Variance-factorized loglinear models for overdispersed and underdispersed counts

A technique is proposed for a more structured approach to modelling over- and under-dispersed counts: variance factorized loglinear (VFL) models are a parameterization of an (enlarged) distribution whereby the overdispersed variance is expressed as a product of the nominal variance and a remainder term involving a highly interpretable loglinear effect with a potentially different set of covariates. This facilitates mean–variance analysis by allowing the variance to be modelled more separately from the mean, especially for generalized linear models (GLMs). Consequently, VFL parameterized models offer substantial advantages over existing methods for investigating overdispersion, such as a unifying framework and greater interpretability. Examples given here of enlarged distributions that supply the equidispersion as a special case include the extended beta-binomial (with a newly proposed ‘clog’ link and special offset) and negative binomial (NB) and generalized Poisson (GP–1 and GP–2 variants). VFL parameterized models are constructed as a vector generalized linear model (VGLM) with a judicious combination of constraint matrices, offsets and link functions. Underdispersed data can be handled by expanding the response by a multiplier and offset adjustment. Useful for disentangling the mean–variance relationship in GLMs, VFL parameterized models are implemented within the very broad framework of the vector generalized additive model (VGAM) R package available from comprehensive r archive network (CRAN). The technique is generally applicable to mean-parameterized distributions.