Beta Regression for Modelling Rates and Proportions

This paper proposes a regression model where the response is beta distributed using a parameterization of the beta law that is indexed by mean and dispersion parameters. The proposed model is useful for situations where the variable of interest is continuous and restricted to the interval (0, 1) and is related to other variables through a regression structure. The regression parameters of the beta regression model are interpretable in terms of the mean of the response and, when the logit link is used, of an odds ratio, unlike the parameters of a linear regression that employs a transformed response. Estimation is performed by maximum likelihood. We provide closed-form expressions for the score function, for Fisher's information matrix and its inverse. Hypothesis testing is performed using approximations obtained from the asymptotic normality of the maximum likelihood estimator. Some diagnostic measures are introduced. Finally, practical applications that employ real data are presented and discussed.

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

[2]  Francisco Cribari-Neto,et al.  Nearly Unbiased Maximum Likelihood Estimation for the Beta Distribution , 2002 .

[3]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[4]  R. Dennis Cook,et al.  Detection of Influential Observation in Linear Regression , 2000, Technometrics.

[5]  Karl V. Bury,et al.  Statistical Distributions in Engineering: Statistics , 1999 .

[6]  W. Fung,et al.  Generalized Leverage and its Applications , 1998 .

[7]  Dean P. Foster,et al.  Fitting Equations to Data , 1998 .

[8]  Jurgen A. Doornik,et al.  Ox: an Object-oriented Matrix Programming Language , 1996 .

[9]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[10]  William E. Griffiths,et al.  Learning and Practicing Econometrics , 1993 .

[11]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[12]  A. Hossain,et al.  A comparative study on detection of influential observations in linear regression , 1991 .

[13]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[14]  Roger Koenker,et al.  A note on studentizing a test for heteroscedasticity , 1981 .

[15]  V. Barnett,et al.  Applied Linear Statistical Models , 1975 .

[16]  J. W. Gorman,et al.  Fitting Equations to Data. , 1973 .

[17]  A. G. Greenhill,et al.  Handbook of Mathematical Functions with Formulas, Graphs, , 1971 .

[18]  N. L. Johnson.  Linear Statistical Inference and Its Applications , 1966 .

[19]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[20]  David M. Miller,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .