Objective
We now show how to test whether a Negative Binomial regression model provides a significant improvement over a Poisson regression model.
Poisson Model
We start by creating a Poisson regression model for the Titanic data in Figure 1 of Negative Binomial Regression Tool (Solver). Figure 1 shows the partial results.
Figure 1 – Poisson regression model
We see that the p-values in cells S4 and S5 are significant, whereas the corresponding p-values for the negative binomial regression model are not significant (see cells S4 and S5 of Figure 2 of Negative Binomial Regression Tool Solver). This indicates that the negative binomial regression is a reasonable fit for the data whereas the Poisson regression model is not.
The AIC value is also lower for the negative binomial regression model (99.43366 vs 107.0595), and similarly for the BIC (102.3431 vs 109.484).
Significance Testing
Since the Poisson regression model is nested in the negative binomial regression model (one extra parameter, namely the alpha dispersion parameter), we can also perform a chi-square test on the differences between the LL values, as shown in Figure 2. Essentially we are testing whether alpha = 0.
Figure 2 – Chi-square comparison test
Here, the Poisson regression values in cells V14 and V15 come from cells R14 and R8 of Figure 1. The corresponding values for the Negative Binomial regression in cells W14 and W15 are copied from cells R14 and R8 of Figure 2 of Negative Binomial Regression Tool (Solver).
The p-value = CHISQ.DIST.RT(X14,X15) = .028248 < .05, which shows there is a significant difference between the models.
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Hilbe, J. M. (2014) Modeling count data. Cambridge University Press
https://www.cambridge.org/core/books/modeling-count-data/BFEB3985905CA70523D9F98DA8E64D08
Hintze, J. L. (2007) Negative binomial regression. NCSS
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Negative_Binomial_Regression.pdf