Discrete Weibull generalized additive model: an application to count fertility data

Fertility plans, measured by the number of planned children, have been found to be affected by education and family background via complex tail dependences. This challenge was previously met with the use of non-parametric jittering approaches. The paper shows how a novel generalized additive model based on a discrete Weibull distribution provides partial effects of the covariates on fertility plans which are comparable with jittering, without the inherent drawback of conditional quantiles crossing. The model has some additional desirable features: both overdispersed and underdispersed data can be modelled by this distribution, the conditional quantiles have a simple analytic form and the likelihood is the same as that of a continuous Weibull distribution with interval-censored data. Because the likelihood is like that of a continuous Weibull distribution, efficient implementations are already available, in the R package gamlss, for a range of models and inferential procedures, and at a fraction of the time compared with the jittering and Conway–Maxwell–Poisson approaches, showing potential for the wide applicability of this approach to the modelling of count data.