  When the primary outcome is difficult to collect, surrogate endpoint is
typically used as a substitute. It is possible that for every individual,
treatment has a positive effect on surrogate, and surrogate has a positive
effect on primary outcome, but for some individuals, treatment has a negative
effect on primary outcome. For example, a treatment may be substantially
effective in preventing the stroke for everyone, and lowering the risk of
stroke is universally beneficial for a longer survival time, however, the
treatment may still cause death for some individuals. We define such
paradoxical phenomenon as individual surrogate paradox. The individual
surrogate paradox is preposed to capture the treatment effect heterogeneity,
which is unable to be described by either the surrogate paradox based on
average causal effect (ACE) (Chen et al., 2007) or that based on distributional
causal effect (DCE) (Ju and Geng, 2010). We investigate existing surrogate
criteria in terms of whether the individual surrogate paradox could manifest.
We find that only the strong binary surrogate can avoid such paradox without
additional assumptions. Utilizing the sharp bounds, we propose novel criteria
to exclude the individual surrogate paradox. Our methods are illustrated in an
application to determine the effect of the intensive glycemia on the risk of
development or progression of diabetic retinopathy.
