Sampling adjusted analysis of dynamic additive regression models for longitudinal data

We consider a modelling approach to longitudinal data that aims at estimating flexible covariate effects in a model where the sampling probabilities are modelled explicitly, The joint modelling yields simple estimators that are easy to compute and analyse, even if the sampling of the longitudinal responses interacts with the response level. An incorrect model for the sampling probabilities results in biased estimates. Non-representative sampling occurs, for example, if patients with an extreme development (based on extreme values of the response) are called in for additional examinations and measurements. We allow covariate effects to be time-varying or time-constant. Estimates of covariate effects are obtained by solving martingale equations locally for the cumulative regression functions. Using Aalen's additive model for the sampling probabilities, we obtain simple expressions for the estimators and their asymptotic variances. The asymptotic distributions for the estimators of the non-parametric components as well as the parametric components of the model are derived drawing on general martingale results. Two applications are presented. We consider the growth of cystic fibrosis patients and the prothrombin index for liver cirrhosis patients. The conclusion about the growth of the cystic fibrosis patients is not altered when adjusting for a possible non-representativeness in the sampling, whereas we reach substantively different conclusions about the treatment effect for the Liver cirrhosis patients.