Validation Using the Information-Theory Approach

Individual-Level Surrogacy

The information-theoretic approach for the evaluation of surrogate endpoints (Alonso and Molenberghs, 2007) is discussed in detail in Chapters 8 and 9. Briefly, this approach allows us to evaluate surrogacy at the individual and trial levels in a general surrogacy setting. In this section, we briefly present the setting and illustrate the use of two SAS macros for a normal-normal and survival-binary setting. We consider a multi-trial setting and the following models for the true endpoint:

Let G² be the likelihood ratio test statistic to compare models M₀ and Mi in (12.50) within the ith trial. The association between both endpoints is quantified using the likelihood reduction factor (LRF) given by:

where N is the total number of the trials, and n* is trial-specific sample size. As pointed out in Chapter 9, the LRF ranges between 0 and 1. The case with LRF=0 indicates that the surrogate and the true endpoint are independent in each trial.

Trial-Level Surrogacy

Trial-level surrogacy can be estimated using a two-stage approach. At the first stage, the following models are formulated for the two endpoints:

Here, p_Ti and p_Si are trial-specific intercepts and a* and в* are trial-specific treatment effects. Note that the models can be fitted with common intercepts (i.e., reduced fixed-effects models). At the second stage, the parameter estimates obtained from (12.52) are used to fit two linear regression models given by

where the error terms e₀i and ?ц are normally distributed with zero mean and constant variance <г² and 2, respectively. When the reduced fixed-effects models are used in (12.52), ps is dropped in (12.53). The trial-level surrogacy is estimated by:

where G² is the likelihood ratio test statistic comparing the two models in (12.53).