Zeroinflated

Zero-inflated refers to a class of statistical models designed to accommodate count data that exhibit more zeros than standard count distributions would predict. These models assume two latent data-generating processes: one governs whether an observation is a zero or a positive count, and a second governs the size of the positive counts. As a result, zeros can arise from both processes, yielding greater zero proportions than conventional models.

The most common formulations are zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB). For ZIP, the probability mass function is P(Y=0) = p + (1-p) e^{-λ}, and for y>0, P(Y=y) = (1-p) e^{-λ} λ^y / y!. Here p is the probability of a structural zero, and λ is the mean of the Poisson count when the non-zero component is in effect. ZINB replaces Poisson with a negative binomial distribution to allow additional overdispersion.

Estimation typically uses maximum likelihood, often via expectation-maximization or direct optimization. The two-process structure yields interpretable parameters: the inflation model (often a logistic regression for p) and the count model (a Poisson or NB regression for λ, possibly with covariates). Distinguishing zero-inflation from overdispersion can be challenging, and model selection commonly uses AIC/BIC or tests such as the Vuong test.

Applications span ecology (species abundance with many zeros), epidemiology, manufacturing, insurance, and social sciences. Software support exists in R (pscl package, zeroinfl function) and Python (statsmodels’ ZeroInflatedPoisson and ZeroInflatedNegativeBinomial models). Zero-inflated models provide a flexible framework for improving fit and inference when zeros are in excess of standard count models.