Novel Criteria to Exclude the Surrogate Paradox and Their Optimalities
  When the primary outcome is hard to collect, surrogate endpoint is typically used as a substitute. However, even when the treatment has a positive average causal effect (ACE) on the surrogate endpoint, which also has a positive ACE on the primary outcome, it is still possible that the treatment has a negative ACE on the primary outcome. Such a phenomenon is called the surrogate paradox and greatly challenges the use of surrogate. In this paper, we provide novel criteria to exclude the surrogate paradox. Unlike other conditions previously proposed, our conditions are testable since they only involve observed data. Furthermore, our criteria are optimal in the sense that they are sufficient and "almost necessary" to exclude the paradox: if the conditions are satisfied, the surrogate paradox is guaranteed to be absent while if the conditions fail, there exists a data generating process with surrogate paradox that can generate the same observed data. That is, our criteria capture all the information in the observed data to exclude the surrogate paradox rather than relying on unverifiable distributional assumptions. 
Yunjian Yin, Lan Liu, Zhi Geng, and Peng Luo
math.ST stat.TH