The frequency of doctor consultations has direct consequences for health care budgets, yet little statistical analysis of the determinants of doctor visits has been reported. We consider the distribution of the number of visits to the doctor and, in particular, we model its dependence on a number of demographic factors. Examination of the Australian 1995 National Health Survey data reveals that generalized linear Poisson or negative binomial models are inadequate for modelling the mean as a function of covariates, because of excessive zero counts, and a mean-variance relationship that varies enormously over covariate values. A negative binomial model is used, with parameter values estimated in subgroups according to the discrete combinations of the covariate values. Smoothing splines are then used to smooth and interpolate the parameter values. In effect the mean and the shape parameters are each modelled as (different) functions of gender, age and geographical factors. The estimated regressions for the mean have simple and intuitive interpretations. However, the dependence of the (negative binomial) shape parameter on the covariates is more difficult to interpret and is subject to influence by extreme observations. We illustrate the use of the model by estimating the distribution of the number of doctor consultations in the Statistical Local Area of Ryde, based on population numbers from the 1996 census.