I have these data:
set.seed(1)
predictor <- rnorm(20)
set.seed(1)
counts <- c(sample(1:1000, 20))
df <- data.frame(counts, predictor)
I ran a poisson regression
poisson_counts <- glm(counts ~ predictor, data = df, family = "poisson")
And a negative binomial regression:
require(MASS)
nb_counts <- glm.nb(counts ~ predictor, data = df)
Then I calculated for dispersion statistics for the poisson regression:
sum(residuals(poisson_counts, type="pearson")^2)/df.residual(poisson_counts)
# [1] 145.4905
And the negative binomial regression:
sum(residuals(nb_counts, type="pearson")^2)/df.residual(nb_counts)
# [1] 0.7650289
Is anyone able to explain, WITHOUT USING EQUATIONS, why the dispersion statistic for the negative binomial regression is considerably smaller than the dispersion statistic for the poisson regression?