Poisson Inverse Gaussian Regression

Abstract

S ome P oints of D iscussion • Why hasn't the PIG model been widely used before this? • What types of data are best modeled using a PIG regression? • How do we model data with a very high initial peak and long right skew? • How do we know whether a PIG is a better-fitted model than negative binomial or Poisson models? P oisson I nverse G aussian M odel A ssumptions The Poisson inverse Gaussian (PIG) model is similar to the negative binomial model in that both are mixture models. The negative binomial model is a mixture of Poisson and gamma distributions , whereas the inverse Gaussian model is a mixture of Poisson and inverse Gaussian distributions . Those of you who are familiar with generalized linear models will notice that there are three GLM continuous distributions: normal, gamma, and inverse Gaussian. The normal distribution is typically parameterized to a lognormal distribution when associated with count models, presumably because the log link forces the distribution to have only nonnegative values. The Poisson and negative binomial (both NB2 and NB1) models have log links. Recall that the negative binomial is a mixture of the Poisson and gamma distributions, with variances of μ and μ 2 / v , respectively. We inverted v so that there is a direct relationship between the mean, dispersion, and variance function. Likewise, the inverse Gaussian is a mixture of Poisson and inverse Gaussian distributions, with an inverse Gaussian variance of μ 3 Φ.