Análisis crítico del coeficiente de determinación (R2), como indicador de la calidad de modelos lineales y no lineales
Resumen
En casi todas las áreas de la ciencia, el modelado de datos se ha utilizado a lo largo de los años con diferentes propósitos, principalmente explicativos y predictivos. Para probar si los modelos aplicados se ajustan a los datos, a menudo se usa el coeficiente de determinación (R2). Este artículo propone los pros y los contras de su uso como indicador de la calidad de los modelos lineales y no lineales y las posibles soluciones a los problemas derivados de su aplicación. También se analiza la necesidad de probar los parámetros del modelo y la aplicabilidad del número de parámetros a considerar dentro del modelo utilizado, aspecto que influye en la calidad del ajuste.
Citas
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Austin, P. C., y Steyerberg, E. W. (2015). The number of subjects per variable required in linear regression analyses. Journal of Clinical Epidemiology, 68(1),627-636. doi: 10.1016/j.jclinepi.2014.12.014
Blanco-Fernández, A., Colubi, A., y González-Rodríguez, G. (2012). Confidence sets in a linear regression model for interval data. Journal of Statistical Planning and Inference, 146(6), 1320-1329. doi: 10.1016/j.jspi.2011.09.017
Carroll, R. J., y Ruppert, D. (1988). Transformation and weighting in regression.Chapman and Hall/CRC.
Charter, R. A. (2009). Differences scores: regression-based reliable difference and the regression-based confidence interval. Journal of Clinical Psychology, 65(4), 456-460. doi: 10.1002/jclp.20554
Cheng, C.-L., Shalabh, y Garg, G. (2014). Coefficient of determination for multiple measurement error models. Journal of Multivariate Analysis, 126(1), 137-152. doi: 10.1016/j.jmva.2014.01.006
Cornell, J. (1987). Factors that influence the value of the coefficient of determination in simple linear and nonlinear regression models. Phytopathology, 776(1), 63-70. doi: 10.1094/Phyto-77-63
Cramer, J. (1987). Mean and variance of R2 in small and moderate samples. Journal of Econometrics, 35(2-3), 253-266. doi: 10.1016/0304-4076(87)90027-3
Dietrich, S. W., Dreyer, N. D., Hansch, C., y Bentley, D. L. (1980). Confidence interval estimators for parameters associated with quantitative structure- activity relationships. Journal of medicinal chemistry, 23(11), 1201-1205. doi: 10.1021/jm00185a010
Esteves, E. (2008). Nonlinear regression using the solver® tool from microsoft excel. Technovision Electronics Series, 18(1), 1-13.
Genc¸, S., y Mendes¸, M. (2021). Evaluating performance and determining optimum sample size for regression tree and automatic linear modeling. Arquivo Brasileiro de Medicina Veterina´ ria e Zootecnia, 73(6), 1391-1402. doi: http://dx.doi.org/10.1590/1678-4162-12413
Hagquist, C., y Stenbeck, M. (1998). Goodness of fit in regression analysis – R2 and G2 reconsidered. Quality and Quantity, 32(1), 229-245. doi: 10.1023/A:1004328601205
Hahn, G. J. (1973). The coefficient of determination exposed. Chemical Technology, 3(10), 609-612.
Jaqaman, K., y Danuser, G. (2006). Linking data to models: data regression. Nature Reviews Molecular Cell Biology, 7(1), 813-819. doi: 10.1038/nrm2030
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Motulsky, H. J., y Ransnas, L. (1987). Fitting curves to data using nonlinear regression: a practical and nonmathematical review. The FASEB Journal, 1(5), 365-374. doi: 10.1096/fasebj.1.5.3315805
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Ribeiro, M. J. B. (2014). Curva de crescimento de codornas ajustadas por modelos näo lineares [Master Thesis].
