Adjusting for baseline information in comparing the efficacy of treatments using bivariate varying-coefficient models

In biomedical studies, patients' reaction to the treatment can be different depending on their health condition at baseline. In this paper, we develop a bivariate varying-coefficient regression model for longitudinal data with the baseline outcome. The proposed model enables the exploration of the dynamic trend of response variables over time and to provide an effective treatment based on an individual's baseline level of disease by allowing the coefficients to vary with time and baseline. The varying coefficients are modelled through basis function approximation and a set of basis functions is selected by the proposed criterion based on the empirical loglikelihood. After the proposed model is fitted to data, the hypothesis test is designed to evaluate the efficacy of treatments across baseline levels. Theoretical and empirical studies confirm that the proposed methods choose the most parsimonious model consistently and compare the treatment effects successfully across baseline levels. The entire procedure is illustrated with depression data analysis.