Published

2026-07-01

Bayesian spatial and mixed-effects modeling of the Saber 11 test data

Modelación bayesiana espacial y de efectos mixtos en los datos de la prueba Saber 11

DOI:

https://doi.org/10.15446/rce.v49n2.122860

Keywords:

Bayesian statistics; , ICFES;, INLA; , Lattice data;, Mixed-effects models; , Spatial statistics. (en)
Datos de retícula;, Estadística bayesiana; , Estadística espacial; , ICFES;, INLA;, Modelos de efectos mixtos. (es)

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Authors

  • Juan Jose Galeano Arenas King Abdullah University of Sciences and Technology https://orcid.org/0009-0001-6274-4633
  • Elias Teixeira Krainski King Abdullah University of Sciences and Technology
  • Haavard Rue King Abdullah University of Sciences and Technology

Fitting high-dimensional random-effect models can be computationally demanding. To address this challenge, methodologies such as the Integrated Nested Laplace Approximation (INLA) provide efficient deterministic approximations for latent Gaussian models. We illustrate this approach using the Saber 11 mathematics scores from Colombia, fitting Bayesian spatial mixed-effects models with municipality-level and school-level random effects. The model accounts for associations between mathematics performance and selected student, household, and institution-related covariates, while capturing spatial structure across municipalities and heterogeneity between schools. Penalised Complexity priors were used to specify prior distributions for the variance components, and model comparison was performed using predictive assessment criteria, including Automatic Leave-Group-Out Cross-Validation. Based on the reported comparison criteria, the model including both municipality-level spatial effects and school-level random effects showed the best predictive performance among the candidate models.

Ajustar modelos de efectos aleatorios en altas dimensiones puede ser computacionalmente exigente. Para abordar este problema, metodologías como Integrated Nested Laplace Approximation (INLA) proporcionan aproximaciones determinísticas e_cientes para modelos gaussianos latentes. Ilustramos este enfoque utilizando los puntajes de matemáticas en la prueba Saber 11 en Colombia, ajustando modelos bayesianos espaciales y de efectos mixtos con efectos aleatorios a nivel municipal y escolar. El modelo permite estudiar asociaciones entre el desempeño en matemáticas y covariables seleccionadas relacionadas con el estudiante, el hogar y la institución, mientras captura la estructura espacial entre municipios y la heterogeneidad entre escuelas. Se utilizaron Prioris Penalizadas por complejidad para especificarlas distribuciones a priori de los componentes de varianza y la comparación de modelos se realizó mediante criterios de evaluación predictiva, incluyendo Automatic Leave-Group-Out Cross-Validation. Con base en los criterios de comparación reportados, el modelo que incluye tanto efectos espaciales a nivel municipal como efectos aleatorios a nivel escolar mostró el mejor desempeño predictivo entre los modelos candidatos.

References

Adin, A., Krainski, E. T., Lenzi, A., Liu, Z., Martínez-Minaya, J. & Rue, H. (2024), 'Automatic cross-validation in structured models: Is it time to leave out leave-one-out?', Spatial Statistics 62, 100843. https://www.sciencedirect.com/science/article/pii/S2211675324000344

Barrett, P., Treves, A., Shmis, T., Ambasz, D. & Ustinova, M. (2019), The Impact of School Infrastructure on Learning: A Synthesis of the Evidence, International Development in Focus, World Bank, Washington, DC. ISBN 978-1-4648-1378-8.

http://hdl.handle.net/10986/30920

Besag, J. & Kooperberg, C. (1995), 'On conditional and intrinsic autoregressions', Biometrika 82(4), 733-746. https://doi.org/10.1093/biomet/82.4.733

Besag, J., York, J. & Mollié, A. (1991), 'Bayesian image restoration, with two applications in spatial statistics', Annals of the Institute of Statistical Mathematics 43, 1-59. https://doi.org/10.1007/BF00116466

