# 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 G^{2} be the likelihood ratio test statistic to compare models *M _{0}* 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*are trial-specific intercepts and

_{Si}*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 *e _{0}i* and

*?ц*are normally distributed with zero mean and constant variance <г

^{2}and

where G^{2} is the likelihood ratio test statistic comparing the two models in (12.53).