view/29751.

Harvard

Hooks, T., Marx, D., Kachman, S. and Pedersen, J. (2009) “CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS”, Revista Colombiana de Estadística, 32(1), pp. 17–31. Available at: https://revistas.unal.edu.co/index.php/estad/article/view/29751 (Accessed: 18 April 2024).

IEEE

[1]
T. Hooks, D. Marx, S. Kachman, and J. Pedersen, “CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS”, Rev. colomb. estad., vol. 32, no. 1, pp. 17–31, Jan. 2009.

MLA

Hooks, T., D. Marx, S. Kachman, and J. Pedersen. “CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS”. Revista Colombiana de Estadística, vol. 32, no. 1, Jan. 2009, pp. 17-31, https://revistas.unal.edu.co/index.php/estad/article/view/29751.

Turabian

Hooks, Tisha, David Marx, Stephen Kachman, and Jeffrey Pedersen. “CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS”. Revista Colombiana de Estadística 32, no. 1 (January 1, 2009): 17–31. Accessed April 18, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/29751.

Vancouver

1.
Hooks T, Marx D, Kachman S, Pedersen J. CRITERIOS DE OPTIMALIDAD PARA LOS MODELOS CON EFECTOS ALEATORIOS. Rev. colomb. estad. [Internet]. 2009 Jan. 1 [cited 2024 Apr. 18];32(1):17-31. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/29751

Download Citation

Article abstract page views

182

Downloads

Download data is not yet available.

License

Copyright (c) 2009 Revista Colombiana de Estadística

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).