ZINB

ZINB refers to the zero-inflated negative binomial distribution, a two-part model designed for count data that show both overdispersion and excess zeros. It combines a binary process that yields additional zeros with a negative binomial process that governs positive counts.

In formal terms, let Y be a nonnegative integer. With probability pi, the observation is an extra zero. With probability 1 - pi, the observation follows a negative binomial distribution. Thus the probability mass function is:

- P(Y = 0) = pi + (1 - pi) NB(0; r, p)

- P(Y = y) = (1 - pi) NB(y; r, p) for y = 1, 2, ...

Here NB(y; r, p) is the negative binomial probability, often parameterized by shape r and success probability p, given by NB(y; r, p) = Gamma(y + r) / (Gamma(r) y!) * p^r * (1 - p)^y. An alternative is to use mean mu and dispersion k, with NB(y; mu, k) reflecting the same distribution in a different parameterization.

Estimation and use

Parameters pi, r (or mu, k) are typically estimated by maximum likelihood or Bayesian methods. Zero-inflation (pi) can be modeled separately from the mean of the NB component, allowing covariates to influence both the probability of extra zeros and the expected count. This leads to ZINB regression models, useful when covariates affect two separate data-generating processes.

Applications and context

ZINB models are commonly applied to ecological and epidemiological data, genomics (notably single-cell sequencing), and insurance claims, where there are many zeros and overdispersed counts. They are often contrasted with hurdle models; model choice depends on whether zeros are generated by a separate process or by the same mechanism governing counts. Potential challenges include identifiability issues and sensitivity to model mis-specification.