Last updated: 2020-04-12
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In its very original form, negative binomial distribution \(NB(r,p)\) has pmf \(P(X=k;r,p) = {k+r-1 \choose k}p^r(1-p)^k\), where \(k\) is number of failures observed in a sequence of Bernoulli trials that are stopped at \(r\) successes, and \(p\) is the paprameter of the Bernnouli dsitrbution. It’s mean is \(\mu = \frac{pr}{1-p}\) and variance is \(\sigma^2 = \frac{pr}{(1-p)^2} = \mu + \frac{1}{r}\mu^2\). On the other hand, if the distribution is charaterized by mean and variance, then \(p = \frac{\sigma^2-\mu}{\sigma^2}\) and \(r=\frac{\mu^2}{\sigma^2-\mu}\).
Sometimes people write \(NB(\mu,\alpha)\), where \(\mu\) is the mean and \(\alpha\) is called dispersion parameter. Obviously, \(\alpha = \frac{1}{r}\).
In R function rnbinom, size is the \(r\), prob is the \(p\) and \(\mu\) is the mean.
It’s well known that the marginal distribution of a Poisson-gamma random variable is negative bionomial. Suppose \(X\sim Poisson(\lambda)\) and the Poisson parameter \(\lambda\) itself is distributed as \(Gamma(r,\frac{1-p}{p})\), where \(r\) is the shape parameter and \(\frac{1-p}{p}\) is the rate parameter of gamma distribution(note: A gamma distribution with shape a and rate b has pdf \(\frac{b^a}{\Gamma(a)}x^{a-1}\exp(-bx)\), mean \(a/b\) and variance \(a/b^2\)), then the marginal distribution of \(X\) is \(NB(r,p)\).
On the other hand, if \(\lambda\sim Gamma(a,\mu)\), where \(\mu=a/b\), then \(X\sim NB(\mu,1/\alpha)\); if further let \(p = \frac{\mu}{\mu+\alpha}\), then \(X\sim NB(\alpha,p)\).
If we have one observation from a NB and \(r\) is known, then mle is \(\hat p = \frac{r}{r+k}\) and an unbiased estimator is \(\hat p = \frac{r-1}{r+k-1}\).
Now we focus on the parameterization with mean \(\mu\) and dispersion parameter \(\alpha\). With \(\alpha\) fixed, this is a member of an exponential dispersion family appropriate for discrete variables, with natural parameter \(\log(\frac{\mu}{\mu+1/\alpha})\).
In nbglm, the dispersion parameter \(\alpha\) is common for all observations. We can test \(H_0: \alpha=0\) to see if Poisson is enough. This quadratuc mean-variance relationship model is called NB2.
NB1(linear) instead uses another paramaterization of the gamma mixture which yields a negative binomial distribution with with mean \(\mu\) and variance \(\frac{\mu(1+r)}{r}\). See Foundations of Linear and Generalized Linear Models, p250.