Published

2013-07-01

Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa

Estimation of Variance Components in Linear Mixed Models with Commutative Orthogonal Block Structure

Keywords:

álgebra conmutativa Jordan, componentes de varianza, modelo mixto (es)
Commutative Jordan algebra, Mixed model, Variance components. (en)

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Authors

  • Sandra S. Ferreira University of Beira Interior
  • Dário Ferreira University of Beira Interior
  • Célia Nunes University of Beira Interior
  • João T. Mexia New University of Lisbon

La segregación y el emparejamiento son técnicas para estimar las componentes de varianza en modelos mixtos. Una pregunta que ha surgido es si la segregación puede ser aplicada en situaciones en las que el emparejamiento no es aplicable. Nuestra motivación para esta investigación se basa en el hecho de que se quiere una respuesta a esta pregunta y se quiere explorar esta importante clase de modelos con el fin de contribuir al desarrollo de los modelos mixtos. Esto es posible utilizando la estructura algebraica de los modelos mixtos con estructura de bloques ortogonal conmutativa. Se presentan dos ejemplos que muestran que la segregación puede ser aplicada en situaciones donde el emparejamiento no es aplicable.

Segregation and matching are techniques to estimate variance components in mixed models. A question arising is whether segregation can be applied in situations where matching does not apply. Our motivation for this research relies on the fact that we want an answer to that question and to explore this important class of models that can contribute to the development of mixed models. That is possible using the algebraic structure of mixed models. We present two examples showing that segregation can be applied in situations where matching does not apply.

Estimation of Variance Components in Linear Mixed Models with Commutative Orthogonal Block Structure

Estimación de lascomponentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa

SANDRA S. FERREIRA1, DÁRIO FERREIRA2, CÉLIA NUNES3, JOÃO T. MEXIA4

1University of Beira Interior, Faculty of Sciences, Department of Mathematics and Center of Mathematics, Covilhã, Portugal. Professor. Email: sandraf@ubi.pt
2University of Beira Interior, Faculty of Sciences, Department of Mathematics and Center of Mathematics, Covilhã, Portugal. Professor. Email: dario@ubi.pt
3University of Beira Interior, Faculty of Sciences, Department of Mathematics and Center of Mathematics, Covilhã, Portugal. Professor. Email: celian@ubi.pt
4New University of Lisbon, Faculty of Science and Technology, Department of Mathematics and Center of Mathematics and Its Applications, Covilhã, Portugal. Professor. Email: jtm@fct.unl.pt

Abstract

Segregation and matching are techniques to estimate variance components in mixed models. A question arising is whether segregation can be applied in situations where matching does not apply. Our motivation for this research relies on the fact that we want an answer to that question and to explore this important class of models that can contribute to the development of mixed models. That is possible using the algebraic structure of mixed models. We present two examples showing that segregation can be applied in situations where matching does not apply.

Key words: Commutative Jordan algebra, Mixed model, Variance components.

Resumen

La segregación y el emparejamiento son técnicas para estimar las componentes de varianza en modelos mixtos. Una pregunta que ha surgido es si la segregación puede ser aplicada en situaciones en las que el emparejamiento no es aplicable. Nuestra motivación para esta investigación se basa en el hecho de que se quiere una respuesta a esta pregunta y se quiere explorar esta importante clase de modelos con el fin de contribuir al desarrollo de los modelos mixtos. Esto es posible utilizando la estructura algebraica de los modelos mixtos con estructura de bloques ortogonal conmutativa. Se presentan dos ejemplos que muestran que la segregación puede ser aplicada en situaciones donde el emparejamiento no es aplicable.

Palabras clave: álgebra conmutativa Jordan, componentes de varianza, modelo mixto.

Texto completo disponible en PDF

References

1. Bailey, R. A. (2004), Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge University Press, Cambridge.

2. Cali'nski, T. & Kageyama, S. (2000), Block Designs: A Randomization Approach Vol. I: Analysis, Springer-Verlag, New York.

3. Cali'nski, T. & Kageyama, S. (2003), Block Designs: A Randomization Approach Vol. II: Analysis, Springer-Verlag, New York.

4. Cox, D. & Solomon, P. (2003), Components of Variance, Chapman and Hall, New York.

5. Fernandes, C., Mexia, J., Ramos, P. & Carvalho, F. (2011), 'Models with stair nesting', AIP Conference Proceedings - Numerical Analysis and Applied Mathematics 1389, 1627-1630.

