Adjusted R2-type measures for Tweedie models

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.

[1]  L. Magee,et al.  R 2 Measures Based on Wald and Likelihood Ratio Joint Significance Tests , 1990 .

[2]  Martina Mittlböck,et al.  MEASURES OF EXPLAINED VARIATION IN GAMMA REGRESSION MODELS , 2002 .

[3]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[4]  Gordon K. Smyth,et al.  Series evaluation of Tweedie exponential dispersion model densities , 2005, Stat. Comput..

[5]  M Schemper,et al.  Explained variation for logistic regression. , 1996, Statistics in medicine.

[6]  ScienceDirect.  Computational statistics & data analysis , 1983 .

[7]  Bent Jørgensen.  The Theory of Exponential Dispersion Models and Analysis of Deviance , 1992 .

[8]  Martina Mittlböck,et al.  Pseudo R-squared measures for Poisson regression models with over- or underdispersion , 2003, Comput. Stat. Data Anal..

[9]  Martina Mittlböck,et al.  Adjustments for R 2 -measures for Poisson regression models , 2000 .

[10]  J. Copas.  Regression, Prediction and Shrinkage , 1983 .

[11]  F. Windmeijer,et al.  An R-squared measure of goodness of fit for some common nonlinear regression models , 1997 .

[12]  Gordon K. Smyth,et al.  Adjusted likelihood methods for modelling dispersion in generalized linear models , 1999 .

[13]  Trevor Hastie,et al.  A Closer Look at the Deviance , 1987 .

[14]  Ananda Sen,et al.  The Theory of Dispersion Models , 1997, Technometrics.

[15]  Martina Mittlböck,et al.  Adjusted R2 Measures for the Inverse Gaussian Regression Model , 2002, Comput. Stat..

[16]  J. Hardin,et al.  Generalized Linear Models and Extensions , 2001 .

[17]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[18]  Gordon K. Smyth,et al.  Generalized linear models with varying dispersion , 1989 .

[19]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[20]  G. Smyth.  The statmod Package , 2007 .

[21]  James K. Lindsey,et al.  Applying Generalized Linear Models , 2000 .