Bolin, D. & Wallin, J. (2023), 'Local scale invariance and robustness of proper scoring rules', Statistical Science 38(1), 140-159. https://doi.org/10.1214/22-STS864

Chaverra Santos, M. (2020), 'Educación y pobreza: una aproximación documental a los procesos educativos en entornos de exclusión y desigualdad social en Chocó, Colombia', Ciencias Sociales y Educación 9(17), 145-161. https://revistas.udem.edu.co/index.php/Ciencias-Sociales/article/view/3409

Collazos Valenzuela, A. C., Quintero Medina, M. V. & Trujillo Caicedo, K. N. (2021), 'Determinantes del rendimiento académico de la prueba Saber 11 en Colombia durante el periodo 2014-2019', Panorama 15(29). https://www.redalyc.org/articulo.oa?id=343967896009

Departamento Administrativo Nacional de Estadística (2024), Boletín técnico: Licencias de construcción (elic), febrero 2024, Technical report, DANE. https://www.dane.gov.co/-les/operaciones/ELIC/bol-ELIC-feb2024.pdf

Duarte, J., Bos, M. S. & Moreno, M. (2012), Quality, equality and equity in Colombian education: Analysis of the Saber 2009 test, Technical note, Inter- American Development Bank. https://doi.org/10.18235/0010419

Gaviria, A. & Barrientos, J. H. (2001), 'Determinantes de la calidad de la educación en Colombia', Archivos de Economía 159, 1-88.

Held, L., Schrodle, B. & Rue, H. v. (2010), Posterior and Cross-validatory Predictive Checks: A Comparison of MCMC and INLA, in 'Statistical Modelling and Regression Structures', Springer, pp. 111-131.

Instituto Colombiano para la Evaluación de la Educación (2024), 'Resultados únicos Saber 11'. Accessed Nov. 26, 2024. https://www.datos.gov.co/Educaci-n/Resultados-nicos-Saber-11/kgxf-xxbe/about-data

Jiang, J. & Nguyen, T. (2007), Linear and generalized linear mixed models and their applications, Vol. 1, Springer.

Liu, Z. & Rue, H. (2024), 'Leave-group-out cross-validation for latent Gaussian models', (submitted). https://arxiv.org/abs/2210.04482

Ozturk, I. (2001), 'The role of education in economic development: a theoretical perspective', Journal of Rural Development and Administration 33(1), 39-47.

Pebesma, E. (2018), 'Simple Features for R: Standardized Support for Spatial Vector Data', The R Journal 10(1), 439-446. https://doi.org/10.32614/RJ-2018-009

R Core Team (2026), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. https://www.Rproject.org/

Rue, H. & Held, L. (2005), Gaussian Markov Random Fields: Theory and Applications, Vol. 104 of Monographs on Statistics and Applied Probability, Chapman & Hall, London. http://dx.doi.org/10.1201/9780203492024

Rue, H., Martino, S. & Chopin, N. (2009), 'Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations', Journal of the Royal Statistical Society Series B 71(2), 319-392.

Rue, H., Riebler, A., Sørbye, S., Illian, J., Simpson, D. & Lindgren, F. (2017), 'Bayesian computing with inla: A review', Annual Review of Statistics and Its Application 4, 395-421.

Simpson, D., Rue, H., Riebler, A., Martins, T. G. & Sørbye, S. H. (2017), 'Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors', Statistical Science 32(1), 1 - 28. https://doi.org/10.1214/16-STS576

Sørbye, S. H. & Rue, H. (2014), 'Scaling intrinsic Gaussian Markov random field priors in spatial modelling', Spatial Statistics 8(3), 39-51.