6. Fernandes, C., Ramos, P. & Mexia, J. (2010), 'Algebraic structure of step nesting designs', Discussiones Mathematicae. Probability and Statistics 30, 221-235.

7. Ferreira, S. S., Ferreira, D. & Mexia, J. T. (2007), 'Cross additivity in balanced cross nesting models', Journal of Statistical Theory and Practice(3), 377-392.

8. Ferreira, S. S., Ferreira, D., Nunes, C. & Mexia, J. T. (2010), 'Nesting segregated mixed models', Journal of Statistical Theory and Practice 4(2), 233-242.

9. Fonseca, M., Mexia, J. T. & Zmy'slony, R. (2003), 'Estimators and tests for variance components in cross nested orthogonal models', Discussiones Mathematicae - Probability and Statistics 23(3), 175-201.

10. Fonseca, M., Mexia, J. T. & Zmy'slony, R. (2007), 'Jordan algebras generating pivot variables and orthogonal normal models', Journal of Interdisciplinary Mathematics(10), 305-326.

11. Fonseca, M., Mexia, J. T. & Zmy'slony, R. (2008), 'Inference in normal models with commutative orthogonal block structure', Acta et Commentationes Universitatis Tartuensis de Mathematica(12), 3-16.

12. Houtman, A. & Speed, T. (1983), 'Balance in designed experiments with orthogonal block structure', Annals of Statistics 11(4), 1069-1085.

13. Mejza, S. (1992), 'On some aspects of general balance in designed experiments', Statistica 52, 263-278.

14. Mexia, J. T., Vaquinhas, R., Fonseca, M. & Zmyslony, R. (2010), 'COBS: Segregation, Matching, Crossing and Nesting', Latest Trends on Applied Mathematics, Simulation, Modeling, 4th International Conference on Applied Mathematics, Simulation, Modelling (ASM'10), 249-255.

15. Nelder, J. (1965a), 'The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance', Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 273(1393), 163-178.

16. Nelder, J. (1965b), '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. Series A, Mathematical and Physical Sciences 283(1393), 147-162.

17. Seely, J. (1971), 'Quadratic subspaces and completeness', The Annals of Mathematical Statistics 42, 710-721.

18. Zmy'slony, R. (1978), 'A characterization of best linear unbiased estimators in the general linear model', Mathematical Statistics and Probability Theory 2, 365-373.

[Recibido en octubre de 2012. Aceptado en septiembre de 2013]

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

@ARTICLE{RCEv36n2a05,
    AUTHOR  = {Ferreira, Sandra S. and Ferreira, Dário and Nunes, Célia and T. Mexia, João},
    TITLE   = {{Estimation of Variance Components in Linear Mixed Models with Commutative Orthogonal Block Structure}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2013},
    volume  = {36},
    number  = {2},
    pages   = {259-269}
}

References

Bailey, R. A. (2004), Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge University Press, Cambridge.

Calinski, T. & Kageyama, S. (2000), Block Designs: A Randomization Approach Vol. I: Analysis, Springer-Verlag, New York.

Calinski, T. & Kageyama, S. (2003), Block Designs: A Randomization Approach Vol. II: Analysis, Springer-Verlag, New York.

Cox, D. & Solomon, P. (2003), Components of Variance, Chapman and Hall, New York.

Fernandes, C., Mexia, J., Ramos, P. & Carvalho, F. (2011), ‘Models with stair nesting’, AIP Conference Proceedings - Numerical Analysis and Applied Mathematics 1389, 1627–1630.

Fernandes, C., Ramos, P. & Mexia, J. (2010), ‘Algebraic structure of step nesting designs’, Discussiones Mathematicae. Probability and Statistics 30, 221–235.

Ferreira, S. S., Ferreira, D. & Mexia, J. T. (2007), ‘Cross additivity in balanced cross nesting models’, Journal of Statistical Theory and Practice (3), 377–392.

Ferreira, S. S., Ferreira, D., Nunes, C. & Mexia, J. T. (2010), ‘Nesting segregated mixed models’, Journal of Statistical Theory and Practice 4(2), 233–242.