Ricci, L. (2010). Adjusted -squared type measure for exponential dispersion models. Statistics and Probability Letters, 80(1), 1365-1368. doi: 10.1016/ j.spl.2010.04.019
Sapra, R. L. (2014). Using R2 with caution. Current Medicine Research and Practice, 4(3), 130-134.
doi: 10.1016/j.cmrp.2014.06.002
Saunders, L. J., Russell, R. A., y Crabb, D. P. (2012). The coefficient of determination: What determines a usefulR2 statistic?. Investigative Opthalmology and Visual Science, 53(11), 6830-6832. doi: 10.1167/iovs.12-10598
Schneider, A., Hommel, G., y Blettner, M. (2010). Linear regression analysis.Deutsches Aerzteblatt Online, 107(44), 776-782. doi: 10.3238/arztebl.2010.0776
Srivastava, A. K., Srivastava, V. K., y Ullah, A. (1995). The coefficient of determination and its adjusted version in linear regression models. Econometric Reviews, 14(2), 229-240. doi: 10.1080/07474939508800317
Vecchia, A. V., y Cooly, R. L. (1987). Simultaneous confidence and prediction intervals for nonlinear regression models with application to a groundwater flow model. Water Resources Research, 23(7), 237-250.
doi: 10.1029/ WR023i007p01237
Yamanaka, S. E. (2018). Modelos näo lineares mistos em estudos de crescimentos de frango de corte
[Master Thesis].
Zulkifli, N., Sorooshian, S., y Anvari, A. (2012). Modeling for regressing variables.
Journal of Statistical and Econometric Methods, 1(2), 1-8.
Austin, P. C., y Steyerberg, E. W. (2015). The number of subjects per variable required in linear regression analyses. Journal of Clinical Epidemiology, 68(1),627-636. doi: 10.1016/j.jclinepi.2014.12.014
Blanco-Fernández, A., Colubi, A., y González-Rodríguez, G. (2012). Confidence sets in a linear regression model for interval data. Journal of Statistical Planning and Inference, 146(6), 1320-1329. doi: 10.1016/j.jspi.2011.09.017
Carroll, R. J., y Ruppert, D. (1988). Transformation and weighting in regression.Chapman and Hall/CRC.
Charter, R. A. (2009). Differences scores: regression-based reliable difference and the regression-based confidence interval. Journal of Clinical Psychology, 65(4), 456-460. doi: 10.1002/jclp.20554
Cheng, C.-L., Shalabh, y Garg, G. (2014). Coefficient of determination for multiple measurement error models. Journal of Multivariate Analysis, 126(1), 137-152. doi: 10.1016/j.jmva.2014.01.006
Cornell, J. (1987). Factors that influence the value of the coefficient of determination in simple linear and nonlinear regression models. Phytopathology, 776(1), 63-70. doi: 10.1094/Phyto-77-63
Cramer, J. (1987). Mean and variance of R2 in small and moderate samples. Journal of Econometrics, 35(2-3), 253-266. doi: 10.1016/0304-4076(87)90027-3
Dietrich, S. W., Dreyer, N. D., Hansch, C., y Bentley, D. L. (1980). Confidence interval estimators for parameters associated with quantitative structure- activity relationships. Journal of medicinal chemistry, 23(11), 1201-1205. doi: 10.1021/jm00185a010
Esteves, E. (2008). Nonlinear regression using the solver® tool from microsoft excel. Technovision Electronics Series, 18(1), 1-13.
Genc¸, S., y Mendes¸, M. (2021). Evaluating performance and determining optimum sample size for regression tree and automatic linear modeling. Arquivo Brasileiro de Medicina Veterina´ ria e Zootecnia, 73(6), 1391-1402. doi: http://dx.doi.org/10.1590/1678-4162-12413
Hagquist, C., y Stenbeck, M. (1998). Goodness of fit in regression analysis – R2 and G2 reconsidered. Quality and Quantity, 32(1), 229-245. doi: 10.1023/A:1004328601205
Hahn, G. J. (1973). The coefficient of determination exposed. Chemical Technology, 3(10), 609-612.
