  We propose a regression model for count data when the classical generalized
linear model approach is too rigid due to a high outcome of zero counts and a
nonlinear influence of continuous covariates. Zero-Inflation is applied to take
into account the presence of excess zeros with separate link functions for the
zero and the nonzero component. Nonlinearity in covariates is captured by
spline functions based on B-splines. Our algorithm relies on maximum-likelihood
estimation and allows for adaptive box-constrained knots, thus improving the
goodness of the spline fit and allowing for detection of sensitivity
changepoints. A simulation study substantiates the numerical stability of the
algorithm to infer such models. The AIC criterion is shown to serve well for
model selection, in particular if nonlinearities are weak such that BIC tends
to overly simplistic models. We fit the introduced models to real data of
children's dental sanity, linking caries counts with the so-called
Body-Mass-Index (BMI) and other socioeconomic factors. This reveals a puzzling
nonmonotonic influence of BMI on caries counts which is yet to be explained by
clinical experts.
