Sufficient conditions for treatment responders to have longer survival than non-responders

In cancer clinical trials, tumor response, defined in terms of predetermined relative decrease in tumor size, and overall survival are often used to evaluate treatment efficacy. Investigators often attempt to demonstrate a positive relationship betwen the two measures by comparing the survival of the responders to that of the non-responders. The biasedness of this comparison is explored mathematically. Sufficient conditions on the hazard functions to response and dying without responding are given so that the responders have stochastically longer survival than the non-responders. The conditions contain no restrictions on how response status might alter the subsequent death hazard. Examples are given where achieving a response has no or adverse impact on survival and yet responders appear to have longer survival. The results are applicable whenever the outcome of interest is the duration between two events and a time dependent covariate is used to define comparison groups.