Jaqaman, K., y Danuser, G. (2006). Linking data to models: data regression. Nature Reviews Molecular Cell Biology, 7(1), 813-819. doi: 10.1038/nrm2030
Jenkins, D. G., y Quintana-Ascencio, P. F. (2020). A solution to minimum sample size for regressions. PLoS ONE, 15(2), 1-15. doi: https://doi.org/10.1371/ journal.pone.0229345
Kumaria, K., y Yadav, S. (2018). Linear regression analysis study. Journal of the Practice of Cardiovascular Sciences, 4(1), 33-36. doi: 10.4103/jpcs.jpcs 818
Lee, K. B., Lee, J. M., Park, T. S., y Lee, S. H. (2010). Construction of classical confidence regions of model parameters in nonlinear regression analyses. Applied Radiation Isotopes, 68(7-8), 1261-1265.
doi: 10.1016/j.apradiso.2009.11.013
Maydeu-Olivares, A., y Garc´ıa-Forero, C. (2010). Goodness-of-fit testing. International Encyclopedia of Education, 7(1), 190-196. doi: 10.1016/B978-0-08-044894-7.01333-6
Motulsky, H. J., y Ransnas, L. (1987). Fitting curves to data using nonlinear regression: a practical and nonmathematical review. The FASEB Journal, 1(5), 365-374. doi: 10.1096/fasebj.1.5.3315805
Ohtani, K., y Hasegawa, H. (1993). On small sample properties of R2 in a linear regression model with multivariate t errors and proxy variables. Econometric Theory, 9(3), 504-515. doi: 10.1017/s0266466600007805
Paz, C. C., Packer, I. U., Freitas, A. R., Tambasco-Talhari, D., Regitano, L. C. A., Alencar, M. M., y Cruz, G. M. (2004). Ajuste de modelos na˜o-lineares em estudos de associac¸a˜o entre polimorfismos gene´ticos e crescimento em bovinos de corte. Revista Brasileira de Zootecnia, 33(6), 1416-1425. doi:10.1590/S1516-35982004000600008
Quinino, R. C., Kings, E. A., y Bessegato, L. F. (2012). Using the coefficient of determination R2 to test the significance of multiple linear regression., journal=Teaching Statistics, volume=35, number=2, pages=”84-88”,
doi =10.1111/j.1467-9639.2012.00525.x,.
Ribeiro, M. J. B. (2014). Curva de crescimento de codornas ajustadas por modelos näo lineares [Master Thesis].
Ricci, L. (2010). Adjusted -squared type measure for exponential dispersion models. Statistics and Probability Letters, 80(1), 1365-1368. doi: 10.1016/ j.spl.2010.04.019
Sapra, R. L. (2014). Using R2 with caution. Current Medicine Research and Practice, 4(3), 130-134.
doi: 10.1016/j.cmrp.2014.06.002
Saunders, L. J., Russell, R. A., y Crabb, D. P. (2012). The coefficient of determination: What determines a usefulR2 statistic?. Investigative Opthalmology and Visual Science, 53(11), 6830-6832. doi: 10.1167/iovs.12-10598
Schneider, A., Hommel, G., y Blettner, M. (2010). Linear regression analysis.Deutsches Aerzteblatt Online, 107(44), 776-782. doi: 10.3238/arztebl.2010.0776
Srivastava, A. K., Srivastava, V. K., y Ullah, A. (1995). The coefficient of determination and its adjusted version in linear regression models. Econometric Reviews, 14(2), 229-240. doi: 10.1080/07474939508800317
Vecchia, A. V., y Cooly, R. L. (1987). Simultaneous confidence and prediction intervals for nonlinear regression models with application to a groundwater flow model. Water Resources Research, 23(7), 237-250.
doi: 10.1029/ WR023i007p01237
Yamanaka, S. E. (2018). Modelos näo lineares mistos em estudos de crescimentos de frango de corte
[Master Thesis].
Zulkifli, N., Sorooshian, S., y Anvari, A. (2012). Modeling for regressing variables.
Journal of Statistical and Econometric Methods, 1(2), 1-8.
