We consider the problem of efficiently estimating survival prognosis under a data structure complicated by the presence of immortal time bias. Where the fundamental concern of survival analysis is the estimation of time-to-event outcomes, possibly subject to left or right censoring, the matter of efficient estimation under a bias induced by time-dependent risks presents a novel challenge that has received surprisingly meager attention in the literature. The present problem examines data on observations potentially subject to multiple survival processes, where individuals in a cohort are followed starting with an indexing event (a risk-increasing survival process) until death, with a subset of individuals shifting from the baseline risk profile to a secondary risk profile prior to death. We compare both parametric and nonparametric estimators of survival, including variations of the Cox proportional hazards model and the Kaplan-Meier estimator, evaluating the efficiency of each in the estimation of the multiple survival processes that occur under this data-generating process. We illustrate the utility of employing the nonparametric estimator that we propose via simulation studies; moreover, we motivate our investigation with examples of inappropriate applications of these estimators in the medical literature, and, time permitting, an analysis of observational medical data that exemplifies the corrections provided by our estimator of choice.