On the individual surrogate paradox.

When the primary outcome is difficult to collect, a surrogate endpoint is typically used as a substitute. It is possible that for every individual, the treatment has a positive effect on the surrogate, and the surrogate has a positive effect on the primary outcome, but for some individuals, the treatment has a negative effect on the 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 life expectancy; however, the treatment may still cause death for some individuals. We define such paradoxical phenomenon as the individual surrogate paradox. The individual surrogate paradox is proposed to capture the treatment effect heterogeneity, which is unable to be described by either the surrogate paradox based on average causal effect or that based on distributional causal effect. We investigate the 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.