BACKGROUND: The validation of intermediate markers as surrogate markers (S) for the true outcome of interest (T) in clinical trials offers the possibility for trials to be run more quickly and cheaply by using the surrogate endpoint in place of the true endpoint. PURPOSE: Working within a principal stratification framework, we propose causal quantities to evaluate surrogacy using a Gaussian copula model for an ordinal surrogate and time-to-event final outcome. The methods are applied to data from four colorectal cancer clinical trials, where S is tumor response and T is overall survival. METHODS: For the Gaussian copula model, a Bayesian estimation strategy is used and, as some parameters are not identifiable from the data, we explore the use of informative priors that are consistent with reasonable assumptions in the surrogate marker setting to aid in estimation. RESULTS: While there is some bias in the estimation of the surrogacy quantities of interest, the estimation procedure does reasonably well at distinguishing between poor and good surrogate markers. LIMITATIONS: Some of the parameters of the proposed model are not identifiable from the data, and therefore, assumptions must be made in order to aid in their estimation. CONCLUSIONS: The proposed quantities can be used in combination to provide evidence about the validity of S as a surrogate marker for T.