The Theil-T index and the KL divergence: A supplementary reply to Prof. R. Hoffmann
DOI:
https://doi.org/10.18222/eae.v36.12362Keywords:
Educational Indicators, Educational Inequalities, Educational evaluation, Learning AssessmentAbstract
The note responds to Professor R. Hoffmann’s criticism of the use of the Kullback-Leibler (KL) divergence to measure educational inequalities. It is shown that the method of relative densities is more general than the measure of divergence in question. Hoffmann’s examples are questioned because they use uniform distributions for comparison and for not considering the Item Response Theory (IRT) construction method. It is argued that a good reference for the proficiency scale represents a desired educational goal. It is proven mathematically, in the Appendix, that the Theil-T index is actually a special case of the KL divergence. This demonstrates that KL is not an inadequate measure, but rather a more general and flexible tool for analyzing educational inequalities.
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Ali, S. M., & Silvey, S. D. (1966). A general class of coefficients of divergence of one distribution from another. Journal of the Royal Statistical Society. Series B (Methodological), 28(1), 131-142. https://www.jstor.org/stable/2984279 DOI: https://doi.org/10.1111/j.2517-6161.1966.tb00626.x
Atkinson, A. B. (2015). Inequality: What can be done? Harvard University Press. DOI: https://doi.org/10.4159/9780674287013
Cruz, L., & Loureiro, A. (2020). Alcançando um nível de educação de excelência em condições socioeconômicas adversas: O caso de Sobral. World Bank Group. http://documents.worldbank.org/curated/en/778741594193637332
Ćwik, J., & Mielniczuk, J. (1989). Estimating density ratio with application to discriminant analysis. Communications in Statistics – Theory and Methods, 18(8), 3057-3069. https://doi.org/10.1080/03610928908830077 DOI: https://doi.org/10.1080/03610928908830077
Ernica, M., Rodrigues, E. C., & Soares, J. F. (2025). Desigualdades educacionais no Brasil contemporâneo: Definição, medida e resultados. Dados, 68(1), Artigo e20220109. https://doi.org/10.1590/dados.2025.68.1.345 DOI: https://doi.org/10.1590/dados.2025.68.1.345
Fletcher, P. R. (1994). A Teoria da Resposta ao Item: Medidas invariantes do desempenho escolar. Ensaio: Avaliação e Políticas Públicas em Educação, 1(2), 21-27. http://educa.fcc.org.br/pdf/ensaio/v01n02/v01n02a04.pdf
Hambleton, R. K., Swaminathan, H., & Rogers, H. J. (1991). Fundamentals of Item Response Theory. Sage.
Handcock, M. S., & Morris, M. (1998). Relative distribution methods. Sociological Methodology, 28(1), 53-97. https://doi.org/10.1111/0081-1750.00042 DOI: https://doi.org/10.1111/0081-1750.00042
Handcock, M. S., & Morris, M. (1999). Relative distribution methods in the social sciences. Springer.
Heckman, J. J., Lochner, L. J., & Todd, P. E. (2006). Earnings functions, rates of return and treatment effects: The Mincer equation and beyond. In E. Hanushek, & F. Welch (Eds.), Handbook of the economics of education (Vol. I; pp. 307-458). North Holland. https://doi.org/10.1016/S1574-0692(06)01007-5 DOI: https://doi.org/10.1016/S1574-0692(06)01007-5
Hoffmann, R. (2025). Sobre como medir diferenças de resultados no ensino. Estudos em Avaliação Educacional, 36, Artigo e10663. https://doi.org/10.18222/eae.v36.10663 DOI: https://doi.org/10.18222/eae.v36.10663
Hoffmann, R., Botassio, D. C., & Jesus, J. G. de. (2019). Distribuição de renda: Medidas de desigualdade, pobreza, concentração, segregação e polarização. Edusp.
Instituto Brasileiro de Geografia e Estatística (IBGE). (2023). Pnad Contínua – Pesquisa Nacional por Amostra de Domicílios Contínua 2022: Microdados. IBGE. https://ftp.ibge.gov.br/Trabalho_e_Rendimento/Pesquisa_Nacional_por_Amostra_de_Domicilios_continua/Trimestral/Microdados/2022/
Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira (Inep). (2020). Microdados do Saeb 2019. Inep. https://www.gov.br/inep/pt-br/acesso-a-informacao/dados-abertos/microdados/saeb
Klein, R. (2013). Alguns aspectos da Teoria de Resposta ao Item relativos à estimação das proficiências. Ensaio: Avaliação e Políticas Públicas em Educação, 21(78), 35-56. https://doi.org/10.1590/S0104-40362013005000003 DOI: https://doi.org/10.1590/S0104-40362013005000003
Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. The Annals of Mathematical Statistics, 22(1), 79-86. https://www.jstor.org/stable/2236703 DOI: https://doi.org/10.1214/aoms/1177729694
Oliveira, A. M. H. C. de, & Guimarães, R. R. de M. (2009). Trends in the relative distribution of wages by gender and cohorts in Brazil (1981-2005). In Proceedings of the Population Association of America 2009 Annual Meeting (pp. 1-21). https://paa2009.populationassociation.org/papers/90744
Organisation for Economic Co-operation and Development (OECD). (2009). PISA Data Analysis Manual: SPSS® (2nd ed.). Pisa; OECD Publishing. https://doi.org/10.1787/9789264056275-en DOI: https://doi.org/10.1787/9789264056275-en
Organisation for Economic Co-operation and Development (OECD). (2023). PISA 2022 results: Learning during – and from – disruption (Vol. II). Pisa; OECD Publishing. https://doi.org/10.1787/a97db61c-en DOI: https://doi.org/10.1787/a97db61c-en
Pasquali, L., & Primi, R. (2003). Fundamentos da Teoria da Resposta ao Item – TRI. Avaliação Psicológica, 2(2), 99-110. https://pepsic.bvsalud.org/pdf/avp/v2n2/v2n2a02.pdf
Rodrigues, C. G. (2009). A relação entre a expansão do acesso ao ensino e o desempenho escolar no Brasil: Evidências com base no Saeb para o período de 1997 a 2005 [Tese de doutorado, Universidade Federal de Minas Gerais]. Repositório Institucional UFMG. https://hdl.handle.net/1843/AMSA-84CKF5
Rodrigues, C. G., Rios-Neto, E. L. G., & Pinto, C. C. de X. (2013). Changes in test scores distribution for students of the fourth grade in Brazil: A relative distribution analysis for the years 1997-2005. Economics of Education Review, 34, 227-242. https://doi.org/10.1016/j.econedurev.2012.12.006 DOI: https://doi.org/10.1016/j.econedurev.2012.12.006
Shlens, J. (2014). Notes on Kullback-Leibler divergence and likelihood. ArXiv. https://doi.org/10.48550/arXiv.1404.2000
Soares, J. F., & Delgado, V. M. S. (2016). Medida das desigualdades de aprendizado entre estudantes de ensino fundamental. Estudos em Avaliação Educacional, 27(66), 754-780. https://doi.org/10.18222/eae.v27i66.4101 DOI: https://doi.org/10.18222/eae.v27i66.4101
Soares, J. F., & Ernica, M. (2025). O conceito rege o método: Uma resposta a Hoffmann. Estudos em Avaliação Educacional, 36, Debate e12270. https://doi.org/10.18222/eae.v36.12270 DOI: https://doi.org/10.18222/eae.v36.12270
Wallmark, J., Josefsson, M., & Wiberg, M. (2024). Introducing flexible monotone multiple choice Item Response Theory models and bit scales. ArXiv. https://doi.org/10.48550/arXiv.2410.01480
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