Validation Using the Information-Theory Approach
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 can be estimated using a two-stage approach. At the first stage, the following models are formulated for the two endpoints:
Here, pTi and pSi 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
where G2 is the likelihood ratio test statistic comparing the two models in (12.53).