Sufficient conditions for concluding surrogacy based on observed data

In this paper, we show that Prentice's criteria for surrogates can be strengthened to remedy some weaknesses of the criteria, and we also propose some sufficient conditions under which the treatment effects on a surrogate and on the true endpoint have qualitative implication and equivalence relations. With or without requiring Prentice's criteria, we discuss what conditions are required to qualitatively assess the causal effect of treatment on an unobserved endpoint in terms of an observed surrogate. Rather than a correlation between a surrogate and an endpoint, we require stricter measurements of association for the qualitative assessment. Further we show that these conditions can be satisfied by commonly used models, such as generalized linear models and Cox's proportional hazard models.