Services on Demand
Journal
Article
English (pdf)
Article in xml format
Article references
Automatic translation
Send this article by e-mailIndicators
-
Cited by SciELO -
Access statistics
Related links
-
Cited by Google -
Similars in
SciELO -
Similars in Google
Share
Permalink
Revista Colombiana de Estadística
Print version ISSN 0120-1751
Rev.Colomb.Estad. vol.28 no.1 Bogotá Jan./June 2005
1Instituto de Cibernética, Matemática y Física. Ciudad Habana. Cuba. E-mail: minerva@icmf.inf.cu
2Instituto de Cibernética, Matemática y Física. Ciudad Habana. Cuba
Montero et al. (2002) proposed a strategy to formulate multilevel models related to a contingency table sample. This methodology is based on the application of the general linear model to hierarchical categorical data. In this paper we applied the method to a multilevel logistic regression model using simulated data. We find that the estimates of the random parameters are inadmissible in some circumstances; large bias and negative estimates of the variance are expected for unbalanced data sets. In order to correct the estimates we propose to use a numerical technique based on the Truncated Singular Value Decomposition (TSVD) in the solution of the problem of generalized least squares associated to the estimation of the random parameters. Finally a simulation study is presented to shows the effectiveness of this technique for reducing the bias of the estimates.
Keywords: Multilevel models, Generalized least squares, Truncated Singular Value.
Montero, Castell & Ojeda (2002) propusieron una estrategia para formular modelos multinivel para tablas de contingencia basada en la aplicación del modelo lineal general a datos categóricos jerárquicos. Aplicando el método a un modelo de regresión logística multinivel con datos simulados, encontramos que las estimaciones de los parámetros aleatorios son inadmisibles en ciertas situaciones, con sesgos grandes y estimaciones negativas de la varianza cuando los conjuntos de datos son desbalanceados. Para corregir los estimadores proponemos una técnica basada en descomposición de valores singulares truncados en la solución de mínimos cuadrados generalizados para estimar los parámetros aleatorios. Mediante simulación mostramos la efectividad de la técnica en cuanto a la reducción del sesgo de los estimadores.
Palabras Clave: Modelos multinivel, mínimos cuadrados generalizados, valores singulares truncados.
Texto completo disponible en PDF
References
1. Breslow, N. E. & Clayton, D. G. (1993), "Approximate inference in generalized linear mixed models", American Statistical Association 88, 9-25. [ Links ]
2. Forthofer, R. N. & Koch, G. G. (1973), "An analysis for compounded functions of categorical data", Biometrics 29, 143- 159. [ Links ]
3. Goldstein, H. (1987), Multilevel Models in Educational and Social Research, Charles Griffin. [ Links ]
4. Goldstein, H. (1991), "Nonlinear multilevel models, with an application to discrete response data", Biometrika 78(1), 45- 51. [ Links ]
5. Goldstein, H. (1995), Multilevel Statistical Models, 2 edn, Halsted Press. [ Links ]
6. Goldstein, H. & Rasbash, J. (1992), "Efficient computational procedures for the estimation of parameters in multilevel models based on iterative generalized least squares", Computational Statistics and Data Analysis 13, 63- 71. [ Links ]
7. Golub, G. & Loan, C. F. V. (1996), Matrix Computations, 3 edn. [ Links ]
8. Grizzle, J. E., Starmerc, F. & Koch, G. (1969), "Analysis of categorical data by linear models", Biometrics 25, 489- 504. [ Links ]
9. Hansen, P. C. (1998), Rank-deficient and discrete ill-posed problems: Numerical aspects and linear inversion, Society for Industrial and Applied Mathematics. [ Links ]
10. Lee, Y. & Nelder, J. A. (1996), "Hierarchical generalized linear models", Royal Statistics Society B(58), 619- 678. [ Links ]
11. Lee, Y. & Nelder, J. A. (2001), "Hierarchical generalized linear models: a synthesis of generalized linear model and structured dispersion",Biometrika 88, 987- 1006. [ Links ]
12. Longford, N. (1994), "Logistic regression with random coefficients", Computational Statistics and Data Analysis97, 1- 15. [ Links ]
13. Montero, M., Castell, E. & Ojeda, M. M. (2002), Modelos multinivel de una muestra de tablas de contingencia utilizando el enfoque gsk, Technical Report 2002- 167, Reporte de investigación del ICIMAF. [ Links ]
14. Montero, M., Castell, E. & Ojeda, M. M. (2003), Modelos multinivel para una muestra de tablas de contingencia: un estudio por simulación, Technical Re- port 2003- 228, Reporte de investigación del ICIMAF. [ Links ]
15. Paige, C. C. (1979), "Fast numerically stable computations for generalizad lin- ear least squares problems", Society for Industrial and Applied Mathematics 1(1), 165- 171. [ Links ]
