Convergence Theorems in Multinomial Saturated and Logistic Models

• Autores: Erick Orozco Acosta, Humberto Jesús Llinás Solano, Javier Fonseca-Rodríguez
• Localización: Revista Colombiana de Estadística, ISSN-e 2389-8976, ISSN 0120-1751, Vol. 43, Nº. 2, 2020, págs. 211-231
• Idioma: inglés
• DOI: 10.15446/rce.v43n2.79151
• Títulos paralelos:
  • Teoremas de convergencias en los modelos saturados y logísticos multinomiales
• Enlaces
  • Texto completo
• Resumen
  • español

    Resumen En este artículo se desarrolla un estudio teórico de los modelos logísticos y saturados multinomiales cuando la variable de respuesta toma uno de R ≥ 2 niveles. Se presentan y demuestran teoremas sobre la existencia y cálculos de las estimaciones de máxima verosimilitud (ML-estimaciones) de los parámetros de ambos modelos. Se encuentran sus propiedades y, usando teoría asintótica, se prueban teoremas de convergencia para los vectores de puntajes y para las matrices de información. Se presenta y analiza una aplicación de esta teoría con datos tomados de la librería aplore3 del programa R.

  • English

    Abstract In this paper, we develop a theoretical study about the logistic and saturated multinomial models when the response variable takes one of R ≥ 2 levels. Several theorems on the existence and calculations of the maximum likelihood (ML) estimates of the parameters of both models are presented and demonstrated. Furthermore, properties are identified and, based on an asymptotic theory, convergence theorems are tested for score vectors and information matrices of both models. Finally, an application of this theory is presented and assessed using data from the R statistical program.

• Referencias bibliográficas
  • Agresti, A.. (2013). Probability and Statistics. 3. John Wiley and Sons.
  • Anderson, C.,Verkuilen, J.,Peyton, B.. (2010). 'Modeling polytomous item responses using simultaneously estimated multinomial logistic...
  • Begg, C. B.,Gray, R.. (1984). 'Calculation of polychotomous logistic regression parameters using individualized regressions'. Biometrika....
  • Chan, Y. H.. (2005). 'Biostatistics 305. multinomial logistic regression'. Singapore medical journal. 46. 259
  • Chen, Y.-H.,Kao, J.-T.. (2006). 'Multinomial logistic regression approach to haplotype association analysis in population-based case-control...
  • Chuang, H.-L.. (1997). 'High school youths' dropout and re-enrollment behavior'. Economics of Education Review. 16. 171
  • Cox, D. R.. (1958). 'The regression analysis of binary sequences'. Journal of the Royal Statistical Society: Series B (Methodological)....
  • Darlington, R. B.. (1990). Regression and linear models. McGraw-Hill College.
  • Dessens, J.,Jansen, W.,G. B. Jansen, W.,Ganzeboom, B.,Van der Heijden, P.. (2003). 'Patterns and trends in occupational attainment of...
  • Díaz, L.,Morales, M.. (2009). Análisis estadístico de datos categóricos. Editorial Universidad Nacional de Colombia.
  • Ekström, M.,Esseen, P.-A.,Westerlund, B.,Grafström, A.,Jonsson, B.,Stáhl, G.. (2018). 'Logistic regression for clustered data from environmental...
  • Exavery, A.,Mbaruku, G.,Mbuyita, S.,Makemba, A.,Kinyonge, I.,Kweka, H.. (2014). 'Factors affecting uptake of optimal doses of sulphadoxine-pyrimethamine...
  • Fahrmeir, L.,Kaufmann, H.. (1985). 'Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models'....
  • Fontanella, C. A.,Early, T. J.,Phillips, G.. (2008). 'Need or availability? modeling aftercare decisions for psychiatrically hospitalized...
  • Harrell, F.. (2015). Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis....
  • Harville, D.. (1997). Matrix algebra from a statistician's perspective. Springer-Verlag.
  • Hosmer, D.,Lemeshow, S.. (2000). Applied logistic regression, Wiley Series in Probability and Statistics. John Wiley & Sons Inc.
  • Kim, J.. (2015). 'School socioeconomic composition and adolescent sexual initiation in malawi'. Studies in Family Planning. 46. 263
  • Kleinbaum, D.,Klein, M.. (2010). Statistics for Biology and Health. Springer-Verlag. New York.
  • LLinás, H.. (2006). 'Precisiones en la teoría de los modelos logísticos'. Revista Colombiana de Estadística. 29. 239
  • LLinás, H.,Arteta, M.,Tilano, J.. (2016). 'El modelo de regresión logística para el caso en que la variable de respuesta puede asumir...
  • LLinás, H.,Carreño, C.. (2012). 'The multinomial logistic model for the case inwhich the response variable can assume one of three levels...
  • Lloyd, C. J.,Frommer, D. J.. (2008). 'An application of multinomial logistic regression to estimating performance of a multiple-screening...
  • Long, J. S.. (1987). 'A graphical method for the interpretation of multinomial logit analysis'. Sociological Methods & Research....
  • McCullagh, P.. (1983). 'Quasi-likelihood functions'. The Annals of Statistics. 11. 59-67
  • McCullagh, P.,Neider, J.. (2018). Generalized Linear Models. CRC Press.
  • McFadden, D.. (1973). Conditional Logit Analysis of Qualitative Choice Behavior, BART impact studies final report series: Traveler behavior...
  • McNevin, D.,Santos, C.,Gamez-Tato, A.,Álvarez-Dios, J.,Casares, M.,Daniel, R.,Phillips, C.,Lareu, M.. (2013). 'An assessment of bayesian...
  • Monroy, L.,Morales, M.,Dávila, L.. (2018). Análisis estadístico de datos categóricos. Universidad Nacional de Colombia.
  • Monyai, S.,Lesaoana, M.,Darikwa, T.,Nyamugure, P.. (2016). 'Application of multinomial logistic regression to educational factors of the...
  • Peng, C.-Y. J.,Lee, K. L.,Ingersoll, G. M.. (2002). 'An introduction to logistic regression analysis and reporting'. The journal ofeducational...
  • Pohlman, J.,Leitner, D.. (2003). 'A comparison of ordinary least squares and logistic regression'. The Ohio journal of science. 103....
  • Rao, C. R.,Rao, C. R.,Statistiker, M.,Rao, C. R.,Rao, C. R.. (1973). Linear statistical inference and its applications. Wiley New York.
  • Schnor, C.,Vanassche, S.,Bavel, J.. (2017). 'Stepfather or biological father? education-specific pathways of postdivorce fatherhood'....
  • Tan, Q.,Christiansen, L.,Christensen, K.,Kruse, T. A.,Bathum, L.. (2004). 'Apolipoprotein e genotype frequency patterns in aged danes...
  • Tilano, J.,Arteta, M.. (2012). Modelos logísticos multinomiales: estimaciones, pruebas y selección de modelos. Universidad del Norte.
  • Wedderburn, R. W. M.. (1974). 'Quasi-likelihood functions, generalized linear models, and the gauss-newton method'. Biometrika. 61....
  • Wedderburn, R. W. M.. (1976). 'On the existence and uniqueness of the maximum likelihood estimates for certain generalized linear models'....
  • Zacks, S.. (1971). The Theory of Statistical Inference, Wiley series in probability and mathematical statistics. John Wiley.