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Adjusted \(R^2\)-type measures for tweedie models. (English) Zbl 05564594
Summary: \(R^{2}\)-type measures are commonly used tools for assessing the predictive power of linear regression models; for generalized linear models this measure is based on deviances. For small samples, it is necessary to adjust \(R^{2}\) for the number of covariates. This article shows that some of the adjustments that have been proposed for logistic, Poisson, gamma and inverse Gaussian models, can be expressed in a general form that covers all exponential dispersion models with power variance function (also called Tweedie models). The adjusted measure is asymptotically unbiased. Numerical results of a simulation study are presented to explore the estimator behavior under different sample sizes, different number of covariates and different population values of the adjusted deviance based measure. The performances of the adjusted measures are also investigated on a data set used by Box and Cox.

MSC:
62-XX Statistics
Software:
R; Statmod; Tweedie
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[1] Box, G.E.P.; Cox, D.R., An analysis of transformations (with discussion), J. roy. statist. soc. B, 26, 211-252, (1964)
[2] Cameron, A.C.; Windmeijer, F.A.G., An \(R\)-squared measure of goodness-of-fit for some common nonlinear regression models, J. econometrics, 77, 329-342, (1997) · Zbl 0877.62097
[3] Copas, J.B., Regression, prediction and shrinkage, J. roy. statist. soc. B, 45, 311-354, (1983) · Zbl 0532.62048
[4] Dunn, P., 2004. Tweedie: Tweedie exponential family models. R package version1.02, \(\langle\)http://www.r-project.org/⟩.
[5] Dunn, P.K.; Smyth, G.K., Series evaluation of Tweedie exponential dispersion model densities, Statist. comput., 15, 4, 267-280, (2005)
[6] Feller, W., 1978. An introduction to Probability Theory and its Applications, vol. II, third ed. Wiley, New York. · Zbl 0138.10207
[7] Hardin, J.; Hilbe, J., Generalized linear models and extensions, (2001), Stata Press
[8] Hastie, T., A closer look at the deviance, Amer. statist., 41, 16-20, (1987) · Zbl 0607.62084
[9] Heinzl, H.; Mittlböck, M., Adjusted \(R^2\) measures for the inverse Gaussian regression model, Comput. statist., 17, 4, 525-544, (2002) · Zbl 1037.62068
[10] Heinzl, H.; Mittlböck, M., Pseudo \(R\)-squared measures for Poisson regression models with over- or underdispersion, Comput. statist. data anal., 44, 253-271, (2003) · Zbl 1429.62320
[11] Jørgensen, B., 1992. The theory of exponential dispersion models and analysis of deviance. Monografías de Matemática, vol. 51, IMPA, Rio de Janeiro, Brazil · Zbl 0983.62502
[12] Jørgensen, B., The theory of dispersion models, (1997), Chapman & Hall London · Zbl 0928.62052
[13] Lindsey, J., Applying generalized linear models, (1997), Springer Texts in Statistics · Zbl 0883.62074
[14] Magee, L., \(R^2\) measures based on Wald and likelihood ratio joint significance tests, Amer. statist., 44, 250-253, (1990)
[15] McCullagh, P.; Nelder, J.A., Generalized linear models, (1989), Chapman & Hall London · Zbl 0744.62098
[16] Mittlböck, M.; Heinzl, H., Measures of explained variation in gamma regression models, Commun. statist. B simulation comput., 31, 1, 61-73, (2002) · Zbl 1081.62542
[17] Mittlböck, M.; Schemper, M., Explained variation for logistic regression, Statist. med., 15, 1987-1997, (1996)
[18] Mittlböck, M.; Waldhor, T., Adjustments for \(R^2\)-measures for Poisson regression models, Comput. statist. data anal., 34, 4, 461-472, (2000) · Zbl 1046.62090
[19] R Development Core Team, 2006. R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, \(\langle\)http://www.r-project.org/⟩.
[20] Ricci, L., 2000. Estudio de las medidas de tipo \(R^2\) en modelos exponenciales con dispersión. Unpublished Magister Thesis, Magister en Estadística, Universidad Nacional de Córdoba.
[21] Smyth, G.K., Generalized linear models with varying dispersion, J. roy. statist. soc. B, 51, 47-60, (1989)
[22] Smyth, G.K., 2006. The statmod package. Available from \(\langle\)http://www.r-project.org/⟩.
[23] Smyth, G.K.; Verbyla, A.P., Adjusted likelihood methods for modelling dispersion in generalized linear models, Environmetrics, 10, 695-709, (1999)
[24] Tweedie, M., 1984. An index which distinguishes between some important exponential families. Statistics: applications and new directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference, pp. 579-604.
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