Tavera-Cifuentes, M. C. & Otero, J. (2025). ColOpenData: Download Colombian Demographic, Climate and Geospatial Data. R package version 0.3.1. https://github.com/epiverse-trace/colopendata

van Niekerk, J., Krainksi, E., Rustand, D. & Rue, H. (2022), 'A new avenue for Bayesian inference with INLA', (submitted). https://arxiv.org/abs/2204.06797

van Niekerk, J. & Rue, H. (2024), 'Low-rank variational Bayes correction to the Laplace method', Journal of Machine Learning Research 25(62), 1-25. http://jmlr.org/papers/v25/21-1405.html

Wickham, H., Averick, M., Bryan, J., Chang, W., McGowan, L. D., François, R., Grolemund, G., Hayes, A., Henry, L., Hester, J., Kuhn, M., Pedersen, T. L., Miller, E., Bache, S. M., Müller, K., Ooms, J., Robinson, D., Seidel, D. P., Spinu, V., Takahashi, K., Vaughan, D., Wilke, C., Woo, K. & Yutani, H. (2019), 'Welcome to the tidyverse', Journal of Open Source Software 4(43), 1686.

How to Cite

APA

Galeano Arenas, J. J., Teixeira Krainski, E. & Rue, H. (2026). Bayesian spatial and mixed-effects modeling of the Saber 11 test data. Revista Colombiana de Estadística, 49(2), 499–526. https://doi.org/10.15446/rce.v49n2.122860

ACM

[1]
Galeano Arenas, J.J., Teixeira Krainski, E. and Rue, H. 2026. Bayesian spatial and mixed-effects modeling of the Saber 11 test data. Revista Colombiana de Estadística. 49, 2 (Jul. 2026), 499–526. DOI:https://doi.org/10.15446/rce.v49n2.122860.

ACS

(1)
Galeano Arenas, J. J.; Teixeira Krainski, E.; Rue, H. Bayesian spatial and mixed-effects modeling of the Saber 11 test data. Rev. colomb. estad. 2026, 49, 499-526.

ABNT

GALEANO ARENAS, J. J.; TEIXEIRA KRAINSKI, E.; RUE, H. Bayesian spatial and mixed-effects modeling of the Saber 11 test data. Revista Colombiana de Estadística, [S. l.], v. 49, n. 2, p. 499–526, 2026. DOI: 10.15446/rce.v49n2.122860. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/122860. Acesso em: 19 aug. 2026.

Chicago

Galeano Arenas, Juan Jose, Elias Teixeira Krainski, and Haavard Rue. 2026. “Bayesian spatial and mixed-effects modeling of the Saber 11 test data”. Revista Colombiana De Estadística 49 (2):499-526. https://doi.org/10.15446/rce.v49n2.122860.

Harvard

Galeano Arenas, J. J., Teixeira Krainski, E. and Rue, H. (2026) “Bayesian spatial and mixed-effects modeling of the Saber 11 test data”, Revista Colombiana de Estadística, 49(2), pp. 499–526. doi: 10.15446/rce.v49n2.122860.

IEEE

[1]
J. J. Galeano Arenas, E. Teixeira Krainski, and H. Rue, “Bayesian spatial and mixed-effects modeling of the Saber 11 test data”, Rev. colomb. estad., vol. 49, no. 2, pp. 499–526, Jul. 2026.

MLA

Galeano Arenas, J. J., E. Teixeira Krainski, and H. Rue. “Bayesian spatial and mixed-effects modeling of the Saber 11 test data”. Revista Colombiana de Estadística, vol. 49, no. 2, July 2026, pp. 499-26, doi:10.15446/rce.v49n2.122860.

Turabian

Galeano Arenas, Juan Jose, Elias Teixeira Krainski, and Haavard Rue. “Bayesian spatial and mixed-effects modeling of the Saber 11 test data”. Revista Colombiana de Estadística 49, no. 2 (July 16, 2026): 499–526. Accessed August 19, 2026. https://revistas.unal.edu.co/index.php/estad/article/view/122860.

Vancouver

1.
Galeano Arenas JJ, Teixeira Krainski E, Rue H. Bayesian spatial and mixed-effects modeling of the Saber 11 test data. Rev. colomb. estad. [Internet]. 2026 Jul. 16 [cited 2026 Aug. 19];49(2):499-526. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/122860

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