doi:10.1016/S0167-9473(03)00062-8 
Copyright © 2003 Elsevier B.V. All rights reserved.
Pseudo R-squared measures for Poisson regression models with over- or underdispersion
Department of Medical Computer Sciences, University of Vienna, Spitalgasse 23, A-1090, Vienna, Austria
Received 14 August 2002;
revised 15 March 2003.
Available online 15 April 2003.
References and further reading may be available for this article. To view references and further reading you must
purchase this article.
Abstract
The Poisson regression model is frequently used to analyze count data. Pseudo R-squared measures for Poisson regression models have recently been proposed and bias adjustments recommended in the presence of small samples and/or a large number of covariates. In practice, however, data are often over- or sometimes even underdispersed as compared to the standard Poisson model. The definition of Poisson R-squared measures can be applied in these situations as well, albeit with bias adjustments accordingly adapted. These adjustments are motivated by arguments of quasi-likelihood theory. Properties of unadjusted and adjusted R-squared measures are studied by simulation under standard Poisson; over- and underdispersed Poisson regression models and their use is exemplified and discussed with popcorn data.
Author Keywords: Adjusted R-squared measure; Poisson regression; Overdispersion; Underdispersion; Extra-Poisson variation; Deviance residuals; Likelihood ratio; Shrinkage; Quasi-likelihood
Fig. 1. Boxplots of the distributions of 1000 simulated samples for unadjusted and adjusted pseudo R-squared values (in percent) for n=16, k=5, , 1, 2 and 3, respectively. Horizontal lines are drawn for true population value RD,l2=0 percent.
Fig. 2. Boxplots of the distributions of 1000 simulated samples for unadjusted and adjusted pseudo
R-squared values (in percent) for
n=16,
k=5, , 1, 2 and 3, respectively. Horizontal lines are drawn for true population value
RD,l=20 percent.
Fig. 3. Boxplots of the distributions of 1000 simulated samples for unadjusted and adjusted pseudo
R-squared values (in percent) for
n=16,
k=5, , 1, 2 and 3, respectively. Horizontal lines are drawn for true population value
RD,l2=40 percent.
Fig. 4. Boxplots of the distributions of 1000 simulated samples for unadjusted and adjusted pseudo
R-squared values (in percent) for
n=16,
k=5, , 1, 2 and 3, respectively. Horizontal lines are drawn for true population value
RD,l2=60 percent.
Fig. 5. Boxplots of the distributions of 1000 simulated samples for unadjusted and adjusted pseudo
R-squared values (in percent) for
n=16,
k=5, , 1, 2 and 3, respectively. Horizontal lines are drawn for true population value
RD,l2=80 percent.
Table 1. Chosen values for the regression coefficient β1 in the simulations studies
Table 2. Simulated results for underdispersed Poisson model with 
, 1000 replications. Mean values (in percent) and mean squared error ratios of pseudo R-squared measures are given
Table 3. Simulated results for standard Poisson model with φ=1, 1000 replications. Mean values (in percent) and mean squared error ratios of pseudo R-squared measures are given
Table 4. Simulated results for overdispersed Poisson model with 
, 1000 replications. Mean values (in percent) and mean squared error ratios of pseudo R-squared measures are given
Table 5. Simulated results for overdispersed Poisson model with 
, 1000 replications. Mean values (in percent) and mean squared error ratios of pseudo R-squared measures are given
Table 6. Simulated results for all models with RD,l2=0 and 20 percent, 1000 replications. Ratios of truncated mean squared errors of pseudo R-squared measures are given
Table 7. Results of Poisson regression models for the popcorn experiment of [Myers 2002]: (a) overdispersion is ignored, and (b) overdispersion is taken into account and the dispersion parameter estimate Not-foundP is used which is based on generalized Pearson statistic (the results for the deviance-based estimate Not-foundD are similar and therefore not shown)
Table 8. Estimated pseudo R-squared measures (in percent) and shrinkage factors for the popcorn experiment of [Myers 2002]: (a) overdispersion is ignored, and (b) overdispersion is taken into account
Abstract
Purchase PDF (639 K)
E-mail Article
Add to my Quick Links
Cited By in Scopus (9)
Purchase PDF (106 K)
