  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.
