Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing

TLDR

Zero‑inflated Poisson regression models count data with excess zeros, such as manufacturing defects that are almost absent when equipment is aligned but follow a Poisson distribution when misaligned. The model posits that with probability p the observation is zero, and with probability 1‑p a Poisson(λ) variable is observed, where p and λ may depend on covariates and can be related or independent. ZIP regression is straightforward to fit; its maximum‑likelihood estimates are approximately normal in large samples, enabling confidence intervals via likelihood‑ratio tests or normal approximations, as confirmed by simulation studies.

Abstract

Zero-inflated Poisson (ZIP) regression is a model for count data with excess zeros. It assumes that with probability p the only possible observation is 0, and with probability 1 – p, a Poisson(λ) random variable is observed. For example, when manufacturing equipment is properly aligned, defects may be nearly impossible. But when it is misaligned, defects may occur according to a Poisson(λ) distribution. Both the probability p of the perfect, zero defect state and the mean number of defects λ in the imperfect state may depend on covariates. Sometimes p and λ are unrelated; other times p is a simple function of λ such as p = l/(1 + λ T ) for an unknown constant T . In either case, ZIP regression models are easy to fit. The maximum likelihood estimates (MLE's) are approximately normal in large samples, and confidence intervals can be constructed by inverting likelihood ratio tests or using the approximate normality of the MLE's. Simulations suggest that the confidence intervals based on likelihood ratio test...