Truncated survival data arise when the event time is observed only if it falls within a subject specific region. The conventional risk-set adjusted Kaplan–Meier estimator or Cox model can be used for estimation of the event time distribution or regression coefficient. However, the validity of these approaches relies on the assumption of quasi-independence between truncation and event times. One model that can be used for the estimation of the survival function under dependent truncation is a structural transformation model that relates a latent, quasi-independent truncation time to the observed dependent truncation time and the event time. The transformation model approach is appealing for its simple interpretation, computational simplicity and flexibility. In this paper, we extend the transformation model approach to the regression setting. We propose three methods based on this model, in addition to a piecewise transformation model that adds greater flexibility. We investigate the performance of the proposed models through simulation studies and apply them to a study on cognitive decline in Alzheimer’s disease from the National Alzheimer’s Coordinating Center. We have developed an R package, tranSurv, for implementation of our method.