Semicompeting risks data are commonly seen in biomedical applications in which a terminal event censors a non-terminal event. Possible dependent censoring complicates statistical analysis. We consider regression analysis based on a non-terminal event, say disease progression, which is subject to censoring by death. The methodology proposed is developed for discrete covariates under two types of assumption. First, separate copula models are assumed for each covariate group and then a flexible regression model is imposed on the progression time which is of major interest. Model checking procedures are also proposed to help to choose a best-fitted model. Under a two-sample setting, Lin and co-workers proposed a competing method which requires an additional marginal assumption on the terminal event and implicitly assumes that the dependence structures in the two groups are the same. Using simulations, we compare the two approaches on the basis of their finite sample performances and robustness properties under model misspecification. The method proposed is applied to a bone marrow transplant data set. Copyright 2008 Royal Statistical Society.