Abstract: An observed surrogate endpoint is often used to predict a treatment effect on an unobserved true endpoint when it is difficult or expensive to measure the true endpoint. Although several criteria have been proposed for identifying surrogate endpoints, they all suffer from the surrogate paradox: a treatment has a positive effect on the surrogate and the surrogate has a positive effect on the endpoint; however the treatment has a negative effect on the endpoint. To avoid this paradox, criteria have been proposed for a single surrogate that blocks the path from the treatment to the endpoint. This requires that there is a single path from the treatment to the endpoint and that the surrogate can block this path. However, in many applications, a treatment may affect an endpoint through several paths. Therefore, we use stochastic orders of random vectors to derive criteria for multiple surrogates that avoid the surrogate paradox and can be used to predict the sign of the treatment effect on the unobserved true endpoint. Furthermore under the conditional independence of the treatment and the true endpoint, given the multiple surrogates, we propose sufficient conditions for the sign-equivalence of the treatment effects on the surrogates and on the true endpoint. Lastly, we illustrate how these criteria can be applied to several commonly used models.

Key words and phrases: Average causal effect, prentice criteria, stochastic order, surrogate paradox.