Fonseca, M., Mexia, J. T. & Zmyslony, R. (2003), ‘Estimators and tests for variance components in cross nested orthogonal models’, Discussiones Mathematicae - Probability and Statistics 23(3), 175–201.

Fonseca, M., Mexia, J. T. & Zmyslony, R. (2007), ‘Jordan algebras generating pivot variables and orthogonal normal models’, Journal of Interdisciplinary Mathematics (10), 305–326.

Fonseca, M., Mexia, J. T. & Zmyslony, R. (2008), ‘Inference in normal models with commutative orthogonal block structure’, Acta et Commentationes Universitatis Tartuensis de Mathematica (12), 3–16.

Houtman, A. & Speed, T. (1983), ‘Balance in designed experiments with ortogonal block structure’, Annals of Statistics 11(4), 1069–1085.

Mejza, S. (1992), ‘On some aspects of general balance in designed experiments’, Statistica 52, 263–278.

Mexia, J. T., Vaquinhas, R., Fonseca, M. & Zmyslony, R. (2010), ‘COBS: Segregation, Matching, Crossing and Nesting’, Latest Trends on Applied Mathematics, Simulation, Modeling, 4th International Conference on Applied Mathematics, Simulation, Modelling (ASM’10) pp. 249–255.

Nelder, J. (1965a), ‘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. Series A, Mathematical and Physical Sciences 283(1393), 147–162.

Nelder, J. (1965b), ‘The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance’, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 273(1393), 163–178.

Seely, J. (1971), ‘Quadratic subspaces and completeness’, The Annals of Mathematical Statistics 42, 710–721.

Zmyslony, R. (1978), ‘A characterization of best linear unbiased estimators in the general linear model’, Mathematical Statistics and Probability Theory 2, 365–373.

How to Cite

APA

Ferreira, S. S., Ferreira, D., Nunes, C. and Mexia, J. T. (2013). Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa. Revista Colombiana de Estadística, 36(2), 259–269. https://revistas.unal.edu.co/index.php/estad/article/view/44347

ACM

[1]
Ferreira, S.S., Ferreira, D., Nunes, C. and Mexia, J.T. 2013. Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa. Revista Colombiana de Estadística. 36, 2 (Jul. 2013), 259–269.

ACS

(1)
Ferreira, S. S.; Ferreira, D.; Nunes, C.; Mexia, J. T. Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa. Rev. colomb. estad. 2013, 36, 259-269.

ABNT

FERREIRA, S. S.; FERREIRA, D.; NUNES, C.; MEXIA, J. T. Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa. Revista Colombiana de Estadística, [S. l.], v. 36, n. 2, p. 259–269, 2013. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44347. Acesso em: 28 mar. 2025.

Chicago

Ferreira, Sandra S., Dário Ferreira, Célia Nunes, and João T. Mexia. 2013. “Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa”. Revista Colombiana De Estadística 36 (2):259-69. https://revistas.unal.edu.co/index.php/estad/article/view/44347.

Harvard

Ferreira, S. S., Ferreira, D., Nunes, C. and Mexia, J. T. (2013) “Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa”, Revista Colombiana de Estadística, 36(2), pp. 259–269. Available at: https://revistas.unal.edu.co/index.php/estad/article/view/44347 (Accessed: 28 March 2025).

IEEE

[1]
S. S. Ferreira, D. Ferreira, C. Nunes, and J. T. Mexia, “Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa”, Rev. colomb. estad., vol. 36, no. 2, pp. 259–269, Jul. 2013.

MLA

Ferreira, S. S., D. Ferreira, C. Nunes, and J. T. Mexia. “Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa”. Revista Colombiana de Estadística, vol. 36, no. 2, July 2013, pp. 259-6, https://revistas.unal.edu.co/index.php/estad/article/view/44347.

Turabian

Ferreira, Sandra S., Dário Ferreira, Célia Nunes, and João T. Mexia. “Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa”. Revista Colombiana de Estadística 36, no. 2 (July 1, 2013): 259–269. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44347.

Vancouver

1.
Ferreira SS, Ferreira D, Nunes C, Mexia JT. Estimación de las componentes de varianza en modelos lineales mixtos con estructura de bloques ortogonal conmutativa. Rev. colomb. estad. [Internet]. 2013 Jul. 1 [cited 2025 Mar. 28];36(2):259-6. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44347

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